Tipton Will Expert Heads Up No Limit Hold Vol2

April 14, 2018 | Author: Enrique Barrios | Category: Betting In Poker, Games Of Chance, Gaming, Gambling, Poker
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Expert Heads Up No Limit Hold’em

Advanced Limit-Hold’em Strategy

Secrets of Sit’n’gos

by Will Tipton 978-1-904468-94-3, 336pp, $29.95 / £19.99

by Barry Tanenbaum 978-1-904468-36-3, 256pp, $24.95 / £14.99

by Phil Shaw 978-1-904468-43-1, 224pp, $24.95 / £14.99

Secrets of Pro Tournament Poker, Vol. 1

Secrets of Pro Tournament Poker, Vol. 2

Secrets of Pro Tournament Poker, Vol. 3

by Jonathan Little 978-1-904468-94-3, 336pp, $29.95 / £19.99

by Jonathan Little 978-1-904468-58-5, 272pp, $27.95 / £19.99

by Jonathan Little 978-1-904468-95-0, 432pp, $29.95 / £19.99

Positive Poker

Secrets of Professional Pot-Limit Omaha

The Education of a Modern Poker Player

by Dr. Patricia Cardner with Jonathan Little 978-1-909457-07-2, 272pp, $29.95 / £19.99

by Rolf Slotboom 978-1-904468-30-6, 240pp, $25.95 / £15.99

by John Billingham, Emanuel Cinca & Thomas Tiroch, 978-1-909457-11-9, 272 pages, $29.95 / £19.99

Will Tipton began playing poker online in 2007. He steadily moved up in stakes in online HUNL tournaments to become a regular winner in the high stake games. He recently completed his PhD work at Cornell University and has taken a position as a software engineer at Google.

Expert Heads Up No Limit Hold’Em Volume 2: Strategies for Multiple Streets

Will Tipton


First published in 2014 by D & B Publishing

Copyright © 2014 Will Tipton

The right of Will Tipton to be identified as the author of this work has been asserted in accordance with the Copyrights, Designs and Patents Act 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, electrostatic, magnetic tape, photocopying, recording or otherwise, without prior permission of the publisher.  

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

eISBN: 978-1-90945-7-034

All sales enquiries should be directed to D&B Publishing: e-mail: [email protected]; website: www.dandbpoker.com       Cover design by Horatio Monteverde. Printed and bound by Versa Press in the US.


Please note that this book starts at Chapter 9 since it follows on directly from Volume 1. The two books can be considered together as one complete work.

Preface 9


9.1 9.2 9.3 9.4 9.5

Strategic Play: a Quick Review The Game Plan A River Refresher Useful Tips for Estimating GTO Strategies You Should Now …

10 Turn Play: Polar Versus Bluff-catchers Redux 10.1 The Two Street PvBC Model 10.2 Weird Plays and Refinements of the Nash Equilibrium Solution Concept 10.3 Bet Sizing and Geometric Pot Growth 10.4 Example: The SB Checks Back Bluff-catchers on K♣-7♥-3♦-K♦ 10.5 Exploitative Play 10.6 Designing Statistics for Effective Decision Making 10.7 You Should Now…

9 13 13 17 24 28 49 50 51 66 69 72 83 97 105

11 Nearly Static, Nearly PvBC Turn Play 11.1 Range Splitting in the Presence of Draws and Mediocre Made Hands 11.2 The Bluffing Range 11.3 Applying the Bluffing Indifference to Find the BB’s Calling Range 11.4 Betting for Value and Protection 11.5 Wrapping it up: Exploitative Bluffing 11.6 EV Distributions 11.7 You Should Now …

12 Initiative and Less Common Turn Lines 12.1 12.2 12.3 12.4 12.5 12.6 12.7

Changes of Initiative K♣-7♥-3♦-K♦ Part Deux: The Computational Solution The Delayed c-bet Versus the Turn Check-raise Estimating Mixed Equilibria with Matrix Games River Leads in Checked-down Pots Lessons so Far: The c-bet Polar Dynamic Example: The BB Check-calls Bluff-catchers and Some Traps on 9♥-2♠-9♦-A♠ 12.8 The Turn Protection Raise 12.9 You Should Now …

107 108 114 125 128 135 139 142 144 144 147 152 155 166 168 171 177 192

13 Turn Play: Volatile Boards and Capture Factors 194 13.1 Example: Turn Play after a Flop c-bet on K♣-Q♠-8♥-5♥ 100BB deep 13.2 A Philosophical Note on the Program of Solving Subgames 13.3 Example: Turn Play 145BB deep on A♣-4♥-2♥-7♦ 13.4 Where the Money Comes From: Turn Play with River Capture Factors 13.5 Low-variance Strategies Versus Unknown Opponents 13.6 Example: Turn Play 73BB deep on K♣-10♣-5♣-J♦ 13.7 Example: Turn Play 66BB deep on Q♣-10♠-8♦-9♥

195 201 202 208 223 225 230

13.8 Conclusions and Foreshadowing: The SB Flop c-betting Range 13.9 You Should Now ... 13.10Appendix: Solving the 1-bet-behind River Game

14 Flop Play and the C-bet Dynamic 14.1 14.2 14.3 14.4 14.5 14.6 14.7

The Singly-raised Pots 7♠-6♦-5♠ Redux The Limped Pots The 3-bet Pots The c-bet Dynamic Lessons and Conclusions You Should Now …

15 Pre-flop Play 15.1 15.2 15.3 15.4

Exploitative Opening Short-stacked Pre-flop Play Lessons and Extensions You Should Now …

16 Win-rate Maximizing Play 16.1 16.2 16.3 16.4

Tournaments Versus Cash Games Recursive Games Passing up Marginal Spots You Should Now …

17 Putting it all Together 17.1 Figuring Out our Opponents’ Play 17.2 Adjustment 17.3 Final Thoughts

237 240 240 242 245 277 282 295 315 321 325 326 327 335 367 370 371 371 373 383 389 390 390 411 423


The best time to plant a tree was 20 years ago. The second best time is now. – Chinese proverb Welcome back to Expert Heads Up No Limit Hold’em – it’s new and improved with flops, turns, and way more draws! Having covered a lot of basics in Volume 1, we’re now ready to tackle the complexities of multi-street play. We’ll start with a quick review of the tools and big ideas we developed in the first volume before starting on turn play. Starting in Chapter 10, we will extend our study of polar-versus-bluff-catcher (PvBC) play to multiple streets. As on the river, it captures the core of poker: value-betting and bluffing versus some poor schmuck who wants nothing more than to show down. Our models will become more and more sophisticated from there. If we can solve a game by hand, it is relatively easy to understand exactly why the players adopt the strategies they do. However, we must turn to computational methods to solve very large decision trees that accurately represent complex situations. Examining these solutions will often lead to new insights about strategic play. We will soon take our first steps outside of the PvBC model. Early street play often leads to “nearly-static, nearly-PvBC” play on the turn. Here, only weak draws are possible, and the players’ ranges overlap in only a few easily understood ways. In Chapter 11, we will focus on the effect of these


Expert Heads Up No-Limit Hold ’em, Vol 2 complications in the context of the PvBC decision tree. This decision tree is actually quite fundamental, since it describes play where the “betting initiative” does not change hands. In Chapter 12, we will consider more complex starting distributions and board textures. These factors lead to changes in initiative and much more interesting strategies in general. We will discuss specific plays such as the delayed c-bet, the turn check-raise, and the protection raise, among others, and see how to estimate unexploitable frequencies for making these plays with matrix models. Finally, we will wrap up our discussion of turn play in Chapter 13 with a look at computationally-generated solutions to a variety of turn spots. Why do we have several chapters on the turn but only one each on flop and pre-flop play? For one thing, the deeper we get into a hand, the more board run-outs are possible, and the more chances the players have had to split their ranges. So, there are many more distinct situations on the later streets. More importantly, we will see many aspects of multi-street play in the context of two-street (i.e., turn-onward) situations. So, when we study the solutions of very large early-street games, it will not take long to understand them. The strategies we find will, however, have significant, actionable implications for many aspects of modern HUNL play, including SB opening and flop c-betting. The final two chapters cover extra topics relating to match play. In Chapter 16 we will develop the tools to decide when we can pass up small edges to wait for better spots. The short answer: quite frequently if Villain is bad enough. Finally, we’ll wrap up with a discussion of building reads and adjusting to opponents. I believe in carefully considering every bit of information available early in a match to begin making exploitative adjustments as quickly as possible. However, it is important we do not overreact to shaky intel. Many questions are included in this book. When a question has a simple, black-and-white answer, I try to mention it in the text, but many do not. Some are merely meant to test your understanding of a topic we have covered, while others call for much more independent thought. We have used different symbols to differentiate these. 10

Preface These questions should be relatively easy to answer if you’ve understood the text. Definitely give them a shot before proceeding, and if you get stuck, consider re-reading. These are harder and often quite open-ended. If the correct approach is not immediately obvious, don’t worry, and don’t feel compelled to struggle with these unless you want to know the answer! We’ve done something similar with the book sections themselves. We’ve marked several with a star (*) to indicate that they are in some sense optional. These sections are both challenging and can be omitted without seriously hampering your understanding of later content. It’s not that they aren’t interesting or important – they are. However, if you’re turned off by math, it might be better to skip them, at least initially, rather than risk getting stuck. As far as that goes, this book is a bit top-heavy in its theory content. The material gets somewhat less challenging as the book proceeds and we focus more on computational solutions to large games and less on analytical models. In addition to everyone I noted in this work’s first volume, I’d really like to express my gratitude to Dário Abdulrehman, Rick Atman and Yoann Brient. This book has benefited greatly from their careful readings and insightful suggestions.


Chapter 9 Preliminaries

If you want to build a ship, don’t drum up the people to gather wood, divide the work, and give orders. Instead, teach them to yearn for the vast and endless sea. – Antoine de Saint-Exupéry


Strategic Play: a Quick Review

To begin, what is “strategic play”? In a strategic situation, our payoffs depend directly on the decisions of other people. At the same time, those people are seeking the best for themselves, and their results depend on our actions. This is the case in Rock-Paper-Scissors, it is often true of business or politics, and it is definitely the case in poker. In contrast, think of other crucial decision-making situations, such as what to eat for lunch or where to go to college. Here, our results will depend on our own choices and preferences, and maybe some dumb luck as well, but we generally do not think of them as depending directly on the self-interested actions of other scheming individuals. The breakthrough of game theory in the 20th century was in modelling the interactions of rational decision makers, allowing people to understand and plan in strategic situations.


Expert Heads Up No-Limit Hold ’em, Volume 2 In HUNL, our profit depends on our opponent’s strategy. This has practical consequences. Since our results depend on Villain’s play, and his play depends on his payoffs, we must understand those payoffs. Since Villain’s payoffs depend on Hero’s ranges, we must keep our own ranges in mind while we play. This is a fundamental tenet of the strategic or gametheoretic approach to poker – to play strategically, we must think in terms of our whole strategy. In poker, this means that we must think of the entire ranges with which we take certain actions. Villain is making his decisions based on those ranges, and our results depend on his decisions. We began Chapter 2 by asking how to play most profitably versus an opponent with a particular strategy. That is, we supposed that our opponent’s strategy was fixed and that we knew it somehow, and then we found our maximally exploitative response. This was a natural starting point, since playing as profitably as possible is the whole point of the game. In fact, if an approach to playing poker were not grounded, at least on some level, in an attempt to make as much money as possible, then it would be a pretty poor theory. Now, a distinction is often made between exploitative and unexploitable play. “Play exploitatively” really just means “try to make as much money as possible”, so the value of an exploitative approach is self-evident. However, we cannot forget about the strategic aspect of poker. Unexploitable play also arises from the attempt to play as profitably as possible – when our opponent is doing the same. We saw that there is really no magic behind equilibrium play – it simply means that both players are playing maximally exploitatively, at the same time. We have analyzed many situations by first thinking in terms of unexploitable play. Intuitively, this shows what the strategies would look like in a match between two very strong players. Then, we consider the ways Villain can deviate from equilibrium play, and how we could notice and then exploit those tendencies. Occasionally, we will say that we make a certain play for “balance-related reasons”. This is often used when describing equilibrium strategies, so of course it does not mean that we are taking an action while believing another is more profitable. Rather, this phrase is always meant to recall the


Preliminaries discussion in Section 2.3.1: Balance. There, we saw some situations where multiple lines could be best depending on Villain’s tendencies. If Hero always made the first choice, then Villain’s response made the second more profitable, and vice versa. Often, unexploitable play will involve a mixed strategy in these spots, and without reads, either line is reasonable. That is, making a play for “balance-related reasons” still means to try to play as profitably as possible, by taking a line that could be best, depending on Villain’s unknown strategy. Now, you may think that the strategic approach to the game gives some of your opponents too much credit. Game-theoretic arguments often rely on the assumption that Villain recognizes and adjusts to Hero’s strategy. However, in poker, statements like “the other party is intelligent and acts rationally in his best interest” do not always ring true. In the most extreme case that an opponent is completely transparent to us, then we can maximize our expected profit one hand at a time, i.e., without needing to keep our entire range in mind. We simply take the highest-EV line in every spot, as described in Section 2.2. Unfortunately, this is never really the case in practice. Of course, we generally don’t know much at all about an opponent the first time we play him, and there are so many unique spots in HUNL that it is impossible to have a complete understanding of his play even after playing a significant number of hands. Furthermore, almost all players make some attempt to adjust to their opponents, even if they do it poorly. Additionally, here are some reasons to play strategically, even in situations where we think a purely exploitative approach is perfectly reasonable:

It will help us identify spots where we are losing money and need to adjust. For example, if we arrive at some spot with a mediocre hand, face a bet, and decide it is best to fold, then that’s that. But if we realize that we get to that spot with a range full of mediocre hands, face a bet, and decide it is best to fold all of them, it provides strong indication that we need to make some adjustments. At the very least, we can now consider the possibility that we are being exploited, and if it happens a couple more times, we can be more 15

Expert Heads Up No-Limit Hold ’em, Volume 2 certain that we need to adjust. (Keep in mind that Villain does not need to be intelligent for his strategy to be exploiting ours. Sometimes players’ default strategies will just happen to stack up well versus our own. It does not mean he is consciously taking advantage of us, but we still must be able to realize what’s going on and then adjust to prevent it.)

We will learn much more about the game. In the situation described in the last point, not only did we find a spot where we could adjust to make more money versus a particular opponent. We also learned something about a situation where people can be exploited and added it to our mental list of spots where we may be able to exploit other players in the future.

It is an important skill that we will need eventually if we want to play in tough games, and it becomes much easier with practice, so it is best to go ahead and get in the habit as early as possible.

An awareness of our own ranges is an extremely important prerequisite for using most of the ideas we encounter in these books. A study of game theory can give us lots of insight into what our frequencies and ranges should be in various spots. However, there is no way to use that information unless we know what our current frequencies are and how our adjustments will affect them. For example, knowing that we should call a bet X percent of the time provides very little benefit if we do not know what our range looks like when we face the bet or where our actual holding fits into that range. Staying aware of our own range in every hand we play is not as hard as it might sound at first. You are likely already in the habit of thinking about your opponents’ ranges. Knowing your own ranges should be much easier, since you know yourself, and your default play does not change drastically with every new opponent. It may be difficult at first, and if you are a multitabling online player, you might even need to cut down on the number of tables you play until you get the hang of it.


Preliminaries It is the first hand of the match in your favorite HUNL format. You’re in the SB – which hands do you open-raise? The BB calls, the flop comes Q♠-10♥-4♠, and the BB checks to you. Which hands do you c-bet and which do you check back? Given those ranges, what are the frequencies with which you take each action? Suppose you c-bet and get check-raised half-pot. What are your flatting, 3-betting, and folding ranges, and what are the corresponding frequencies? If you check back the flop and face a turn lead on the J♥, what are your calling, folding, and raising ranges, and what are the respective frequencies? Practicing this skill is likely to pay off as much as anything else in this book.


The Game Plan

Now, let us review a few more specifics. We can organize all of the possible decisions in a HUNL hand into a decision tree made up of decision points linked together by player actions. Each point in the tree, except for the leaves, which represent the end of a hand, represents a particular state of the game and is a spot where a player (or Nature) has to make a decision. The game will move into one of several new states depending on the choice. We saw that the full decision tree representing HUNL at any appreciable stack size is too large to handle. However, there is a lot to learn from approximate games. For example, a tree representing a river situation where there is just one bet left in the remaining stacks is shown in Figure 9.1. A player’s strategy specifies how he will make any decision he can face in a game. In practice, a strategy must specify the range of hands with which the player takes each action at each of his decision points. We can visualize this as follows. Both players start the hand with a range consisting of 100% of each of the 1,326 distinct hold ’em hands. At each of his decision points, a player partitions or splits his range into several portions, one for each of his strategic options. In this way, a player’s range tends to get smaller and 17

Expert Heads Up No-Limit Hold ’em, Volume 2 more clearly defined as the players get deeper and deeper into a hand. This splitting of ranges is illustrated in the “sun-burst” tree on the cover of this book.

Figure 9.1: Decision tree representing river play with one bet left behind.

The expected value or EV of a holding for a player at a particular decision point is his total stack size at the end of the hand, averaged over all the ways the hand can play out from that point onward. Remember that our convention for EV is different than that of some other authors. We work in terms of total stack sizes, as opposed to changes in stack sizes. The basic approach to decision-making at any point is to consider the EV of each of our available options and then go with the largest. A best response or maximally exploitative strategy is one that maximizes a player’s EV in this way with every hand in every spot. Given a game described by a decision tree and Villain’s strategy for playing that game, we saw how to compute Hero’s best response in Chapter 2. When both players are employing maximally exploitative strategies simul18

Preliminaries taneously, we have a Nash equilibrium. When two players adopt their equilibrium strategies in HUNL, neither has any incentive to deviate. They cannot improve their expectations by doing so, since they are already playing maximally exploitatively. An equilibrium strategy is also known as unexploitable, since it is the best a player can do against an opponent who is aware of his strategy and capable of quickly adjusting to it. When two sufficiently smart players face each other, they can do no better than to play their equilibrium strategies. Thus, when we find a game’s equilibrium, we say we have solved it. In this book, we use the terms GTO, unexploitable, and equilibrium as synonyms to refer to such strategies. A solution for the full game of HUNL is not known, but the result of an attempt to get close is called pseudo-optimal or near-optimal play. We have seen that pseudo-optimal play is appropriate not only against mindreading super-geniuses, but also against more run-of-the-mill opponents whose strategies are simply unknown to us. When facing a new opponent, many different exploitative strategies could be best depending on his tendencies. When these tendencies are unknown, however, any deviation from GTO play on our part is just about as likely to hurt as to help us. Without knowledge of a player’s weaknesses, we cannot expect any particular deviation from equilibrium to increase our EV. Although it is not entirely rigorous, we can think of unexploitable play as our best response given complete uncertainty about our opponent. Furthermore, understanding unexploitable play can help us recognize exploitable tendencies in our opponents and understand how to adjust our own ranges to take advantage. For example, recall one of the simplest river situations we looked at in Volume 1: the PvBC game. One player’s range is made up of the nuts and air, and his opponent holds only hands that beat the air but lose to the nuts. We saw that under many conditions, the equilibrium strategies here are for the first player to bet all-in with all of his nut hands and enough bluffs so that his opponent’s EV if he calls is the same as if he folds. Similarly, the second player’s GTO play is to call enough to keep the first indifferent to bluffing. What about exploitative play? If the polar player bluffs a bit too much, his opponent should always call, but if he bluffs even slightly too little, the


Expert Heads Up No-Limit Hold ’em, Volume 2 bluff-catcher should always fold. On the other hand, if the bluff-catcher calls too much, his opponent should never bluff, and vice versa. Of course, “too much” and “too little” are defined in terms of the unexploitable strategies. So, our understanding of GTO play makes it very easy to understand and describe all of the opportunities for exploitative play in this situation. Despite the fact that HUNL’s true equilibrium is likely too large to memorize and too complicated to fully understand (and not even the best approach versus most opponents), the players with the best knowledge of game-theoretic play are also some of the best exploitative players because of their understanding of the game. With this in mind, we have focused on learning about equilibrium strategies to develop intuition and understanding of the structure of HUNL play. In this volume, we will continue our careful consideration of a variety of spots and how we might want to split our ranges when we encounter them. Although we will describe refinements later, our general approach to match play begins by playing pseudo-optimally. From this defensive posture, Hero can observe his opponent’s tendencies and determine appropriate adjustments. Of course, it is rare that a new opponent is a complete unknown. In practice, we may do well to make some pre-game adjustments based on our knowledge of population tendencies – the tendencies of an average individual in our player pool. However, this caveat does not give us a free pass to just make “standard” plays without good reason. Any deviation from equilibrium play should be justified by reference to a particular exploitable tendency, whether of the population on average or of a particular opponent. Although this is a book on heads up play, it’s worth noting that many of the properties that make Nash equilibria so useful do not hold in games with three or more players. In particular, if we play an equilibrium strategy in HUNL, we are guaranteed to at least break even (neglecting rake) on average over both positions. That is not the case in 3-or-more player games, where playing an equilibrium strategy provides no lower bound on our expected winnings. Thus, the Nash equilibrium is much less useful outside of heads up play, and anyone selling the idea of “GTO” strategies for 3-ormore player games should be viewed with suspicion.


Preliminaries Now, we developed a couple of tools that are useful for reasoning about strategies. The equilibration exercise gives us a framework for thinking about how exploitative players adjust to each other in a particular spot. We begin by supposing that Villain is playing a particular strategy. Hero then adopts his best response. We then consider how Villain could readjust to take advantage of our new strategy and what Hero might notice if Villain does make such an adjustment. We can then repeat the process, alternately finding the maximally exploitative strategies for each player and answering some additional questions at each step. The focus of the equilibration exercise is on the first few adjustments. However, this process has a lot in common with a computer algorithm for finding Nash equilibriums known as fictitious play. It turns out that if we continue this sort of process long enough, the players’ strategies may converge to equilibrium. In practice, a scheme involving averaging of the players’ strategies in each step is usually necessary to ensure convergence. The Indifference Principle states that if a player is playing a hand in more than one way at the equilibrium, the EVs of both actions must be equal. Consequently, his opponent must be playing in such a way as to enforce that equality. We will often use the intuition we gain from the equilibration exercise in combination with the Indifference Principle to draw conclusions about unexploitable play. For example, suppose Hero is out of position on the flop with a draw to the nuts that is fairly likely to come in, such as an open-ended straight draw. The effective stacks are moderately deep. For simplicity, suppose Hero is restricted to two options: check-raise and check-call. If Villain is playing exploitably, one of these two options is likely more profitable than the other. However, we can make a pretty convincing argument that at the equilibrium, the two EVs are equal and Hero takes both actions with nonzero probability. Suppose Hero always check-calls with this nut draw. Then, when he does check-raise in that spot and then the draw comes in, his range is capped – he cannot have the straight. When Hero’s range is capped, it implies that Villain’s range has a significant number of hands that are effectively the nuts, and we saw that Villain’s best response in this case usually involves 21

Expert Heads Up No-Limit Hold ’em, Volume 2 making frequent, large bets with much of his range. Since Villain is putting so much money in the pot after Hero check-raises and the draw comes in, being able to show up with the straight after check-raising becomes very profitable for Hero. Thus, Hero is motivated to begin check-raising with the draw on the flop. On the other hand, we can use a similar argument to see that if Hero were always check-raising with the draw, Villain’s response would incentivize him to begin check-calling it. Thus, if Hero takes either option 100% of the time, Villain’s response makes him prefer the other. Thus, neither pure strategy is unexploitable, and GTO play must involve doing both with some frequency. Thus, we have used the equilibration exercise to identify a spot where Hero plays a hand more than one way at equilibrium. The Indifference Principle immediately tells us that the equilibrium EVs of the two actions must be equal. Villain’s play must make this true, and thus we can leverage this knowledge to learn about Villain’s equilibrium strategy. Indifference is not a fundamental property of equilibrium play. We have seen solutions to games where neither player is indifferent between any actions with any hands. Indifference should certainly not be taken for granted when studying a new spot. However, mixed strategies and thus indifference often emerge when both players adjust to each other to try to play as profitably as possible. A thought process like the one above will frequently help us to identify these useful relationships. If Hero plays one pure strategy, Villain’s response incentivizes him to switch to the other. But when he plays the other pure strategy, Villain’s response incentivizes him to switch back to the first. Since neither pure strategy is unexploitable, play must be mixed at the equilibrium, and thus the two lines have equal EV. In Volume 1, we also developed some tools for analyzing and describing ranges. You draw a hand-distribution plot by ranking all hands in a range from strongest to weakest and then plotting how often each hand is contained in the range. This ranking is somewhat ill defined, since there is no clearly correct ordering of hand strengths before the river, nor even then due to card removal effects, but this construction is nonetheless useful for quickly visualizing the strengths of the various hands that make up a range. The shapes of these plots motivated some of the words, such as “po-


Preliminaries lar” and “capped”, that we use to describe some strategically important properties of the ranges themselves. Of course, to draw strategically relevant conclusions, we really need to describe our hands’ strengths in comparison to our opponent’s holdings. Equity distributions allow us to visualize this information directly – they are plots of the equities of all the hands in our range. We found these very useful for our analysis of river play. We can forget about the particular holdings involved and reference holdings only by their percentile in our own range and equity versus Villain’s. Thus, we solved many situations in general and then mapped real hand ranges onto the results. With this abstraction, we lost the ability to deal exactly with card-removal and chopping effects. However, it allowed us to deal with many situations at once and see connections between them. Our convention for drawing equity distributions might need a bit of explanation. Suppose we have a hand from the hth percentile of our distribution. For example, if h=0.9, then our hand is in the top 10% or so of our range. The equity distribution is a representation of a function, EQ(h), which takes a percentile and tells us our equity. For example, taking the symmetric distributions case means that we have the particular function EQ(h)=h. Then, if h=0 (i.e., we have a 0th percentile holding, or the worst hand in our range) then we have EQ(0)=0, that is, no equity. If we have the nuts, we have equity EQ(1)=100%. In general, the equity function for symmetric distributions is a straight line, y=x. However, we plot equity distributions by drawing this function backwards. Instead of plotting EQ(h), we plot EQ(1−h), so that the nut hand with equity 1 shows up at the point (1−1,1)=(0,1), and an air hand shows up at (1−0,0)=(1,0). In effect, the function we care about is EQ(h), but we plot EQ(1−h). The graph of one of these is essentially the mirror image or the other, obtained by flipping it around the vertical line x=0.5. This was done for historical reasons and is perhaps not the clearest choice. For consistency, we will keep the same convention in this volume. However, since equity distributions do not account for the possibility of changing hand values (i.e., draws), which are important on earlier streets, their use in the next few chapters will be fairly limited.


Expert Heads Up No-Limit Hold ’em, Volume 2


A River Refresher

Taylor Caby, a successful no-limit player and prolific poker educator among other things, used to constantly remind students to ask themselves while playing, “Where is my money coming from?” In many settings, the answer may be as simple as the identification of a weak player at the table, and it may be reasonable to make an extra effort to play pots with that player. As sophisticated players in a HU game, we might think a bit more deeply – what exploitable errors is our opponent making and what are we doing, exactly, to profit from them? To address these questions, we generally focus on analyzing the EV of each of the options available to us in individual situations. However, we can gain some important insights by taking a more holistic view of things. So, where does the money come from in a hand? Well, payoffs come at the end of a hand: after a fold or at showdown after the river. After a fold, our payoff is the whole pot or none of the pot, depending on who folded. After an all-in or a showdown, our payoff is essentially just our equity in the pot. In a big picture sense, our early-street actions can be thought of simply as attempts to guide play to leaves of the tree where we have the largest payoffs, on average. Identifying the spots from which we are profiting in a match is important, because it is the first step in developing a plan to get there. Understanding where it is valuable to have some hands and not others can help us organize the play of our whole range, especially on the earlier streets. In particular, we are about to study turn play, and a solid understanding of the values of holdings in certain river spots is necessary for evaluating our options on the turn. Consider all the types of hands you might hold at the beginning of river play. How valuable is each of them? It is not immediately obvious. For example, we have seen situations where a complete air ball is just as valuable as a legitimate value hand at equilibrium. In other spots, good-butnot-excellent value hands are essentially bluff-catchers, and they expect to capture little more of the pot than a good high card. Perhaps we should try 24

Preliminaries to play earlier streets so as to get to the first spot with more bluffs and to the second with fewer marginal value hands. Let us look at the value of various hands on the river in the context of pseudo-optimal play. Of course, a particular holding cannot be considered in a vacuum. The value of taking an action with a hand depends on how Villain reacts to it. That, in turn, depends strongly on the entire range with which we take the action. So, the value of an individual hand on the river is strongly tied to its place in our range and how that range stacks up against Villain’s. Suppose we start river play with remaining effective stacks S and a pot size of P. Call the size of a first river bet B and additional money put in by the first raise C. Remember first of all that we need never play a strategy that gives any hand an EV less than S. We can always just achieve at least that by simply giving up and putting no more money in the pot. Also, the average value of all our hands at equilibrium can be no greater than S+P, since Villain could limit us to this amount by simply giving up with all of his range. Of course, our nut hands will usually expect better than S+P at the equilibrium – they win the pot in addition to any river bets. Now suppose that Hero bets, putting Villain to a decision. If Hero is in the SB, his betting range facing a check on the river should generally be polar. That is, his betting range is composed of some amount of his strongest and weakest holdings, and if he does check back with any hands, it is those in between. These conclusions about the structure of the SB’s strategy in this spot came out of our study of river situations. Of course he wants to bet his strongest hands to try to get called by worse. However, if those are the only hands he bets, Villain folds a lot, so that Hero is incentivized to bet his weak hands, too, to avoid a showdown. We saw that, at the equilibrium, Hero can only “get away with” a limited number of bluffs, so he might as well use those that win the pot as little as possible at showdown. If Hero is in the BB, however, he usually cannot get away with so extreme an approach. If he only checks mediocre holdings, he must contend with the possibility of a SB bet, and the SB’s play will generally incentivize him to start checking some strong hands, for balance-related reasons. (Playing well in the BB is much harder than playing well in position!) Still, we have 25

Expert Heads Up No-Limit Hold ’em, Volume 2 seen that the hands that the BB does bet generally fall into clear valuebetting and bluffing regions. A correctly balanced polar betting range has a clear effect on Villain’s distribution. It splits his range into three portions: those ahead of some of Hero’s value-betting hands, those that lose to even some of Hero’s bluffs, and those that are stronger than all of Hero’s bluffs but weaker than all of his value-bets. The hands in this last group are all effectively bluff-catchers and are, up to card removal effects, indifferent between calling and folding. At the equilibrium, some of these will call and others will fold. Either way, they have an EV of S – when facing a bet they are no better than complete air! Bluff-catchers do perform better than air in general, since they can show down and capture some of the pot if they do not face a bet. However, finding oneself with a range composed primarily of this type of hand is unfortunate, especially when Villain’s range is sufficiently strong that he can bet and put us in such a spot with high frequency. In practice, exploitative call-or-fold play with bluff-catchers involves figuring out whether an opponent is bluffing too much or not enough in a particular spot. In the first case, bluff-catchers all become profitable calls, and in the second, they are all clear folds. In Chapter 7, we discussed common river decision-making thought processes to try to understand situations in which opponents are likely to bluff too much or too little. What about the two other types of hands Villain might hold when facing a bet? With holdings worse than bluff-catchers (i.e., those that lose to some bluffs), folding is clearly better than calling. Thus, when facing a bet, these hands also have an EV of S. Again, however, these hands do not always face a bet, and if they have some showdown value, they may occasionally win at showdown. This is much more likely when in position. When the BB checks a weak hand on the river, it is true that the SB holds some worse ones. However, he will often use these to bluff, and the BB will have to check-fold. When the SB does check behind, he generally has the winner. On the other hand, the SB can more likely average better than S with weak holdings. He can show down to capture a bit of equity, or use it to bluff, if bluffing does at least as well. This is one reason why many weak hands have more value for the SB on the river and an example of the general ob-


Preliminaries servation that it is easier to realize the equity of our hands with position. Thirdly, when Villain faces a bet with a hand that is ahead of some of our value hands, calling is clearly better than folding. In particular, the EV of calling is (S−B)+(2B+P)EQ where EQ is the portion of the pot he expects to win at showdown. Of course, raising could be even better. The range of hands with which he raises at equilibrium has a lot in common with the distribution with which a player in the SB would choose to make a bet in the first place. It will include all of his strongest hands as well as some bluffs for balance. There is no need to balance the more passive option with slow-plays, since calling gets him immediately to showdown. The effect of a raise on Hero’s range is analogous as well. All of our bluffs likely have to now fold, and many of our old value hands become bluff-catchers such that EV(call)=EV(fold)=S−B. Now consider the value of hands in Hero’s initial betting range. Our bluffs have an EV of S in the worst case. However, if we could check those hands with some chance of winning at showdown, then the value of checking is greater than S, and so the value of using these hands to bluff must be greater as well. Again, this is the case for the SB quite a bit more frequently than for the BB. If the strongest bluffing hand is indifferent to checking, weaker holdings usually do strictly better by bluffing. At the other extreme, Hero’s nuts win the pot plus the result of any river betting. Weaker value hands capture an expectation somewhere in between. Finally, what can we say about bet sizing? In perhaps the simplest river situation, Hero’s entire range is polar with respect to his opponent’s. Here, the GTO play is to bet all-in with our whole betting range. To complicate things a bit, we imagined giving Villain a few trapping hands as well, but Hero’s best sizing was still all-in unless Villain just had too many slowplays or stacks were too deep. In another simple model, the players hold identical ranges, the SB faces a decision over whether to bet or check back on the river, and if he bets, the BB can only call or fold. In this case, Hero’s unexploitable strategy involves using different sizings for each of his value hands – the better the hand, the bigger the bet. Of course, these bets were balanced with an appropriate number of bluffs. However, when he had to use a single sizing with all of his betting range, the GTO choice was P.


Expert Heads Up No-Limit Hold ’em, Volume 2 By examining computationally-generated solutions to more complex situations, we learned a few more river bet-sizing principles. Distributions that contain many nut and near-nut hands but also plenty of other holdings that “want” to bluff often used large bet sizings. Distributions containing many value-bettable but non-nut hands and fewer hands with incentive to bluff often made smaller bets. Distributions containing many types of hands tended to use multiple sizings. Better hands generally tended to bet larger, but many hands were played with mixed strategies for balance-related reasons. However, sometimes it was not worth it to balance by, for example, sometimes betting smaller with a nut hand, if Villain did not hold many re-raising hands or if there was not enough money behind for a re-raise to pose a significant threat. Suppose Hero’s holding has equity E at the beginning of river play. If betting were disallowed and hands simply shown down, then Hero would capture the E of the pot on average, so he would have an EV of S+PE. In real play, which holdings have an expectation greater than this and which have less? How does the answer depend on position and the features of the players’ river starting distributions?


Useful Tips for Estimating GTO Strategies

Certain parts of a player’s range are very often played with a mixed strategy. Combined with the Indifference Principle, this fact can be used to draw some conclusions about unexploitable play without having to actually solve the whole game.

9.4.1 The Bluffing Indifference Consider Villain’s weak hands. He will often use these to bluff. If he does 28

Preliminaries not, then Hero’s tight exploitative response will strongly motivate him to start bluffing. On the other hand, in most spots, Villain will have enough weak hands that he cannot get away with bluffing with all of them. If he tries, then Hero can profit by calling with a very high frequency, thus motivating Villain to stop bluffing altogether. Thus, in a wide variety of situations, Villain will play a mixed strategy at the equilibrium with some of his weakest hands – sometimes he will use these to bluff, and sometimes he will not. In this case, we have (9.1)

In the simplest case, not bluffing just means losing the pot, and the EV of not bluffing is easy to find. EVVillain(bluff), on the other hand, depends directly on Hero’s strategy – in particular, how often he folds to a bet. We can solve for this folding frequency to find the one that makes Villain indifferent to bluffing. As usual, an EV equation that applies to one of Villain’s hands tells us something about Hero’s equilibrium play. There is a simple interpretation of Equation 9.1. Basically, it says that Hero has to play back enough versus a potential bluff to prevent Villain from making a profit with it. For this reason, computations based on it are sometimes called non-auto-profit calculations. Of course, it is important to remember the flip side as well – Hero cannot continue too much versus the bet. To be unexploitable, he should call (and raise) just enough to enforce the indifference, but no more. We will call Equation 9.1 the bluffing indifference, since it applies to a player’s bluffing hands. Let us take a few examples. First of all, think about the polar-versus-bluffcatchers (PvBC) river situation that we reviewed in the previous section. Suppose Hero is the bluff-catching player. Then, his only decision in the game is whether to call or fold when facing a bet. An application of the bluffing indifference was essentially all that was necessary to find his equilibrium strategy – he called just enough so that Villain’s EV(bluffing) equalled his EV(not bluffing). Here, “not bluffing” means showing down and losing the pot every time, and “bluffing” means winning the pot if and only if Hero folds. So, we have


Expert Heads Up No-Limit Hold ’em, Volume 2 where F is Hero’s folding frequency when facing a bet, and S, P, and B are the stack, pot, and bet sizes as usual. Solving, we find F=B/(B+P). For example, if the bet is size of the pot, Hero should call half the time and fold half the time. Of course, this is all based on the assumption that a bluffing indifference holds. In Section 4.2, we saw a couple ways that indifference could break down in a flop c-betting spot. In the river PvBC game, air hands are not indifferent to bluffing at equilibrium when Villain’s range contains too many nuts. In that case, he can bet his entire range, including all of his bluffs, and we will still prefer to fold all of our bluff-catchers. There is still an equilibrium, of course, but a non-auto-profit calculation does not give us any useful information about it. For the rest of this section, we will generally assume that some bluffing indifference holds, but it is important to verify this in practical applications. Great – in the simple PvBC situation, Hero’s bluff-catching frequency is P/(B+P). We will often refer to this result as the naive bluff-catching frequency. This is useful to know, but it is often abused by players trying to answer more complicated questions such as, “How often should Hero continue on the flop facing a continuation bet?” The result does not directly apply here, and trying to use it will lead to incorrect results. We can use non-auto-profit reasoning to reason about more complicated situations, but we must account for some additional complexities. Recall another simple river game: the SB bet-or-check situation with symmetric distributions. Both players are dealt a hand between 0 and 1 from a uniform distribution, and higher numbers represent better hands. The initial showdown equity of a hand h is thus its own numerical value: EQ(h)=h. The SB can bet or check. If he checks, we see a showdown, and if he bets, the BB can call or fold. We solved this game in Section 7.3.2. If the bet is the size of the pot, the SB value-bets the top 2/9 of his hands and bluffs the bottom 1/9, while the BB check-calls 4/9 of his range and check-folds the rest. Notice that the BB continues versus the bet with a frequency that is smaller than the 50% that we would have expected using the previous result. Why is this? Well, Equation 9.1 still holds here, but not to all of Villain’s weak hands. 30

Preliminaries Previously, Villain had only one type of potential bluff: complete air with no chance to win at showdown. Now he has holdings with many different values. It is not possible that the indifference holds for all of them. Villain’s EV(bluffing) is the same with all of his weak hands, since none of them ever win when called, but his EV(not bluffing) grows steadily with the numerical value of his holding. Thus, the indifference only holds for the very strongest of his bluffs (or equivalently, the weakest of his checking hands). For Villain’s very strongest bluff, “not bluffing” means something very different here than in the PvBC case. When Villain’s weak hands were all air, not bluffing meant always losing the pot. Here, it means going to showdown and winning some of the time. How does this affect the solutions of the bluffing indifference? Because Villain’s bluffs have some showdown value, the EV of not bluffing is clearly higher than in the naive case. For this equation to hold, the value of bluffing must be higher as well. To make Villain’s EV(bluff) higher, Hero needs to fold more, and indeed he does. He folds with frequency 5/9 instead of 1/2. How should we apply the bluffing indifference to estimate a BB’s bluffcatching frequency in a real river situation? Imagine the BB has checked to the SB on the river. We make our best guess at the players’ ranges, plot the resulting equity distributions using a utility such as the one on this book’s webpage, and find those shown in Figure 9.2. In this case, the top 18% or so of the SB’s distribution consists of hands with about 90% or more equity, but the rest of his hands fall below 70% equity. Thus, it is reasonable to expect the SB to value-bet just that clearly defined set of high-equity holdings. If we stick with the case of a pot-sized bet, the SB must bet one bluff for every two value bets at the equilibrium to make the majority of the BB’s hands indifferent bluff-catchers. Since he is betting about 18% of his range for value, he will bet something like the bottom 9% as bluffs. (Card removal also plays a role in bluff selection, but we’ll neglect that here.) Inspecting the bottom-right corner of Figure 9.2, we see that his 9th percentile hand has about 10% equity.


Expert Heads Up No-Limit Hold ’em, Volume 2

Figure 9.2: River equity distributions example.

What effect does this have on the BB’s equilibrium bluff-catching frequency? Returning to the bluffing indifference, we have

where F, S, P, and B are the same as above, and P=B by assumption. The new term here is the 0.10P on the left side of the equation. It arises from the fact that the SB’s strongest bluff expects to capture about 0.10 of the pot when it chooses to show down. Solving, we find the BB’s folding frequency is F=(B+0.1P)/(B+P). As expected, this is a bit higher than the naive calculation would suggest. Generally, if our opponent’s distribution does not contain enough pure air to reach his desired bluffing frequency, he must start to draw from hands with showdown value. Turning these hands into a bluff “wastes” their showdown value, since they never win when called. Thus, to maintain the indifference, we must decrease our bluff-catching frequency. If we continued calling according to the naive result, Villain would be motivated to bluff less. In particular, only his 0-equity air would be indifferent to bluffing – very few holdings in the case of the distributions in Figure 9.2. Of 32

Preliminaries course, if Villain only bluffed with this small set of hands, then Hero would be strongly incentivized to bluff-catch less, thus moving his play back in the direction of the equilibrium. Now, consider a common pre-flop situation. We min-raise from the SB. How often might we continue facing a re-raise (say, to 5 BB total) at equilibrium? If we fold too much, Villain can happily 3-bet bluff lots of weak hands, which will presumably make us want to defend more often. If we continue too often, the BB can take advantage by removing bluffs from his re-raising range. Thus, it is likely that some of the BB’s weak hands will be played with a mixed strategy (and thus made indifferent) at equilibrium. In fact, it turns out that lots of them are. Most 3-bet bluffs will be played with a mixed strategy, since bluff-raising with small fractions of lots of different hands, as opposed to large fractions of a smaller number of hand combinations, will lead to a more balanced range post-flop, which is playable on a variety of boards. We will assume that the mixing is between bluff-raising and folding, as opposed to, say, bluff-raising and calling. Now, apply the bluffing indifference – for Villain’s indifferent hand combinations, EV(bluffing) equals EV(not bluffing). Here, not bluffing means always losing the hand, like in the naive PvBC case but unlike the SB bet-orcheck game. Bluffing, however, does not simply mean winning if and only if Hero folds, in contrast to both of those river situations. Here, if Villain 3bet bluffs and gets called, he still has a chance to win post-flop. He can hit a lucky board, find profitable semi-bluffing opportunities, etc. This effect increases Villain’s EV(bluffing) compared to the naive case. We can account for it by adding an extra contribution to the right side of Equation 9.1. The extra EV of a bluffing hand when it sees a flop is something like the size of the pot times the fraction of the pot it expects to capture. The raw equity of a 3-bet bluffing hand versus the SB’s calling range figures to be around 35%, but these hands will not be able to realize all of that, since they tend to make hands that are relatively hard to play out of position post-flop. The actual fraction of the pot captured by 3-bet bluffing hands post-flop changes with the stack depth, and the total contribution to the BB’s EV(bluffing) also depends very strongly on how often the SB continues with a 4-bet as opposed to a call. We will understand these details more fully in


Expert Heads Up No-Limit Hold ’em, Volume 2 the future, but for the sake of simplicity, we’ll assume here that a BB bluff captures 15% of the pot on average whenever the SB does not fold. Thus, the bluffing indifference for a 5BB 3-bet over a min-raise looks something like

Here, S is the effective stack size at the beginning of the hand, and Hero folds to a 3-bet F of the time. When we fold, the BB ends up with a stack of S+2, and otherwise, he ends pre-flop play with S−5 behind but wins (0.15)(10) back post-flop. Solving, we find that Hero’s folding frequency is F=45.5%. This is significantly less than the 57.1% that we would have found if we had neglected the BB’s ability to realize some equity post-flop. The lesson here is that if Villain’s bluffs retain some equity in the pot when called, Hero has to call more to keep them indifferent to bluffing. Calling with the naive frequency implied by the PvBC game only succeeds in making Villain indifferent to bluffing with hands that have 0 equity when called. Of course, any hand can flop quads, so Villain would strictly prefer 3-bet bluffing to folding with all of his range. Of course in this case, Hero will almost certainly be motivated to begin defending, thus moving towards equilibrium. This sort of effect will usually be significant when we apply the bluffing indifference on early streets. It is often the case that a player’s weak hands will be indifferent to bluffing at equilibrium, and the bluffing indifference can let us estimate some unexploitable frequencies without solving entire games. In this section, we began with the naive, single-street PvBC situation and found the wellknown bluff-catching frequency of P/(B+P). We then looked at a spot where Villain’s potential bluffs had the opportunity to show down and occasionally win. This made an extra contribution to the EV(not bluff) side of the bluffing indifference necessary. The effect here was to make Hero bluffcatch less often – the more attractive checking was, the more Hero needed to fold to a bet to keep Villain bluffing. Thirdly, we saw a spot where Villain’s bluffs could still win when called. This increased EV(bluffing) and led to looser equilibrium bluff-catching.


Preliminaries Thus, there are several complications when applying the bluffing indifference. Identifying a hand that is indifferent between bluffing and not bluffing can be error-prone. It is certainly incorrect to assume a completely valueless holding by default. Second, the pair of actions between which our hand is indifferent is not always easy. For example, if the BB is facing a min-raise, when does not-bluffing mean calling and when does it mean folding? Finally, it can be tricky to estimate the EVs of actions that do not quickly end the hand. In the 3-betting situation, we assumed that not bluffing meant folding. What SB play might make the BB indifferent between calling and 3-bet bluffing a min-raise? In this book, getting the details right will sometimes require a bit of work. In my opinion, it’s worth it. We’ve seen that details can have a great impact on the results, so using bad or lazy theory is probably worse than using no theory at all. If we simply apply the naive result whenever we want to estimate an unexploitable frequency, we are unlikely to get a more accurate result than if we just asked a decent player for his gut feeling. However, we might put too much faith in a weak strategy that we do not fully understand if we think it is grounded in math, and end up playing poorly. As always, we’ll try to be clever in how we attack problems, but when it comes to poker theory, if it’s worth doing, it’s worth doing right, and sometimes that requires some effort. Suppose you open in the SB and Villain calls. The flop comes K♥-J♣-5♠. How can the BB discourage you from c-betting all of your weak hands? Here, bluffing means c-betting, and not bluffing means checking back. Your weak hands will expect to capture some fraction of the pot in either case. If you keep a database of hands, you can use it to estimate these fractions. Pick your favorite bet sizings. Note that your EV(bluffing) depends on both Villain’s calling and raising frequencies, and there is not just one pair of frequencies that he can choose to make you indifferent. Exploitatively speaking, should you always c-bet or always check here against an unknown opponent in your games? Do you continue facing a c-bet here often enough to prevent SBs from always bluffing? 35

Expert Heads Up No-Limit Hold ’em, Volume 2

9.4.2 The Bluff-catching Indifference and Odds The bluffing indifference is probably the one most commonly used for estimating equilibrium frequencies. It helps us choose good continuing frequencies when facing a bet and thus provides guidance for how to play our mediocre holdings. Playing bluff-catchers is often difficult, so it makes sense to turn to game-theoretic ideas for guidance. However, other sorts of indifferences are also common at equilibrium, and these also provide information about GTO play. Suppose Hero is facing a bet. On the river, a properly-constructed betting range will make many of Hero’s hands indifferent between calling and folding – everything below the bottom of his value range and better than his bluffs, neglecting card removal. The effect of a bet is not quite as clear on earlier streets because of the presence of draws and the need for balance. However, Villain’s early-street bets still often make many of Hero’s hands at least nearly indifferent between calling and folding, and if we think carefully, we can see that this is reasonable. First, even on early streets, betting ranges tend to be somewhat polarized. It makes sense to bet with strong hands in the hope of getting called by worse, but if we only bet strong, we will see lots of folds and be incentivized to bluff as well. If Villain’s betting range is polar, we will still have holdings that are stronger than the bluffs and weaker than the value, and Villain’s equilibrium bluffto-value ratio will still need to make many of these mediocre hands more or less indifferent. If he bluffs too much, we will call a lot, and he will be motivated to bluff less, and vice versa. Thus, the stable point will still occur when bluff-catchers can sometimes call and sometimes fold, i.e., when the two choices have the same EV. There are caveats here. We will encounter “reasons for betting” other than for value and to bluff, and these can lead to non-polarized betting ranges that do not create indifferent bluff-catchers. Also, even when we do face a polarized bet, few early-street holdings are really equivalent. Different hand combinations have more or fewer outs and backdoor outs, play differently on possible future cards, etc. Thus, facing a flop c-bet, it may be 36

Preliminaries most accurate to say that just one of our hands is actually indifferent between calling and folding while the rest strictly prefer one or the other. All that said, hopefully you are convinced that early-street betting ranges often make many hands at least nearly indifferent to calling. The conditions under which this indifference holds and the particular hands that are made indifferent will become clearer as we take some examples. The following holds for Hero’s hands that are indifferent to bluff-catching when facing a bet: This is called the bluff-catching indifference, since it applies to bluffcatchers. Don’t get confused here – the bluffing indifference applies to a player’s bluffs (and usually gives us information about his opponent’s bluff-catchers), while the bluff-catching indifference applies to bluffcatchers (but usually provides information about a bluffing frequency). How can we apply the bluff-catching indifference? Again, the naive case is embodied by the simple river PvBC game. There are only three types of hands here – Villain’s value hands that are obviously going to bet, Hero’s bluff-catching hands whose play we determined by applying the bluffing indifference to Villain’s air in the previous section, and Villain’s air, whose play we can find through the bluff-catching indifference. For Hero’s bluffcatchers, we have

Here, V is the fraction of the Villain’s betting range that consists of value bets. If we bluff-catch, then V of the time we lose the bet B, and the other (1−V) of the time, we end up with the pot and Villain’s bet in addition to our starting stack. Solving, we find V equals (B+P)/(2B+P), and this specifies Villain’s play with his air hands. As usual, Hero’s indifference has given us information about Villain’s strategy. By applying the bluff-catching indifference to a player’s bluff-catchers, we can find his opponent’s bluff-tovalue ratio. Now, as with the bluffing indifference, applying the bluff-catching indifference can be challenging in practice. Finding EV(folding) is easy – we 37

Expert Heads Up No-Limit Hold ’em, Volume 2 know the size of our stack at the end of the hand if we immediately fold – but, what is the EV of bluff-catching? It is often something like in the PvBC case – our stack after we call, plus the pot after we call, times the fraction of the pot we capture after we call. In the PvBC case, and on the river in general, the fraction of the pot we capture is just our raw equity. However, if calling does not guarantee a showdown, then the amount of the pot we can capture (and thus our EV) will depend on future action for several reasons: 1. Our hand can improve to beat some of Villain’s value hands, and we may be able to bet for value. 2. Villain’s bluffs might improve, allowing him to extract more value from us. 3. Even if hand values do not change, we can face future value bets and bluffs. One of the main lessons from our study of the river was that mediocre hands have a hard time realizing all of their equity. If both players just checked down, they would both capture exactly their equity. But, at the equilibrium of the PvBC model, for example, Villain’s bluffs are indifferent to checking down while his strong holdings do much better by betting. All the EV that Villain gains with value bets is EV lost by Hero’s bluff-catchers. Thus, the possibility of future action will have a large effect on the EV of bluff-catching even in very static situations. The possibility of changing hand values complicates things further. These issues arise in traditional exploitative decision-making as well, and it may be helpful to make that connection. Generally, we look to estimate the EV of bluff-catching when we have a call-or-fold decision with a borderline hand. We want to choose the action with the highest EV. The EV of folding is fixed, whereas the issues mentioned above affect the EV of calling. A simple way to approach this decision is to compare our equity to our pot odds. Pot odds are the ratio of the size of the pot after Villain’s bet (how much we stand to win) to Villain’s bet size (how much we stand to lose). Calling when our equity is greater than this ratio (appropriately converted to decimal) is equivalent to calling when EV(calling) is greater than EV(folding), if calling means that we realize exactly our equity in the pot. 38

Preliminaries An example will help make this clear. Suppose we face a half-pot bet. Then, the ratio of how much we stand to win versus how much it costs to call is 3:1. So, a pot-odds calculation tells us that we need to win one time for every three times we lose in order to have a profitable call. That is, we need at least 25% equity. On the river, calling gets us to showdown immediately, so a call allows us to capture exactly our equity in the pot, and in the PvBC game, Hero’s equity is the fraction of Villain’s range that is not value hands, i.e., 1−V. Plugging in to the naive bluff-catching indifference above, the equity we need to be indifferent is 1−V=1−(B+P)/(2B+P)=0.25. With any more equity, a pot-odds analysis tells us to call, and with any less, it tells us to fold. If there is no more betting after our call, a pot odds-based approach is all we need to make a correct exploitative call-or-fold decision (regardless of whether we hold a draw, a made hand, or some combination of the two). Otherwise, we need to account for the possibilities enumerated above. The first deals with the possibility we improve, so it is most relevant when we have a hand that is likely to do so, i.e., a draw. Again, pot odds are essentially the ratio of what we stand to lose to what we stand to gain with a call, and we can adjust for the possibility of winning more on later streets by increasing the amount we stand to win. For example, if we face a bet with a draw to the nuts, the amount we must risk stays the same, but the amount we stand to win must increase, since we expect additional value if we make our hand. The ratio of the costs to this improved estimate of how much we stand to win is known as our implied odds, and it takes the role of the equity in the previous decision-making process. If we make a call-orfold decision by comparing our implied odds to the odds offered by the pot, we have accounted for much of point 1 above. Accounting for the other two points is a bit more difficult. When we have a pure draw to the nuts, we will be certain about whether or not we have the best hand on later streets. However, with a non-nut draw or made hand, we will be in danger of calling with the worse of it or folding with the better, either because Villain has the best hand now or because he improves. We adjust the odds to make them less favorable in an attempt to account for these possibilities (thus obtaining reverse implied odds), but it’s difficult to be quantitative about it. However, these effects are actually quite 39

Expert Heads Up No-Limit Hold ’em, Volume 2 large in no-limit games. For example, if it weren’t for them, we could defend pre-flop min-raises with 100% of hands against almost all opening ranges. We cannot treat early streets in a vacuum – to strategize for multiple streets, we must take into account the possibility of future action. Although it will be a couple of chapters before we get to it in full, we’ll take an approach that is similar in spirit to the implied and reverse implied odds methods. We’ll replace raw equity in the pot odds decision-making process with another number that more accurately describes the amount of the pot we can expect to capture on later streets, called the capture factor. If we become good at estimating this quantity, we will be able to sweep a lot of the details of multi-street play under the covers, which is useful for making quick decisions at the tables. Suppose we are facing a bet on the turn with a mediocre hand that is unlikely to improve. Do these multi-street effects make us want to fold more or less than in a comparable single-street situation? How can Villain change his betting range to keep us indifferent? Consider the SB bet-or-check river game from Section 7.3.2 and the pre-flop 3-betting situation we saw in Section 9.4.1. Use the bluff-catching indifference to estimate the fraction of the bettor’s range that consists of bluffs at equilibrium. In the pre-flop case, again refer to your hand database, if you keep one, to find extra contributions to the EV of bluff-catching. How do your results compare with your own play and that of your regular opponents?

9.4.3 The Nuts, Near-nuts, and Nut Draws In Section 2.3.1, we discussed the BB’s play with a flopped flush draw and focused on check-calling and check-raising. Under certain conditions, we argued that he would take both lines with some frequency at the equilib40

Preliminaries rium. If he took one action too frequently, the SB’s response would incentivize him to take the other. This indicates that neither pure strategy is unexploitable, a mixed strategy must be played at the equilibrium, and the EVs of the two lines must be equal. The same sort of argument can be made in many spots where some play can lead to our holding a capped range later in the hand. If we work through the equilibration exercise, we can often convince ourselves that Villain’s response will make it profitable to start showing up in those spots with our strong hands. However, if we play all of our strong holdings that way, ranges in other parts of our strategy will be neglected, and we will be incentivized to readjust. Of course, this style of argument does not only apply to strong draws. Often, nut hands on early streets have a high chance of remaining the nuts after future cards arrive, and playing them one way all the time would cap our range in other spots. The number of hand combinations involved is important. For example, excluding top set from your range following a K♠7♥-3♥ flop does not cap your range as severely as, say, excluding all heart draws. However we’ll see that the argument does not only apply to the pure nuts. Rather, it frequently applies to many strong or near-nut hands for reasons we will see shortly. I believe it is most accurate to understand this indifference in terms of distributions rather than particular hand combinations: whenever we cap our distribution, Villain’s counterstrategy might incentivize us to stop. Now, we do need to take a bit of care. Taken too far, this sort of argument can lead to conclusions such as that unexploitable play will involve having some chance of showing up with the nuts in any spot. That’s certainly not the case. For example, any pre-flop hand can end up being the nuts on later streets, but of course we should not take every possible pre-flop line with every starting hand with positive frequency. The remote chance of a particularly favorable board will not make up for putting a lot of money in bad. Generally, the shorter the effective stack size, the less important is the threat of future action. If we always check-raise the flop with our draws, we will usually be left with a capped range when we check-call and then the draws come in, thus allowing the SB to play aggressively with much of his range. However, if the stack-to-pot ratio on the flop is sufficiently small, the value we gain from showing up with a made draw after check-calling 41

Expert Heads Up No-Limit Hold ’em, Volume 2 may simply not be enough to outweigh the advantages of semi-bluffing. Or, if the chance of our draw coming in is sufficiently remote, checkfolding might be best. We can usually be happy to fold all our 5-4 combinations to a flop c-bet on A♥-Q♦-9♠, despite the possibility that the board runs off 2♥-3♣. That simply does not happen often enough (and Villain will not put in enough action when it does) to make up for the money we lose by drawing. On the other hand, missing a bet with a strong hand can sometimes be costly enough to prevent us from ever slow-playing. Other times, Villain’s distribution just doesn’t let him put in enough action against our passive lines to incentivize us to start slow-playing. These indifferences are emergent properties of the complex dynamics when both players try to play HUNL games as profitably as possible, but they are not really fundamental. In future chapters, we will see a variety of computationally-generated equilibria of multi-street situations and will make note of the sorts of spots where nuts and draws to the nuts are made indifferent and nearly indifferent. Pay attention to these to build intuition for attacking these spots. In practice, we often need to proceed carefully through the equilibration exercise, keeping all options and all parts of players’ ranges in mind. Only then can we convince ourselves that none of the relevant options can be played all the time at equilibrium and identify which are used with a non-zero frequency. Review the solutions to the river examples described in 7.4. In which cases did the players play their nut hands in more than one way at the equilibrium? In which did they not? Why? Now, all that said, players play mixed strategies with nut, near-nut, and potential-nut hands in many cases for balance-related reasons, and we can use the corresponding indifference relationships to learn about the equilibrium. The indifferences applying to draws turn out to be difficult to leverage in practice. To see why, let us continue with the example of a BB’s flopped flush draws. We expect the BB’s indifference to tell us something about SB’s GTO strategy. We have, for the BB’s flush draws:


Preliminaries What are these EVs? The EV(check−calling) depends on a lot of things – how often the draw comes in, how often the SB barrels the turn when it does and does not, and so on. Hero’s EV(check−raise) depends on how often the SB folds to the raise and to additional barrels, how he plays versus a turn check after calling, how much action he gives in general after the draw comes in and does not, etc. We can estimate these EVs if we’re clever, but there is no escaping the fact that this indifference equation is a rather complicated relationship between many different aspects of the SB’s play. This is in contrast with the bluffing and bluff-catching indifferences above, which happily depended on just one of a player’s frequencies, which allowed us to immediately solve for it. Our potential-nuts indifferences may be simplified in short-stack situations (for example, if a check-raise is necessarily an all-in so that there is no future action), but we noted earlier that the indifference might not even hold in such a short-stacked spot. We will spot the indifference of draws in games’ solutions and use them as landmarks to make sure we understand what’s going on, but we will rarely use them to solve for unexploitable frequencies from scratch. Continue with our example of the BB’s flopped flush draws. What does the EV of leading the flop depend on? When might the BB be indifferent between leading and check-calling, while check-raising is worse than either? When might EV(leading)=EV(check−raising)>EV(check−calling)? It turns out the indifference facing actual nut hands is quite a bit easier to deal with. Draws want to get as much money as possible in the pot when they make their hand, but when they miss, they benefit from having kept the pot as small as possible, except when they are able to win without showdown. Of course, trying to win without showdown often involves the risk of putting in a lot of money and still not forcing an opponent to fold. The tradeoffs here are a bit complicated. Playing the nuts is much easier, at least when we are not worried about being drawn out on, i.e., on the river or on static boards. Then, the only consideration facing these hands is how to get as much money in the pot as possible. If our hand can’t lose (or chop), its EV is simply all the money in Hero’s stack, plus the entire pot, plus whatever additional money Villain 43

Expert Heads Up No-Limit Hold ’em, Volume 2 will put in the pot, on average. At any particular decision point, the first two terms are fixed, so we maximize our EV by taking whatever line causes Villain to put the most money in the pot, on average. Thus, if we are indifferent between two lines, it tells us something very intuitive about Villain’s play following both actions – Villain puts the same amount of money in the pot, on average, against both of them. The simplicity of this statement makes it very powerful, as we will see in an example shortly. First, notice that the same reasoning often also applies to strong but nonnut holdings, especially on static boards. Suppose we have a hand strong enough that we are happy to get all-in and strong enough that, whenever Villain has better, he too will try to get all-in. Then, whenever we are losing, we are sure to get all-in and go broke. Since the outcome when Villain has a better hand is already fixed, we maximize our overall EV by playing to maximize it in the case that Villain has a worse hand. When Villain has a losing hand, we do best by simply trying to get as much money into the pot as possible. So, if a near-nut hand is indifferent between two lines, it must be the case that Villain puts in the same amount of action versus both of them on average. That is, the sort of indifference relationships we have been discussing in this section apply to near-nut hands as well. Also, although our argument has assumed the board is static, we can imagine that when it is only nearly static, then players’ nut and near-nut hands will often be nearly indifferent, still a useful relationship for making approximations. Consider the following useful example. The decision tree for the 1-betbehind full-street river game is shown in Figure 9.1. As the name suggests, it models river situations where the stack-to-pot ratio is small enough at the beginning of river play that any bet is all-in. In complexity, this game lies sort of in between the bet-or-check games and the two-bets-behind games we focused on in Chapter 7. However, it is useful in its own right for understanding river play, since players often size their early street bets so as to end up with just one bet left behind on the river. We will study it further in Section 13.4.2. Take for simplicity the symmetric distributions case with one pot-sized bet behind at the beginning of river play. Suppose Hero is in the BB and holds a


Preliminaries hand that is the nuts or the near-nuts in the sense that it will always get all-in versus a better hand. These holdings actually make up a significant portion of his range. For example, with our assumptions, it turns out that the top one-third of his river starting distribution always gets it in versus better hands at the equilibrium. So, with his near-nuts, Hero’s options are to jam or to check with the intention of calling a bet all-in, and we can convince ourselves that he cannot be taking either line 100% of the time at the equilibrium. Why is it that Hero’s GTO strategy cannot involve either always checking or always leading here with all his near-nut hands? Because of this indifference, Villain must put in the same amount of money, on average, versus both of Hero’s actions. In particular, the SB must be jamming when checked to with exactly the same frequency as he calls a lead all-in. This turns out to be a useful observation when trying to understand river play with one bet behind. In fact, combined with the previous indifferences, we now make a lot of progress towards solving this game. Consider giving it a try before continuing. First, we already knew how to estimate how often the SB calls a lead in this spot – we apply the bluffing indifference to the BB’s bluffs to find that the SB must call 50% of the time in order to keep the BB’s weak hands indifferent to bluffing, and he builds this calling range out of his strongest 50% of hands, since it does not make sense to call with a weaker hand while folding a stronger one. Previously, we had no short-cuts that allowed us to determine how frequently the SB bet when checked to, but now, we know that his GTO betting frequency in that spot must also be 50%! To drive this point home, suppose the SB bet less than half the time when checked to. Then, Hero would lead with all his strong hands, Villain would counter by playing aggressively versus Hero’s severely capped checking range, and Hero would be motivated to begin checking strong hands. So, the SB is jamming half the time facing a check. Additionally, since the bet is the size of the pot, his betting range should contain two value-bets for every one bluff in order to make Hero’s bluff-catchers indifferent to calling. Thus, 2/3 of his betting range should be value, and 1/3 should be 45

Expert Heads Up No-Limit Hold ’em, Volume 2 bluffs. Logically, we know that the SB will build his value range from his strongest hands, and he will pull his bluffs from his weakest. So, we can conclude that the SB’s equilibrium strategy involves betting when checked to with the top 1/3 and bottom 1/6 of his river starting distribution, and we have completely determined the SB’s equilibrium play of this game! A similar indifference between checking and leading holds for the BB’s nut hands in the two-bets-behind river situation we covered in Section 7.3.3, at least in the symmetric-distributions case. Of course, when the BB leads with the nuts, it is with the intention of bet-calling, and when he checks, he will check-raise if given the chance. Thus, it is not necessarily the case that the SB bets when checked to with exactly the same frequency that he calls a bet, but he puts in the same amount of money on average versus the BB’s checkraises and bet-calls. If he bets when checked to more than he calls a lead, he must make up for that by calling a check-raise more than he raises a bet. Suppose we have flopped the nuts. We are indifferent between slow-playing and fast-playing, but draws are possible and some of Villain’s range has some chance to improve to beat our hand. How does this affect the relative amounts of money Villain must put in versus our two lines on average when the draws come in and when they do not?

9.4.4 Putting it all Together: Blocking Bets Finally, we can combine some of these ideas to learn a bit more about block-betting from the BB on the river. In Section 7.3.3, we focused on using them exploitatively, since NLHE players’ response to small bets varies extremely widely and is often quite poor, and it can be difficult to see exactly how they might be useful in equilibrium play. For one thing, when we use multiple bet sizings in a spot, the solutions can rarely be broken down into clearly defined action regions. The solutions almost always end up being highly mixed for balance-related reasons, making it difficult to see quite what is happening. However, in the context of the 1-bet-behind river situation, it is relatively easy to see how blocking bets might fit into our GTO strategy with certain 46

Preliminaries river starting distributions. So suppose we are playing a game like the one shown in Figure 9.1, except that the BB may also lead for a blocking-sized bet, and, facing such a bet, the SB can fold, call, or jam. That is, any bet is all-in except for the BB’s blocking bets. We start the river with one potsized bet left in the effective stacks. Suppose the BB’s blocking bets are 1/5 the size of the pot, and suppose the starting distributions are such that all the indifferences we have been discussing indeed hold. First of all, many aspects of the solutions to this game are going to be similar to the solutions to the basic 1-pot-sized-bet-behind game we saw in the last section. The SB will still probably call a lead all-in 50% of the time in order to keep Hero’s air indifferent to bluffing. Then, the SB must also jam 50% of the time when checked to in order to keep Hero’s nuts indifferent between check-calling and leading all-in. Now, how should the SB play when facing a blocking bet? How often does he continue versus the bet? We can apply the bluffing indifference to find out, but first, it should seem reasonable that the SB cannot fold very much at all or else the BB could very profitably use the small sizing to bluff. In particular, if P and S are the sizes of the pot and effective stack at the beginning of the river, L is the size of a blocking bet, and the SB folds to a block F of the time, then we have for the BB’s potential bluffs:

so that the SB’s folding frequency is F=1/6. So, the SB must indeed be continuing a lot versus a blocking bet. This assumes that the BB’s river starting distribution actually contains a sufficient number of relatively-weak hands that can be used as bluffs. How often does the SB raise a blocking bet? We can use what we learned in the previous section to figure it out! First of all, before doing the math, a bit of logic will give the main point: the SB cannot raise a block-bet as often as he calls a lead all-in or bets all-in when checked to. Why not? Well, if he did, then the BB’s nut and strong value hands would all start blockbetting, since it would allow them to make more money on average than 47

Expert Heads Up No-Limit Hold ’em, Volume 2 the other options. The BB’s nuts would get all-in just as often as if they took another line, but they would also get extra value from all the other hands the SB has to call with when facing a blocking-bet. We can find the SB’s raising frequency necessary to make the BB indifferent to blocking with near-nuts. For these, if G is the SB’s raising frequency when facing a blocking-bet, we have

and we find G=5/12 . So, the SB indeed gets all-in versus a blocking bet somewhat less than the 50% of the time he gets it in versus a check or a jam. We have implicitly assumed here that the BB’s river starting distribution contains some strong value hands he can sometimes block bet with – no leading with any sizing is going to work well for him if his range is capped low. Now we know that, when facing a blocking bet, the SB is calling a ton and getting it in fairly often but not as often as if he faced a check or a jam. This is a perfect environment for the BB’s weak value hands, the sort that stereotypically might try a small bet to “block” the SB from making a bigger bet. These hands can make a small lead, actually get a bit of value from the SB’s very wide calling range, and also not face an all-in as often as if they had checked. Avoiding the jam is something to be happy about for these mediocre holdings, since it usually makes them indifferent between a call and a fold, i.e., it makes it very difficult for them to realize their equity. So, as we saw in the solutions to the examples in chapter 7, the BB’s GTO river strategy frequently includes blocking bets, and now we can see why. The blocking range is usually comprised primarily of his hands with mediocre showdown value. If he were only blocking with these, the SB would raise him very frequently, so he is motivated to add strong value hands into his blocking range until the SB raises less often and block-betting the nuts is no more profitable than his other options. But if the BB were only blocking with weak value and strong value hands, the SB would fold a lot, so the BB can take the opportunity to block as a bluff with some very weak hands as well. If the BB’s river starting distribution does not contain all of 48

Preliminaries these three types of hands, he might not be able to incorporate blockbetting in his river strategy. Lastly, remember what we learned about block-betting when studying computational solutions of river spots. The BB rarely preferred to block bet with any of his holdings. More often, his blocking hands were indifferent to playing “normally”. Nonetheless, he improved his overall EV by opening a blocking range, and the EVs of his mediocre made hands were particularly improved. This implies that moving some hands to his blocking range improved his EV when he checked those hands as well. What was going on there? Basically, the SB could no longer take advantage of the BB’s checking range being quite as bluff-catcher-heavy as it was before. Thus, he could not value-bet and bluff quite as much when checked to, and the BB’s mediocre made hands were able to see a showdown more often than they did before. These indifferences will show up all the time – indifferences of bluffs, bluff-catchers, near-nuts, and draws. Identifying these will help us make sense of complicated strategies. It is not always easy to untangle the complicated interactions between players’ equilibrium ranges, and these observations will give us some ways to understand what we see and give confident answers to questions about why GTO play is the way it is. Of course, these tools are also useful for estimating unexploitable frequencies for use in our own play.


You Should Now …

Know what equilibrium and exploitative play are and how we use them.

Understand the importance of reading your own hand for playing strategically.

Remember the Indifference Principle and its applications to players’ bluffs, bluff-catchers, and nut hands.

Understand river play with one bet behind and know the properties of the equity distributions that make block-betting reasonable in that context. 49

Chapter 10 Turn play: Polar Versus Bluff-catchers Redux

Essentially, all models are wrong, but some are useful. – George Box Many aspects of multi-street play will be familiar from our study of river and pre-flop-only situations. Certainly the governing principles have not changed – we’ll still look to find the EVs of each of our options and choose the largest. However, the flop and turn have a number of surprises in store for us as well. Multi-street games cannot just be thought of as a collection of single-street spots played in sequence. Equilibrium frequencies on early streets will be strongly affected by the threat of future action, and the possibility of changing hand values motivates plays that wouldn’t make any sense on the river. To get started, we will take another look at a situation that has served us well so far: the play of a polar range versus a range of bluff-catchers. These PvBC situations actually come up quite frequently on the turn, at least approximately, by virtue of flop play’s affect on the players’ distributions. We will begin with the simplest case and gradually introduce the complications of real play.


Turn Play: Polar Versus Bluff-catchers Redux

10.1 The Two Street PvBC Model Suppose we are on the turn, the SB holds a range of bluff-catchers, and the BB’s distribution is relatively polar. That is, each of the BB’s holdings is either better or worse than all of the SB’s range. Additionally, assume for now that hand values are completely static: hands which are the nuts on the turn will still be the nuts on the river, hands that are bluff-catchers on the turn will still be bluff-catchers on the river, and similarly for air. We will analyze this situation in some detail, but first, when is this model useful? That is, when do these two assumptions accurately describe real play? The assumption of static hand values is much more reasonable on some boards than others prior to the river (recall Section 6.2.1). Do the PvBC distributions often arise on the turn? Holding a polar distribution essentially means that we know whether or not we have the best hand, and we saw on the river that it allows us to take advantage of a bluff-catching opponent. We can keep him from realizing a lot of his showdown value, often by using large bets. We can imagine that this is not really a stable situation once we expand our view to incorporate the early streets. Villain will probably be sufficiently motivated to change his earlystreet play so that he no longer finds himself purely bluff-catching. Thus, it might seem unlikely that pure PvBC situations show up very often in the equilibrium solutions to the full HUNL game. On the other hand, since the average pot size (and thus a nut hand’s winnings) grows so quickly with the number of bets that go in, Villain’s strong hands may find that slow-playing is rarely worth it, even if it means that his passive lines are taken with capped distributions. Realistically, when Villain is motivated to slow-play, he often only needs to do so with a small number of hands to achieve balance. Additionally, we will see later that he usually does so with his very strongest holdings, while less nutted value hands prefer to play aggressively for what we’ll identify as protectionrelated reasons. Thus, even well-balanced ranges can end up resembling the PvBC case, perhaps with a few traps added to the bluff-catcher’s range.


Expert Heads Up No-Limit Hold ’em, Volume 2 (And, remember that in Section 7.2.5, we found that in the river case, adding a small number of traps to the bluff-catcher’s range usually did not change the players’ strategies.) Whether or not they arise in GTO play of the full game, these situations certainly arise in real play, at least approximately. Balancing is difficult (and not always necessary) and standard play often leads players to define their ranges as composed of weak made hands on the flop. Then, if the board is static, we will arrive at the turn in an approximately-PvBC spot. Suppose we are in a singly-raised pot – the SB raised pre-flop and was called. A simplified description of standard flop play might be as follows. The BB starts out with a check, and the SB c-bets most of his hands. Standard c-betting criteria vary with board texture and stack sizes, but something like second pair or better might bet for value, while holdings worse than A-high bet as a bluff. Facing the c-bet, the BB value-raises his strongest hands, folds his weakest, and calls with many single pairs. With this dynamic, there are a couple ways we can arrive at a PvBC spot on the turn. If he checks back the flop, the SB has essentially just turned his hand face-up as weak showdown value – something from A-high to a low pair. He would have bet for value with any better hand and bluffed with anything worse. Insofar as the majority of the BB’s holdings will be either better than bottom pair or worse than A-high, his distribution is polar relative to the SB’s narrowly defined range. Checking back mediocre holdings is not necessarily bad in itself – in fact, it makes a lot of sense – but if the SB does not play some other types of hands the same way, he can find himself on the bad end of a PvBC situation. The BB can find himself holding a range of bluff-catchers on the turn as well. When he check-calls the flop, his holdings range from bottom pairs to weak top pairs, while the SB holds everything except bottom pair and Ahigh. Then, the players can both hold middle pairs and weak top pairs, but otherwise, all of the SB’s holdings are either better than all of BB’s range or worse than all of his range. That is, to good approximation, the SB is relatively polar at the beginning of turn play. In practice, check-calling the flop does not usually define the BB’s range quite so narrowly as checking back the flop does for SB’s. However, the SB’s 52

Turn Play: Polar Versus Bluff-catchers Redux checking back only affects his own range, while both players narrow their ranges when the flop goes bet-call. In combination, c-betting somewhat polar and check-calling somewhat bluff-catcher-heavy often lead to an approximately-PvBC turn spot. If either player’s flop play follows the stereotype to an extreme degree, it could be enough, and it is often sufficient that both players have some tendency in that direction. As in the SB case, the BB’s passive flop play with his mediocre hands is not bad in and of itself, but his failure to keep other types of hands in his distribution can be exploited. Keep in mind that we are focusing here on the case of static hand values. First, this allows us to strategize on the turn knowing how strong our hand will be on the river. Second, and perhaps more importantly, if a turn card is likely to significantly mix up hand values (i.e., if the board is volatile), then the strategies we described above are less likely to lead to a PvBC situation on the turn. Even if we hold primarily mediocre hands at the end of flop play, some will become significantly stronger or weaker on many turn cards. For this reason, checking back mediocre hands on a volatile board, at least some of which have the possibility of improving (e.g., 10♠-6♥ on an 8♣-7♦-6♠ flop), has fewer of the downsides associated with turning your range face-up on a static board. Take a minute to look at some turn starting distributions. Pick a static flop and determine the SB c-betting and BB check-calling ranges you’d use against an unknown opponent. Pick a random turn card and graph the turn starting distributions using a computer utility such as the one available on this book’s website. Does the SB’s distribution look polar? Why or why not? Now, we’re about to solve the two-street, static PvBC game. As in the single-street case, some indifference relationships will be the key to finding the strategies. The indifferences that hold are likely those you would guess. In fact, we could just assume them, solve for the strategies, and then verify that both players are playing best responses. Guess-and-check is a perfectly legitimate way to find an equilibrium! However, there’s a lot of value in proceeding more carefully. It’ll help us understand exploitative strategies in the PvBC context, and it’ll prepare us to understand what’s going 53

Expert Heads Up No-Limit Hold ’em, Volume 2 on once we break out of the PvBC game and into situations that are not so simple. That said, the following argument is probably the most challenging portion of the book (we’re getting it out of the way early!) So, it will be helpful to have a bird’s-eye view of the points we’re working towards. Assume the polar player has enough air that he cannot always bluff. Then, GTO play is as follows. The polar player will bet his value on both streets, using a particular bet sizing that puts him all-in on the river. With his air, he will sometimes give up on the turn, sometimes bluff once (a single barrel), and sometimes bluff twice (a double barrel). He will never slow-play by checking the turn with his nuts, so he will never bet the river after the turn checks through. The bluff-catching player will be indifferent to calling whenever he faces a bet. The bulk of the next section will be spent showing that these indifferences hold and also that the polar player cannot slowplay – once we know these things, finding the equilibrium will be easy.

10.1.1 The Solution First, as in PvBC river play, only the polar player will do any betting. Essentially, since any particular holding of his is the nuts or air, he knows whether or not he has the best hand. The bluff-catching player, on the other hand, is in the dark, and there is no reason for him to ever bet, since his opponent is able to play perfectly against it. If we check-call with a bluff-catcher, we always put in a bet against the nuts, but we have some chance of winning a bet from Villain’s air as well. However, if we bet ourselves, we ensure that we put a bet in whenever Villain has the nuts, and we give him the option to get away cheaply with his air. Since the bluff-catching player will never bet at equilibrium, positions are essentially unimportant. The polar player will always get a chance to bet on each street. His opponent will check back if given the chance and will call or fold when facing a bet. This should agree with our experience of real play. For example, if the SB c-bets the flop and the BB check-calls, the BB will generally follow up with a check on the turn unless the turn card happens to have a large effect on the relative hand strengths. 54

Turn Play: Polar Versus Bluff-catchers Redux For the sake of the discussion, we will assume the SB is bluff-catching and the BB is polar, as if the SB has checked back the flop with a range of purely weak showdown hands. We will follow up with an example where this is the case. However, keep in mind that positions do not really matter, at least in the ideal case, due to the hand distributions. Our results will apply just as well when the SB is polar. At the beginning of turn play, the pot size is P and both players have S behind. The polar BB will either check or bet B. If he checks, then so does the SB, and we get to the river with the same pot and stack sizes. If he bets, the SB can fold or call, and if he calls, we get to the river with a pot of P+2B and remaining stacks of S−B. In general, we will need to consider all of the possible river cards separately, but for now, they are all effectively the same due to our assumption of static hand values. So, the decision tree is shown in Figure 10.1. It shows the turn action and the two river situations that arise after a bet does and does not go in on the turn.

Figure 10.1: Decision tree for the PvBC situation on the turn with static hand values.

As on the river, we have something of a special case if the polar player’s range is especially nut-heavy. In this situation, he can bet with all of his 55

Expert Heads Up No-Limit Hold ’em, Volume 2 hands, all-in or even smaller, and the SB will still prefer folding over calling with his bluff-catchers. If this is possible, then this will be an equilibrium strategy for the BB, since it wins him the whole pot all the time, and he can expect no better than that at equilibrium. Practically speaking, it will rarely be the case on the turn that the polar player’s range is much, much stronger than his opponent’s. For example, consider the two situations that we saw lead to a turn PvBC spot. In one, the SB checked back a flop with a range of weak made hands, and in the other, the BB check-called with a fairly similar range. In both cases, the bluff-catching player at least has something, and the majority of the polar player’s range likely falls into the air category by comparison. As they say, it’s hard to make a hand heads up. Perhaps we can imagine turn spots in 3-bet pots or after flop check-raises where ranges are narrower in general, and the strong portion of the polar player’s range is fairly large – possibly large enough that he can get away with betting all of his weak hands as bluffs. We will consider these possibilities in the context of some examples a bit later. The criterion for this special case is similar to that on the river, and we will not go through it here. Once we solve the normal case, it will be easy to visualize the conditions under which our solutions break down. For now, we will proceed with the assumption that the BB’s range is not strong enough to just bet and take down the pot all the time on the turn. In this case, when facing a turn bet, the SB will sometimes call and see a river. If he did not, then the BB would bluff all his air on the turn in order to win the whole pot all the time, but then the SB would certainly want to start calling since we assumed that the BB did not have a strong enough range to make the SB always prefer folding. Now, how do we find the equilibrium? We’ll start by identifying some indifference relationships. They might seem obvious by analogy to the single-street case, but several subtleties arise because of the multi-street nature of the game, and these ideas won’t all hold in more complicated situations, so it’ll be helpful later if we understand exactly what’s going on. Once we have the indifferences, finding the strategies will be easy. Let us start with something we should already understand: play in the two 56

Turn Play: Polar Versus Bluff-catchers Redux river situations. We know the BB will always bet the river with his nuts, and using all-in as his sizing is at least co-optimal. The indifferences that hold, however, are not immediately obvious, since we don’t know how much nuts and air he brings to each river spot. First, we can see that in both river spots, if the BB has a betting range at all, then his air must be indifferent to bluffing. He can’t have a non-empty betting range and still strictly prefer to check with his air at equilibrium, since that would mean he’s betting only with the nuts, and the SB’s response will clearly motivate him to begin bluffing. On the other hand, his air can’t strictly prefer to bluff either. Why not? We’ll consider the two river spots separately. First, consider river play after the turn goes bet/call. After the turn bet, remaining stacks are S−B. If the BB strictly preferred to bluff his air here, then that would mean he bets his entire range. It would also mean that the SB sometimes folds to a bet, since otherwise there is no way the BB could want to bet his bluffs. But if the SB is folding to bets, then his EV when facing a bet is S−B, and since the BB is always betting the river, the SB’s EV with a bluff-catcher on the river is always S−B. If this were so, then the SB’s strategy must involve always folding to the turn bet – why would he call and guarantee himself an EV of S−B when he can just fold the turn and achieve an EV of S? This cannot be the case, since we have already assumed that the BB’s range is in fact not strong enough to always win the pot on the turn. Thus, if the BB’s range is not strong enough to bet and always take it down on the turn, he also cannot strictly prefer to bet his bluffs on the river after the turn goes bet/call. We now know a couple of things. First, the SB sometimes calls when facing a turn bet, and he sometimes calls the subsequent all-in on the river as well. Second, in order to make the SB want to call the turn, the BB must sometimes be getting to this river spot with air that gives up. Now, notice that the BB’s EV with air hands at the beginning of river play here is S−B: that is the EV of giving up on the pot, and his EV of bluffing cannot be any greater or else he would always do so. Since his EV with air is S+P if he bets the turn and gets a fold but is S−B if he bets the turn and gets a call, something which happens with non-zero frequency, his EV of betting the turn with air is necessarily less than S+P. We will use this fact shortly.


Expert Heads Up No-Limit Hold ’em, Volume 2 Now, the BB cannot strictly prefer to bet his bluffs after the turn checks through either. If he did, at equilibrium, that would mean that betting all his bluffs was still not enough to make the SB want to call with bluffcatchers. In other words, the SB always folds, and the BB’s bluffs here have an EV of S+P. We know that this is better than his EV of betting the turn, so the conclusion here would be that the BB saves all his air to bet the river and is still able to make the SB always fold to the bet. However, this can’t be true. An all-in on the river that included all the air in his turn starting range could only be a more attractive call for the SB than the same bet on the turn. Thus, if he could always bluff in this river spot, he could also always bluff the turn, and we know that is not the case. In neither river spot can the BB bet all of his air. At this point, we know that in both river spots, the BB must be indifferent to betting with all of the bluffs with which he gets there, at least if he has a betting range at all (i.e., if he has nuts that need balancing). Great, we can use that. We also saw that the SB is not always folding to a bet in either river situation, but he is clearly not always calling either. (Why not?) Since he sometimes calls and sometimes folds when facing a bet, he must be indifferent between the two options. So, whenever we get to the river, we know how things will go. The BB will bet all-in with all of the nuts he gets there with, and as many bluffs as it takes to make the SB indifferent to bluff-catching, and the SB will call enough to keep the BB indifferent to bluffing. Now, if the BB does not get to one of the river spots with any nuts, he won’t have a betting range there, since he can’t bet only bluffs. However, in river spots where he has a betting range, all the indifference relationships we might hope for turn out to be true. Now what about turn play? The SB cannot always fold to a bet, but can he always call? We need to suppose that he does, figure out the BB’s maximally exploitative response, and then check if it incentivizes the SB to begin folding instead. This should seem reasonable, but things are a bit complicated. If this were the river, then the BB’s best response would be to simply stop bluffing since he is always getting called, and this would certainly make the SB want to stop calling with bluff-catchers. On the turn, however, it is possible that the BB might still find it best to bluff some in order to continue on the river. A turn bluff certainly is not profitable in and of itself, but perhaps he can make up for that by betting again on the river. 58

Turn Play: Polar Versus Bluff-catchers Redux What the BB will certainly not do, however, is just bluff the turn and then give up on the river – if the SB is always calling the turn, he is just throwing money away. So whenever the BB bets the turn here, it must be with the intention of continuing on the river. And whenever he decides to play bet-bet with a bluff, that line must have an EV of at least S, or else he would have just given up on the turn. So again, how might the BB counter a SB who always calls a turn bet, and is the SB motivated to begin folding on the turn? Well, the SB will always fold on the turn if the EV of calling and playing a river is less than S. So, what is the value of the SB’s bluff-catchers on the river after he calls a turn bet, given that the BB is playing maximally exploitatively? It depends on the SB’s river play, of course, but the best possible case for him is that he too is playing a best-response strategy in the river subgame. But if both players are playing best responses, then the players play an equilibrium in the river subgame, and we know what that looks like. The BB bets the right mix of value and bluffs to make the SB indifferent to calling and folding. Thus, the SB’s EV is S−B regardless of his play, whenever he faces a bet. And, he always faces a bet, since the BB never bets the turn and gives up on the river. The BB will bluff just enough on the turn that he can always continue on the river and exactly make the SB indifferent to calling and folding. Thus, S−B is the value of the SB’s hand whenever he calls the turn bet. So, if the SB never folds the turn with his bluff-catchers, and the BB responds maximally exploitatively, the SB’s EV is no more than S−B when facing a turn bet. He could clearly do better by just folding on the turn to end up with S. In other words, if the SB is always calling when facing a turn bet, then even if he plays as well as possible on the river, the BB’s counterstrategy will incentivize him to start folding on the turn. This is what we needed to show that the SB must play a mixed strategy with his bluffcatchers on the turn when facing a bet. We now know that the SB is playing to keep BB indifferent to bluffing in each spot where the BB actually has a betting range. We can apply the bluffing indifference to find the SB’s calling frequencies in these cases. First, the SB calls the turn just enough to make the BB indifferent to bluffing. Thus, for the BB’s air:


Expert Heads Up No-Limit Hold ’em, Volume 2

The right side of the last equation gives the BB’s expected stack size if he bets the turn and then gives up on the river with a bluff. He always ends up with at least S−B, and he gets an additional B+P (i.e., he gets his bet back and wins the pot) whenever the SB folds the turn. Solving, we find that the SB folds to a turn bet with frequency B/(B+P). If he calls any less, then stabbing (at least) once is necessarily better than giving up on the turn for the BB’s air. If the SB is calling any more, then just giving up is better than bluffing once. Bluffing twice, of course, could still be better than either of those options. This result should look familiar. This is the same calling frequency that the bluff-catching player used in the river PvBC game. This is surprising! Calling with a bluff-catcher is much less appealing on the turn because of the possibility of facing a river bet. However, the frequency simply arises from the need to make the BB indifferent to betting once to try to win the pot with hands that will otherwise go to showdown and lose, the same as on the river. If the SB calls any less than this, the BB will prefer to always bet the turn rather than just give up with his bluffs. We now know the SB’s play on the turn, and a couple familiar applications of the bluffing indifference give us his frequencies in the river spots, again, assuming that the BB gets to them with a range containing some value hands. After the turn checks through, the BB’s river bet sizing will be all-in, S, into a pot of P, so the SB will call P/(S+P) of the time. After the turn goes bet/call, the BB will bet S−B into a pot of P+2B, so the SB’s calling frequency will be (P+2B)/(S−B+P+2B). Notice that these expressions are essentially the same except that the river stack and bet sizes are different because of the turn bet. How about the BB’s play? We now have enough information about the SB’s strategy that we can figure out the most profitable way for the BB to play his nuts, and we can then find the amount of bluffs he plays similarly for balance. His two choices on the turn are bet and check – if he holds the nuts, which is better? First, if he checks the turn, he will end up in a river spot where he will jam and get called with a certain frequency. We have 60

Turn Play: Polar Versus Bluff-catchers Redux

since he ends up with S+P all the time and another S the P/(S+P) of the time that the SB calls his river jam. On the other hand, if the BB bets B on the turn and continues all-in on the river, he ends the hand with at least S+P. He wins an additional B when the SB calls the turn and folds the river, and an additional S when the SB calls both streets:


The fractions (S−B)/(S−B+P+2B) and (P+2B)/(S−B+P+2B) are the SB’s folding and calling frequencies, respectively, after he gets to the river and faces a shove. Now that we have the EVs of the BB’s two options with the nuts on the turn, which is bigger? To check, we subtract EVBB(bet turn with nuts)−EVBB(check turn with nuts) and, after some simplification, find a result which is necessarily positive. That means EVBB(bet turn with nuts)>EVBB(check turn with nuts), which is what we wanted to know. It should also agree with our intuition: it is better to bet the turn and river than to slow-play the turn and then over-bet jam the river with our nut hands. The multi-street betting strategy essentially lets us get more value with our strong hands. In some sense, Villain gets sucked in since the bets are smaller and he has to call more frequently to discourage our bluffs. Our opponent has to put more money in the pot on average, which is what we want for our nut hands. As for our bluffs, they still break even at equilibrium. Evaluate EVBB(bet)−EVBB(check) for the BB’s nuts on the turn and verify that it is necessarily positive (Hint: B≤S). If you allow for non-equilibrium SB frequencies, when is checking the turn the correct exploitative play? So, the BB bets the turn and jams the river with all of his nut hands. How many bluffs does he play likewise? Recall that on the river, he bluffs enough that his bluffs make up (S−B)/(2(S−B)+(P+2B)) of his betting range – 61

Expert Heads Up No-Limit Hold ’em, Volume 2 the same as the odds the SB is getting on a call. But now, how many additional bluffs is he betting on the turn and then giving up on the river? It has to be just enough to make the SB indifferent between folding and calling on the turn. (Whether we compare folding to call turn-call river or call turn-fold river does not matter, since the EVs of the SB’s river actions are equal.) We can use this indifference to solve for the BB’s single-barrel bluffing frequency. We have

since, when he plans to call turn and fold river, his stack size at the end of the hand is always at least S−B, and he gains another 2B+P whenever the BB has a bluff that does not follow through. Thus, the fraction of the BB’s turn betting range which is bluffs that bet once and then give up is B/(2B+P).

10.1.2 Summary of the Two-street Static PVCB Situation How should we think about these results? The BB’s value hands always bet the turn and jam the river. Some of his bluffs also bet twice. These make the SB indifferent between calling and folding on the river, i.e., he chooses his double-barrel bluffing frequency so that the ratio of value bets to bluffs equals the SB’s pot odds when he faces the river bet. The BB also has some bluffs that only bet once. These make the SB indifferent between folding the turn and calling once. To accomplish this, the number of these onebarrel bluffs as compared to the total number of hands he shoves on the river (value and bluffs) equals the SB’s pot odds on the turn. Essentially, we can think of the BB’s entire river shoving range (including bluffs) as “value” hands on the turn, and then use the formulas describing his bluffing frequencies from the single-street case to find his one-barrel bluffing frequency on the turn. There are a lot of similarities between the single-street and multi-street PvBC games. This will certainly not always be the case in multi-street play.


Turn Play: Polar Versus Bluff-catchers Redux

Figure 10.2: Range splitting for GTO play of the two-street static PvBC game, drawn for the case where B=P and S=4P. The polar BB bets the turn and jams the river with his nut hands, which make up N of his turn starting distribution. He bluffs once, and twice with some of his air as well. The bluff-catching SB folds half of his range each time he faces a bet.

These strategies are summarized in Figure 10.2. They are to-scale for the case where a pot-sized bet on the turn leaves a pot-sized all-in on the river (i.e., B=P and S=4P). The graphic makes it easy to visualize the relative sizes of the various parts of the players’ ranges. We can see that he has two river value jams for every one river bluff jam, as expected. Additionally, he has one single-barrel bluff for every two river jams. Now, for the pot and bet sizes in the figure, the BB bets the turn with 1.25 bluffs for each value hand in his range. He could bet with even more bluffs if there were more money behind. This contrasts with the river case where a bettor could never have more than one bluff for each value hand, no matter how big the bet. Why is this important? Bluff-catchers have more value than air only by virtue of the fact that they can sometimes show down and win. When they face a bet at equilibrium in our PvBC models, they are made indifferent between calling and folding – they have no more value than air. Being able to bluff more means that the polar player can 63

Expert Heads Up No-Limit Hold ’em, Volume 2 more frequently devalue his opponent’s holdings. This is another way to understand the result that betting the turn and river to get all-in is better than simply jamming a single street. Related to this, notice that the SB folds the same proportion of his range on each street, and he gets to showdown a lot less often than in the single-street case. In effect, the polar player can exert much more leverage over multiple streets, making it even harder for a bluff-catching opponent to realize his equity. In this game, slow-playing is always incorrect, and whenever the BB checks the turn, he has air that is giving up. Although this may feel “weak”, it is not exploitable. He is still playing all of his hands as profitably as possible. Despite the fact that the BB is always betting his nut hands, his opponent’s maximally exploitative response does not motivate him to begin checking them. Exactly how much more value do the BB’s nut hands expect by betting twice rather than slow-playing, if B=P and S=4P? Plugging in to Equation 10.1, we have

Essentially, he is guaranteed his stack and the pot, S+P, since he has the nuts, and he wins one additional bet and the SB’s whole stack another quarter of the time each. Now, EVBB(check−bet with nuts) depends on how the SB plays after the turn goes check-check. We will see shortly that the equilibrium play here is not entirely clear, but for now, assume that the SB tries to make the BB’s bluffs indifferent on the river, i.e., he calls 1/5 of the time. Then, we have

Betting twice allows the BB to win his opponent’s whole stack more often than by slow-playing, and he sometimes wins an extra bet B on top of that. Slow-playing is no good here, but it can become a more viable option when the situation allows Villain to bet when checked to, as we will see. We promised to return to the possibility of the breakdown of indifference. How many nut hands does the BB need to hold to make the SB want to always fold to a single bet on the turn? This is easy to see by examining Fig64

Turn Play: Polar Versus Bluff-catchers Redux ure 10.2 and imagining what happens as we change the bet sizes or the fraction, N, of the BB’s turn starting range that is nuts. Notice that the total size of the BB’s turn betting range is proportional to N – in the case of two pot-sized bets, it is (3/2)(3/2)N=(9/4)N. If we increased the bet sizings, that fraction would increase, and the total fraction of the BB’s turn starting range that bets would expand, to the right in the figure. If we increase N itself, the bluffing ranges will also increase proportionally, and the total betting range will expand. But the BB only has so many hands in his range that he can use as bluffs. If we increase N or the sizings so much that the BB’s betting hands fill up the bar all the way to the right and can go no further, something has to give. The BB will not be able to bet enough bluffs to keep the SB indifferent to his call-or-fold decisions on both streets. If the BB bluffs “enough” on the turn, then he cannot do so on the river, and we have EVSB(fold turn)=EVSB(call−fold)>EVSB(call−call). If he bluffs enough on the river but not on the turn, we find EVSB(fold turn)>EVSB(call−fold)=EVSB(call−call). Of course, if he chooses not to bluff enough on either street, we have EVSB(fold turn)>EVSB(call−fold)>EVSB(call−call). Folding the turn is the SB’s best option in each of these cases, so that’s his unexploitable play when the BB’s range is too nut-heavy. For the case of two pot-sized bets, as illustrated, the largest N can be so that we still have the normal case is where the nuts and two complete bluffing regions just barely comprise his whole turn starting distribution, i.e., where (3/2)(3/2)N=1, or N= 4/9. An even smaller amount of nuts would be necessary if there were more money behind so that the BB could bet larger, but this is already fewer value hands than necessary to win the whole pot in the single-street case. Finally, it is worth noting that the static PvBC game can be extended to more than two streets, and the solutions are qualitatively similar. For example, if flop play starts with PvBC distributions and hand values are static, the solution involves our polar player betting each street with all


Expert Heads Up No-Limit Hold ’em, Volume 2 value hands and some bluffs and giving up with some bluffs on each street. His opponent folds some proportion of his bluff-catchers each time he faces a bet, and the proportions are analogous to the two-street case. The amount of nut holdings the polar player needs to always capture the pot scales geometrically as well. That said, the static hand values approximation tends to be less appropriate when there are more cards to come, and PvBC distributions are rare on early streets in HUNL.

10.2 Weird Plays and Refinements of the Nash Equilibrium Solution Concept Now, what happens after the turn checks through in the two-street static PvBC game? The BB’s range here is entirely composed of bluffs with which he is giving up, so at equilibrium, he will check, and the SB will win the pot. He should never lead out with a bet on the river. But – what if he does? A strategy is supposed to tell us what a player will do at each of his decision points in the game, so to fully define the SB’s strategy, we do need to say how he will play if he does happen to face a bet here. Also, a GTO strategy should certainly provide guidance for play in weird spots. We solved earlier for his calling frequency after the turn goes check-check, but that was under the assumption that BB’s river starting distribution was normal in the sense that it contained some nuts and some air, and too much air to bet with all of it. This turns out not to be the case here. It would be “weird” for the SB to face a bet on the river here, since the BB should never bet at equilibrium. Can weird situations like this happen in single-street games? So, what can we say about the SB’s equilibrium play when he faces such a river bet? The easiest way to think about this is to just go back to the definition: at equilibrium, neither player is able to unilaterally adjust his strategy to improve his expectation. What SB strategy will fulfill this criterion? Since the BB never bets in this spot at equilibrium, no choice of SB calling frequency has any effect whatsoever on the SB’s expectation. No choice of 66

Turn Play: Polar Versus Bluff-catchers Redux SB calling frequency will give the SB the opportunity to improve his expectation. However, some SB calling strategies will make it possible for the BB to change his strategy to increase his expectation, and the SB needs to choose his calling frequency to prevent this if we are to be at equilibrium. In particular, if the SB calls with any frequency smaller than that necessary to keep BB’s air indifferent to bluffing, then the BB can profitably bet his bluffs (despite the fact that he has no nuts for balance). Thus, at equilibrium, the SB definitely has to call at least P/(S+P) of the time. What if the SB calls slightly more than this? His EV does not change, and the BB is not motivated to deviate either. The BB’s bluffs now strictly prefer to give up rather than bluffing, which is fine since they were giving up already. The BB’s EV of checking his nuts on the turn will increase a little, but it will not be enough to make him start slow-playing the turn as long as the SB increases his calling frequency only a little. Of course, if the SB keeps increasing his calling frequency, then at some point the BB will begin to prefer to check the turn and bet the river with his nuts – obviously if the SB calls allin 100% of the time after the turn checks through, the BB’s nuts would definitely want to play that way to maximize their value. Thus, it turns out that there is actually a range of equilibrium SB calling frequencies here. He has to call often enough to keep the BB’s air from bluffing profitably, and he cannot call so much that BB’s nuts want to check the turn. However, anything in between is fine – any of those strategies achieve the same EV for both players at equilibrium and allow no opportunities for either player to improve his expectation by deviating. Now, we are still left with the unanswered question, “exactly how often should the SB call on the river after the turn checks through?” In practice, it is likely that our assumption of PvBC distributions or static hand values was wrong, and in the next chapter, we’ll see that even small violations of either of these assumptions can allow the BB to arrive at the river with some value hands. For now, however, it is instructive to continue within the static PvBC model. To begin thinking about how we should react when facing a bet, we have to decide why we might be facing a bet in the first place. It really should never 67

Expert Heads Up No-Limit Hold ’em, Volume 2 happen given our understanding of the game we are playing and assuming that the BB knows and is trying to play his GTO strategy, so solutions to this problem can get a bit philosophical. One approach here is to assume that BB must have simply made an unintentional error. Assuming he knows and is trying to play his equilibrium strategy, he must either have a value hand that wanted to bet the turn but failed, or an air hand that wanted to check the river but failed. Either way, he just made a random and inadvertent error. In internet poker games, this is known as a “misclick”. If we assume the BB has some chance of misclicking at each of his decision points, then he does get to the river with some small amount of nut hands, we find ourselves back in a simple case, and we can find a precise calling frequency for the SB. This is essentially one approach to what is called a refinement of the Nash equilibrium solution concept. The Nash equilibrium is not uniquely defined, and in fact, games often have many equilibrium strategy pairs. Thus, we can imagine adding extra criteria to the list of properties we want for our strategies so as to pick out the best of the best. Generally, we want one that performs reasonably in weird spots such as the one we saw here. It seems unlikely that there are multiple “interesting” equilibriums in HUNL. For example, a SB having slightly different calling frequencies in a spot we should never even get to is not especially interesting. Since HUNL is a two-player, zero-sum game, all its equilibria must have the same EV for both players, and of course, picking one over the other cannot affect our EV when Villain is also playing unexploitably. (This is not necessarily true for 3+ player games.) However, different equilibrium strategies may have different values when played against exploitable opponents. So, if we do find ourselves with some flexibility in a strategy, we might as well pick the one that seems like it will do well versus the common mistakes of human players rather than worrying much about any further technical requirements. For example, in the situation here after the turn checks through, the BB figures to get to the river with a ton of air in his range. Even if he misclicks the turn with nuts occasionally, he does not have to decide to bet his air very frequently at all before he is bluffing an exploitably high proportion of the time. Perhaps Villain is


Turn Play: Polar Versus Bluff-catchers Redux making a desperate bluff to win the pot and is hoping that we do not realize his range is so air-heavy, or perhaps he does not realize it himself. For this reason, I think it is appropriate to adjust my calling frequency upwards in spots like this and pick one that is on the high side of the reasonable range, even against an unknown or very skilled opponent. I try not to call so often that my opponent can take advantage by beginning to slowplay the turn, but otherwise I call quite frequently. It does not bother me much when an opponent makes a slow-play with a strong hand in spots like this and then bets it and I call on the river. He probably thinks his weird line got him paid, but I know that he would have done better on average if he had just made the standard aggressive play. At the beginning of river play after the turn checks through, if we assume the players are capable of executing their equilibrium strategies without misclicking, the BB always has air, so the SB’s bluff-catchers all become the nuts by comparison! If we give the SB the option to bet the river in this case, can he use it to improve his expectation at equilibrium? How would the players’ GTO strategies change once we added this strategic option to the game? How does this compare to the singlestreet PvBC game where the SB holds all nuts? Suppose the board is not perfectly static and a small amount of the BB’s weak hands improve from air to the nuts due to a favorable river card. How might this change the play on the river after the turn checks through? Could this make the SB want to bet the river at equilibrium? If so – just occasionally or very often?

10.3 Bet Sizing and Geometric Pot Growth Lastly, before we look at a realistic example, what is the BB’s turn bet size B at equilibrium? He gets to choose it, and he will try to maximize his EV.


Expert Heads Up No-Limit Hold ’em, Volume 2 Earlier, we found the BB’s EV when he bets the turn with his value hands – see Equation 10.1. To find the value of B that makes this as large as possible, we take the derivative of this EV with respect to B and set it equal to 0. Taking the positive root of the resulting equation gives us the optimal bet size

This result is a bit messy-looking, but it has a very simple and intuitivelypleasing interpretation. Betting with this sizing has been called, by Chen and Ankenman, betting the geometric growth of the pot (GGOP). The idea here is to bet some fraction of the pot on the turn such that the river shove will be the same fraction of the pot. For example, Figure 10.2 was illustrated for the case where B=P and S=4P. Here, a pot-sized bet on the turn perfectly sets up an exactly pot-sized all-in on the river. GGOP sizing may be significantly larger or smaller than the pot, depending on the pot and stack sizes on the turn. In either case, this bet sizing maximizes the polar player’s EV in static PvBC situations. As mentioned previously, the solution to the PvBC situation generalizes naturally to play over more than two streets. This approach to bet sizing does as well. For example, if we want to use GGOP sizings from the flop onward, we find the fraction of the pot such that we can bet it on three streets and just get all-in on the river. It also makes sense, trivially, in the single-street case. On the river, GGOP just means betting all-in – this is the way to get all-in over one street when our opponent does not raise. Of course, we have seen that all-in is indeed the polar player’s GTO sizing with his whole betting range in river PvBC situations. When should we use this sort of sizing in practice, and what are our alternatives? Essentially, there are two approaches to bet sizing:

Betting in relation to the size of the pot

Betting in relation to the size of the remaining stacks

The GGOP approach is to bet in relation to the remaining stacks. GGOP over multiple streets is analogous to the all-in river bet. It ignores the current pot size completely and simply ensures that, if Villain keeps calling,


Turn Play: Polar Versus Bluff-catchers Redux we can get all-in by the end. As you might guess, this is most appropriate when our distribution is relatively polar. In that case, we know whether or not we have the best hand, and we do best by applying as much pressure as possible (to put the guy to a decision for all his chips, as they say). GGOP sizing is the best way to do this over multiple streets, just as an all-in is the way to accomplish this on the river alone. Over-betting the pot has become more common on the river, but is still rarely used on earlier streets. This is at least partially because it is rare to be very polar on early streets. Players’ ranges become more and more defined over the course of a hand, so it is easier to find PvBC situations on later streets. As we’ll see, however, overbetting certainly has its place on the turn as well. Since a proper betting range is often at least somewhat polarized, GGOP often provides a good starting point for bet sizing. However, when the distributions are less black and white (i.e., we hold some thinner value hands, etc.) we may instead bet in relation to the size of the pot. The bigger we bet, the less often Villain needs to call to prevent us from bluffing, and the more selective he can be with his calling range. So, if a lot of our value hands are non-nutted, then the more we increase our bet sizing, the more often he has the best hand when he does call. Betting in relation to the size of the pot allows us to control the odds we offer our opponent and thus his calling frequency, to make sure our value hands get called by a sufficient number of worse holdings. It also allows us to give ourselves good odds (i.e., a good risk-reward ratio) with our bluffs. The stack depth and possibility of draws also affect sizing. For example, if there is just three-quarters of a pot-sized bet remaining in the stacks on the turn, the GGOP sizing is about 30% of the pot. This sizing works well if hand values are static – not only in an approximately-PvBC situation, but with other sorts of distributions as well. It turns out that the GGOP solution satisfies our risk-reward concerns, since the stacks are so short. If the board is more volatile, however, it can make sense to “charge draws” if many are possible, and in particular, we might just want to jam the turn here for three-quarters pot. Additionally, it turns out that both of these issues, complicated distributions and the possibility of draws, will motivate us to introduce the possibility of raises, especially in deeper-stacked situations, and raises can complicate the bet sizing calculus greatly. 71

Expert Heads Up No-Limit Hold ’em, Volume 2

10.4 Example: The SB Checks Back Bluff-catchers on K♣-7♥-3♦-K♦ Consider the following hand. We encountered it during our discussion of balance in Section 2.3.1, and we promised to come back to it. Hero is the BB, and we expect that he will be relatively polar at the beginning of the turn. Effective stacks: 30BB Pre-flop: SB raises to 2.5BB, BB calls Flop: K♣-7♥-3♦ (5BB) BB checks, SB checks Turn: K♦ (5BB)

The SB checked back on a K♣-7♥-3♦ flop. This is a textbook example of an absolutely weak board for most pre-flop ranges, so many SBs will c-bet all of their air, expecting to frequently win the pot uncontested. In fact, many players c-bet this flop almost always, and the fact that Villain checked indicates that his strategy could be unusual. It is significantly more likely than before that he does not know what a c-bet is or does not understand board textures in general or which textures are good to c-bet. If we have reason to believe that he is, in fact, a thinking player, then there are other possibilities. Perhaps he likes to check back with strong hands to trap on boards he expects us to frequently check-fold, or maybe he suspects we are particularly prone to bluff-raise here, but he does not have the right sort of hand or is not sure enough of his read to play back against a raise. This is not the focus of our example, but something interesting likely happened here, and ingame, we should definitely give some thought to figuring out what it says about Villain’s strategy, especially if we get to see a showdown. All that said, we will assume the SB is some sort of average mid-stakes regular player. In this case, if he has a checking-back range on this flop, it likely consists primarily of weak showdown value holdings. We will assume the SB’s turn starting range is exactly all the A-high and flopped 72

Turn Play: Polar Versus Bluff-catchers Redux pairs of threes from his pre-flop opening range, as well as 2-2 and 4-4 to 66. Given all we talked about in the previous paragraph, this is maybe an over-zealous pruning of an unknown player’s range, but it is approximately correct for many opponents, and it serves as a good example of a realistic polar-versus-bluff-catchers situation. So, the SB holds the following range at the beginning of turn play:

Some SBs will not open-raise pre-flop with all of these hands, but a few more or less 3-x combinations will not have a large effect on our results. What about the BB? With no knowledge of his flop leading tendencies, we will assume he starts the turn with his whole pre-flop calling range, say:

This is around the top 61% of hands, less some holdings he would have 3bet for value. In this case, the turn starting distributions are as shown in Figure 10.3.

Figure 10.3: The turn starting distributions for the K♣-7♥-3♦-K♦ hand indicate a nearly-PvBC situation.


Expert Heads Up No-Limit Hold ’em, Volume 2

10.4.1 The Model Game What is the equilibrium of the ideal turn PvBC model, given the stack sizes and ranges in this hand? How will it compare to the real computationallygenerated equilibrium? First, the pot is P=5 BB at the beginning of turn play, and S=27.5 BB remain behind. Plugging into the GGOP bet sizing formula above, we find that we need to bet about 1.23 times the pot on the turn and river in order to get all-in. That is, we lead the turn for B=6.16BB with all of our betting range, and this leaves S−B=21.34BB to bet into a pot of P+2B=17.32BB on the river. Now, any made hand a pair of sevens or better is ahead of all the SB’s range. About a quarter of the BB’s turn starting distribution falls into this category – let us call it exactly 25% for simplicity. The majority of the rest of his hands lose to A-high. So, as a first approximation, the BB is value-betting a quarter of his turn starting range on the turn and river, and the remainder of his range may be used as bluffs. Referencing the solutions above, we find that at the equilibrium, the BB is jamming enough on the river so that bluffs make up (S−B)/[2(S−B)+(P+2B)] =35.6% of his shoving range. (It would be 33.3% if he were making a pot-sized bet, but the bet here is a bit larger, so Hero can bluff a bit more frequently.) He bluffs enough additional air hands on the turn so that the fraction of turn betting hands that give up on the river is B/(2B+P) =35.6%. Again, the relationship of his single-barrel bluffs to his total river jamming range is the same as the relationship of his river bluff shoves to his river value shoves. So, overall, he bet-bets for value with 25% of his turn starting range, double barrel bluffs with 13.8% of it, and single barrels with 21.5% for a total turn betting frequency of a bit over 60%. His turn leading range contains more bluffs than value hands, and he gives up with a majority of them on the river. How about the SB? Facing a bet on the turn, he folds B/(B+P)=55.2% of the time and calls the other 44.8%. He folds a bit more than half the time since he is facing a somewhat larger than pot-sized bet. Then, if he faces a second bet on the river, he folds an additional (S−B)/(S+P+B)=55.2% of the hands with which he gets to the river. That is, overall, if he faces a bet-bet, he gets to showdown with 20.0% of the bluff-catchers he started with on 74

Turn Play: Polar Versus Bluff-catchers Redux the turn. Of course, he shows down more often overall, since Hero cannot always bet-bet, and he always wins the pot in those cases. What do the EVs look like here? The players begin the turn with 27.5-BB stacks and a 5 BB pot to contest. The BB had the best hand a quarter of the time. Thus, if they had just skipped the turn and river action and showed down their hands, the BB would have ended up on average with 27.5+[(1/4)×5]=28.75 BB and the SB would have expected 27.5+[(3/4)×5]=31.25 BB. However, thanks to the betting, the SB’s EV is only around 29.5 BB, and the BB’s is over 30.5 BB. The BB expects 27.5 BB with his bluffs, of course. This is the same as if they had just checked down – they win none of the pot. His nuts, however, average a bit over 39.5 BB at the equilibrium! They win significantly more than the whole pot at the beginning of the situation. The turn and river betting gained Hero’s nuts tons of value above their raw showdown equity, and in fact he expects to make a tidy profit in this spot despite holding the worst hand three quarters of the time. Here we see again a very important theme of NLHE play: the shape of a player’s distribution has a large effect on his ability to contest and win “his share” of the pot. If we only have a bit of equity, it is best that it be distributed such that we have some very strong and some very weak hands rather than a bunch of mediocre ones. A middling hand is often only a little better than air, while a 100% equity holding is much more valuable than a middling equity one. That is, the relationship between EV and equity is nonlinear. This helps to explain why it is often better to play hands such as 5♥-4♥ preflop rather than less “playable” ones such as K♠-3♦, which may actually have more raw equity versus an opponent’s range. The suited connector is a lot more likely to make us a particularly good or particularly bad hand post-flop, and thus has a higher EV than the other, which is more likely to make a bluff-catcher. We can apply this idea post-flop as well. If we face a bet on the turn, and there is money left behind, we usually much prefer to continue with a draw than a made hand if the two have about the same amount of equity. The draw will sometimes turn into air on the river, in which case it is only a little less valuable than a mediocre made hand.


Expert Heads Up No-Limit Hold ’em, Volume 2 However, the draw can also become very strong, at which point it will be very valuable. One lesson from this work that we can immediately take to the bank has to do with bet sizing. We have focused so far on the case where the BB leads the turn and river using GGOP sizings. These are over-bets (bets larger than the size of the pot). However, many players stick with a smaller sizing with most of their betting range on both streets and especially on the turn. How does the BB’s EV here compare to the case where he uses more standard sizes? In other words, what is these hands’ EV at the equilibrium of the static PvBC game if we constrain the BB to use smaller sizings whenever he bets? Suppose the BB makes 3/4-pot sized bets on the turn but still uses an over-bet all-in on the river. Plugging into Equation 10.1, we quickly find that the EV of the BB’s nuts here is about 39.3BB, down about 20 BB per 100 hands from the optimal choice. Try the case where the BB uses 3/4pot sized bets on both the turn and the river yourself. Suppose the players are playing the same idealized PvBC turn game except that the BB bets 3/4 of the pot on the turn and river with all of his betting range. How much does BB expect to end up with at the equilibrium? Hint: we cannot plug directly into Equation 10.1 since it assumes that we jam the river. However, this situation is equivalent to one where the effective stack is shorter and the two small bets do get us all-in, since none of the additional money behind can ever come into play anyway. In fact, since we are betting the same fraction of the pot on both streets, 3/4-pot is the GGOP sizing for that alternate stack size. Find it, and then apply Equation 10.1. 1

You should have found that the BB’s nuts expect approximately 37.7 BB in the case that he uses the more “standard” sizing on both streets. Although much better than checking down, this strategy loses his nuts over 180 BB/100 relative to the optimal sizing! Certainly there are cases where smaller turn and river leads are appropriate. However, over-bet leading the later streets from the BB versus the flop check back should certainly be a common adjustment to players who check back primarily mediocre made hands. When we are in the SB, we must be capable of balancing our flop checking range when facing a BB who is capable of making such an adjustment. 1


Turn Play: Polar Versus Bluff-catchers Redux

10.4.2 SBs Who Call and Re-evaluate Suppose we are in the BB on the turn in a nearly-PvBC spot. For now, we will bet-bet our value hands, although we’ll find some reasons to get more creative later. However, we face a decision when we hold a potential bluff. SBs who check back the flop with bluff-catchers commonly follow up with exploitable turn play as well. In particular, they do not fold enough to a lead. First, the flop was not very scary, and Villain checked back because he had “something” that he wanted to take to showdown. Then, the king on the turn seemed like the perfect card for any weak showdown-value hand. It could not have improved any hands that were not already ahead, and it even made them less likely because of card removal. (Keep in mind that a reasonable BB pre-flop defending range has a seven or better here over a quarter of the time. I found this number surprisingly high.) Since the SB is ahead of most of his opponent’s range, folding to a single bet seems weak-tight. Thus, many players adopt a plan like, “start out with a call and then re-evaluate river”. Call-and-re-evaluate is a common approach in many spots where players know their range is narrowly defined as bluffcatchers and also know that their opponent holds many potential bluffs. They call at least once without giving the decision much thought and intend to take a more balanced approach if they face additional action. How can we exploit this tendency? It figures to make betting our value hands even better if anything, but how should we play our air? From an equilibration exercise perspective, folding on the turn keeps the BB indifferent between giving up with bluffs and single barrelling. The more the SB folds, the more the BB is incentivized to take a stab with his bluffs. However, if the SB never folds to one bet, then Hero should never bluff once and then give up. We can save our B chips by simply giving up on the turn, although it could still be best to bluff twice. So, if our opponent tends to check back the flop with bluff-catchers and call a turn bet (at least on certain cards), then the most important thing is that we never bluff once and then give up – we should only bluff with the intention of following through on the river.


Expert Heads Up No-Limit Hold ’em, Volume 2 Now, we still need to decide between check-check and bet-bet. This choice depends on how often Villain calls a river bet. If he calls too frequently on both streets, check-check is clearly best. We just have to give up with our bluffs. However, suppose he calls the second bet with a more moderate frequency – suppose he calls just enough to make our bluffs indifferent on the river. What should we do then? The EV of check-check is just S, while our choice to bet-bet means always seeing a river and being made indifferent to betting there. On the river, our EV of checking is S−B, and our EV of continuing the bluff must be the same. Since we always get to that spot if we decide to bet the turn, that is the EV of bet-betting on the turn. Since S−B is much less than S, check-check is still clearly best. Notice what happened here. If Villain is calling too much on the turn and then “playing well” on the river, we still do best by far to just give up with bluffs on the turn. Even if he is folding a bit “too much” on the river so that we prefer bluffing with any air we do happen to bring to the river after betting the turn, it is still better to not get to the river that way in the first place. We prefer an EV of S to one of a bit more than S−B. This is an important symptom of multi-street play and shows that it is important to keep the big picture in mind when developing a strategy. If we are in the SB, we cannot decide to start with a call on the turn and then appeal to the singlestreet river game solutions for help when we face a decision on the river. The BB can exploit this by giving up with his bluffs on the turn, making all our river calls very unprofitable. For completeness, when can we actually bet-bet our bluffs in the BB? Essentially, Villain has to fold enough to make up for our having put B in the pot on the turn with no equity and no chance of winning immediately. We are indifferent between bet-betting and check-checking with a bluff if: (10.2)

So, we prefer to give up with bluffs on the turn unless Villain folds the river at least S/(S+B+P) of the time. This is about 71%, given the sizings in the example, quite a high frequency. By the way, this result should not be surprising. We know from single-street situations that the general form of a folding frequency necessary to keep complete air indifferent to bluffing is 78

Turn Play: Polar Versus Bluff-catchers Redux risk/(risk+reward), where risk is how much we lose when called, and reward is how much we stand to win. In the single-street case, we find a folding fraction of B/(B+P). Here, we essentially have to risk our whole stack (by way of a turn bet and river jam) to try to win the pot plus Villain’s turn bet, and so risk/(risk+reward)=S/(S+B+P). Here’s another way to think about this spot. Players often constrain their own play because of their stylistic preferences or ideas about what constitutes generally good play. This effectively removes options from the game tree. For example, many opponents constrain themselves to playing minraise-or-fold pre-flop from the SB 50-BB deep. Additionally, we described above why a SB might always call when facing a turn lead on the K♣-7♥3♦-K♦ board. Regardless of whether these are good strategies, it is important to study situations under these constraints so that we can effectively strategize in these contexts. We can model our two-street, static-PvBC situation where the bluff-catching SB refuses to fold the turn with the altered decision tree shown in Figure 10.1. This is effectively the same as Figure 10.1 except that the SB is forced to call when facing a turn bet.

Figure 10.4: Altered decision tree for the PvBC situation on the turn with static hand values in the case where the bluff-catching SB constrains his own play with a “call and re-evaluate” strategy.


Expert Heads Up No-Limit Hold ’em, Volume 2 What is the equilibrium of this modified game? Well, Hero bet-bets his nuts, and he bets only bluffs that he can continue with on the river. His betting range on the river makes the SB indifferent to bluff-catching. If he bluffed any less than this, the SB would always fold the river, which would incentivize him to bluff more, and vice versa. Thus, the BB’s strategy is essentially the same as in the single-street PvBC game, except that he now bets twice with all of his betting hands. How about the SB? He only has a river decision here, and his folding frequency at this point is chosen in order to keep Hero indifferent between his two-street pure strategies: bet-bet and check-check. This frequency is the one we just found (Equation 10.2) and is significantly larger than in the single-street river PvBC situation. If the SB folds any less, Hero will not bluff at all. In this two-street game, the SB’s river play must make Hero indifferent between the two-street lines give-up and bet-bet with bluffs. It is not enough for the SB to just make Hero indifferent to bluffing the river after he has already put in B on the turn. He needs to fold more to incentivize Hero to risk that first bet as well. In other words, if you consider just the river play alone, you notice that the SB is not folding enough to make the BB indifferent to bluffing. It appears that the BB can “automatically profit” by betting bluffs on the river, since he strictly prefers bluffing to giving up, given the SB’s high folding frequency. However, the SB is not actually exploitable in the context of this game, since the BB had to put in the turn bet to get to this river spot in the first place. Consider the altered PvBC turn game shown in Figure 10.4. The SB is forced to call a turn bet but can play more intelligently on the river. Suppose the players are playing the GTO strategies we have just discussed. Is this a bad situation for the SB? If we re-introduced his option to fold the turn, could he use it to increase the EV of his bluff-catchers? This spot was a bit contrived, since we forced the SB to always call the turn. However, we will see similar effects in more realistic situations. For example, “floating” is a move where a player calls a bet on an early street with little or no equity, simply because he believes he can take the pot away with a bluff later. Suppose we face a later street bet that we suspect might 80

Turn Play: Polar Versus Bluff-catchers Redux be a float. To discourage such a move, we do not have to call with so high a frequency as to make Villain indifferent to the later street bluff after he has already floated, but only enough to make him indifferent to his whole line, which involved risking an earlier bet also. In general, the frequencies suggested by the bluffing and bluff-catching indifferences in multi-street situations can be different than in the single-street case, and some care must be taken when trying to analyze the river in a vacuum. We can often get a better view of a strategic situation by taking a more holistic view. Now, this does not mean all our previous work on the river in a vacuum was wrong. Let’s think about why. The BB arrives at the river spot at the end of Figure 10.4 with the perfect range such that he can bet all of it and make the SB exactly indifferent to calling. This is a rather unique situation! If he arrived at the river with infinitesimally less air, then he could bet his whole range, and the SB would still strictly prefer to fold. This is a breakdown of indifference that we covered in Chapter 7. On the other hand, if he held infinitesimally more air, then betting his whole range would make the SB strictly prefer calling. Go through the equilibration exercise, and you can convince yourself that the equilibrium will indeed be the normal one from the single-street models we have seen previously. The special thing about the solution in our altered game that lets the SB call sometimes but less than in the single-street solution is that the BB gets to the river with exactly the right range to bet all of it and still make the SB indifferent. Essentially, the SB gets away with folding too much (from a single-street perspective), since the BB cannot easily bluff more, because he is already bluffing all the air he brought to the river, and he cannot easily arrive with more, since he has to call the turn bet to do it. The BB’s river starting range in this unique case lies right on the boundary between starting ranges corresponding to those that lead to breakdown of indifference and those that do not. It turns out that, if we solved this river spot by itself, the SB’s calling frequency would be completely unconstrained – any frequency between 0 and 100 percent would be co-optimal. It’s the additional constraint from the turn spot that fixes things. Thus, our previous river analysis is correct. It’s just that in special cases, play on the river is not completely determined by the river situation alone, and thinking about the larger context can provide more guidance and also inform how the river 81

Expert Heads Up No-Limit Hold ’em, Volume 2 starting distributions should look in the first place. Furthermore, this situation is unlikely to arise in reality, even if the BB tries hard to choose such a range on the turn. Hand values change some from one street to the next, and particular river cards will make him end up with slightly more or fewer value hands. On cards that complete draws or are particularly good for his range, he may have too many value hands, so that the SB can happily fold with bluff-catchers. On blanks, the BB may have an abundance of potentials bluffs, so that the single-street solution applies. It is even harder to deviate from the single-street solution when the distributions are not exactly PvBC. If the SB is folding too much for reasons such as we have seen here, then perhaps the BB will be incentivized to start bluffing with some weak made hands that, under normal circumstances, would try to show down and not contribute much to either his value or bluffing ranges. This additional source of BB bluffs would motivate the SB to begin calling more. Lastly, suppose Hero plays the role of the SB in this K♣-7♥-3♦-K♦ hand. In practice, opponents’ turn and river bluff-leading frequencies from the BB vary widely – take note of them since many will have very exploitable frequencies after we check back the flop. That said, many experienced players will identify the particular turn situation in this example as a poor one in which to make a bluff. They expect the SB to call at least once and often twice with significantly too high a frequency for all the reasons we mentioned previously. As a result, even players with good average turn bluffleading frequencies will often actually have too low a frequency on a board such as this one, while perhaps bluffing too often on others. Against such an opponent, Hero can make exploitative, tight folds with his bluffcatchers when he faces a turn lead. It is important to pay attention not only to opponents’ probing frequencies versus missed c-bets, but also to how these frequencies depend on the board.

10.4.3 Attacking Missed c-bets The previous section was a bit down on playing aggressively versus missed c-bets. However, we’ll see that it is, in fact, a very important part of good 82

Turn Play: Polar Versus Bluff-catchers Redux play in modern games, so some mention of the upsides of the move is in order. Although it never feels great to bluff-lead when we know Villain has something he probably kind of likes, we should remember that, if we’re confident that his flop checking back is bluff-catchers, we’re in the driver’s seat. Villain has essentially given up his positional advantage. With a polar range, we know where we’re at, and he’s the one who has to guess. Additionally, weak showdown value isn’t the only story we can tell about missed c-bets. Occasionally, we’ll run into an opponent who simply c-bets when he likes the flop and checks back when he does not. In this case, we can lead the turn and expect to pick up the pot a lot of the time. (And if we get called, we can expect him to have hit the turn card, which narrows down his range a lot, and we can make an intelligent decision about whether or not to continue a bluff on the river.) Of course, most people know what a c-bet is these days and do so with decent frequencies. More likely, we’ll face opponents who play this way on particular types of boards. For example, if the flop comes 9♠-7♥-6♠, Villain might not expect us to check-fold very often and thus just check back with all of his air, intending to fold to a turn lead. As usual, trying to figure out exactly how opponents are playing, overall and on particular board textures, is paramount.

10.5 Exploitative Play It is important to stay grounded by referring back to real play and to explore all of the situations to which the theory can be applied. Every time we say that Villain’s strategy makes us indifferent between two actions at the equilibrium (and we say it a lot, right?), we need to consider when opponents are likely to deviate in real play, how to notice those deviations at the table, and how to take advantage of them. We hit on some exploitative issues in the context of the hand example, but now we will think about it in more general terms. So far, we have framed our discussion of two-street PvBC play in terms of the situation where the SB checks back the flop with bluff-catchers. How83

Expert Heads Up No-Limit Hold ’em, Volume 2 ever, we mentioned earlier that the SB is often relatively polar after he cbets the flop and the BB check-calls. In fact, this is perhaps the more common situation. This leads to the turn and river situations where the SB can barrel and the BB mostly plays check-and-guess, a scenario that we discussed loosely in Section 6.3. The purpose of this section is two-fold. We will systematically cover exploitative adjustments, and we will develop a feel for the case where the SB is the bettor. The decision tree is still the same as in Figure 10.1 except for the position labels. The SB effectively acts first on each street, since the BB begins each street with a check. Equilibrium play is essentially the same as before. In the usual case where the SB holds relatively few nuts, he will bet-bet with all of them, and with his bluffs, he will sometimes give up, sometimes bet-check, and sometimes bet-bet in order to make the BB indifferent at both of his calling decisions. This leads to the same indifferences, and thus the same frequencies as before. We’ll continue to assume the SB bet-bets with all of his nuts. Although slowplaying could be best against some opponents, it is most reasonable when hand values change and when the BB’s flop check-call does not narrowly define his range, i.e., not in the case of static PvBC play. Thus, we will save discussion of those possibilities for later. In Section 10.5.2, we will see when the SB should give up, bet once, or bet twice with his air, and in Section 10.5.3, whether the BB should fold, call-fold, or call-call with his bluffcatchers.

10.5.1 Useful Single-street Statistics The frequencies with which players take these common lines have standard names. They are used, for example, in online players’ head-up display (HUD) software – applications that display real-time statistics about players’ strategies to aid in decision-making. Suppose the SB c-bets the flop and the BB check-calls and checks the turn. If the SB bets again, this is known as a turn c-bet, and the frequency with which he does so is his turn c-bet frequency. The frequency with which the BB folds facing this bet is his fold to turn c-bet statistic. If the SB c-bets the turn and the BB calls and checks the river, the frequency with which the SB puts in the third barrel is 84

Turn Play: Polar Versus Bluff-catchers Redux known as his river c-bet frequency, and the chance the BB folds is his fold to river c-bet frequency. These are all essentially single-street statistics, and they can be helpful, but we will see that they are not always ideal for decision-making purposes. What are these frequencies at equilibrium in the static, two-street PvBC game? We have already done the work to find these, so here we will just summarize. The BB’s unexploitable fold-to-turn and river c-bet frequencies are B/(P+B) and Br/(Pr+Br), where Br and Pr are the river bet and pot sizes, respectively. (In the case of a river all-in after a turn bet of B, we have Br=S−B and Pr=P+2B.) Notice that these two fold-to-c-bet frequencies are equal whenever the same fraction of the pot is bet on the turn and river, as in the case of the GGOP sizings. As for the SB, his unexploitable turn c-bet frequency is not completely determined by the solution to this game. The number of hands he bets depends on how many nut holdings he arrives with on the turn. It is A×B×N, where N is the nut fraction of the SB’s turn starting distribution, A=(P+2B)/(P+B), and B=(Pr+2Br)/(Pr+Br). For example, in the case of two potsized bets, we saw that A=B=3/2. If N is large, the SB may be able to bet a lot of the time at equilibrium, but if not, then his GTO turn c-bet frequency may be quite low. His river c-bet frequency, on the other hand, is fixed. He gives up with a variable amount of bluffs on the turn, but his river starting distribution contains a certain amount of air with which he gives up on the river and a certain amount with which he continues to bluff. This is illustrated in Figure 10.2. In particular, the SB got to the river with A×B×N of his turn starting range, and he continues betting with B×N of it, so his GTO river c-bet frequency is (BN)/(ABN) = 1/A. Notice that it is the SB’s turn bet sizing that determines his river betting frequency. His river sizing has no effect on his unexploitable river frequency! Why is this? Let bt and br be the SB’s turn and river bet sizes expressed as fractions of the pot at the beginning of the turn and the river, respectively. What are the formulas for the GTO single-street statistics in terms of these variables? 85

Expert Heads Up No-Limit Hold ’em, Volume 2 For reference, we have tabulated the GTO frequencies for a few turn and river bet sizes – see Figure 10.5. Keep these in mind as we discuss exploitative strategies and as you try to decide whether particular opponents take certain actions too frequently or too little. Of course, care must be taken when comparing these frequencies to the statistics we read off a HUD. They should be different for a couple of reasons. HUDs generally do not distinguish opponents’ play versus different bet sizes, nor are they usually aware of board textures or players’ equity distributions, and these frequencies were found after making some fairly strong assumptions about the texture and the distributions. That said, we will see that these can provide a pretty good starting point for strategizing in a wider variety of spots.

Figure 10.5: GTO statistics for the two-street static PvBC game for a few selected bet sizes. Sizes are given as fractions of the pot.

Finally, since position is unimportant in PvBC games, these numbers apply when the BB is polar as well. Some single-street statistics are commonly used to describe the strategies after the SB declines to c-bet the flop. The BB’s turn leading frequency is called his probe turn frequency (and sometimes also his bet versus missed c-bet frequency). The chance he continues betting on the river is his probe turn and barrel frequency. The chance that a player in the SB folds to the turn and river bets are his fold to turn probe and fold to turn probe and barrel statistics, respectively. Consider the SB flop-check-back and BB flop-check-call lines.

1. Which more often leads to PvBC situations on the turn and river in your games? 2. In which of these does the polar player tend to arrive at the turn with fewer value hands, proportionally? Thus, how should the players’ turn betting frequencies in these spots compare? 86

Turn Play: Polar Versus Bluff-catchers Redux 3. In the ideal case, GTO turn and river play does not depend on the players’ positions. Does it play out more or less the same in your games? How might the two complexities of real play – changing hand values and non-perfect-PvBC distributions – affect the two situations differently? 4. What other early-street lines might lead to similar turn starting distributions?

10.5.2 *Hero is the Polar SB Suppose Hero is in the SB. What is his best play versus an exploitable, bluff-catching opponent? In exploitative play, our bluffs and value bets are no longer necessarily related by balance considerations, so we can consider our two types of hands separately, and here we will focus on Hero’s air. Should he check-check (i.e., give up), bet-check, or bet-bet? (We have again labeled Hero’s pure strategies according to his play on the two streets.) Finding the best play can be a bit tricky due to the two-street nature of the situation. For example, perhaps Villain is not folding enough on the turn for us to immediately profit with a bluff there – we may still find it best to bet two streets if Villain folds enough on the end. The best way to figure out which of our three pure strategies is most profitable is to start at the bottom of the decision tree and work our way up. So, we begin with the river situation. Is checking or jamming best, once we arrive there with our air? This is the one-street calculation that we have pretty much done to death. Villain’s GTO calling frequency is given by the bluffing indifference, and if he calls too little, we should always bluff, and vice versa. Finding the unexploitable frequency is easy, and the rest is a matter of hand reading. So now, move up the tree and consider our turn decision, keeping in mind which action we will take if we get to the river. There are two cases, depending on our plan for river play if our turn bluff gets called. If our plan is to give up on the river, then our turn decision, bet or check, is actually a decision between the two-street lines check-check and bet-check. The EV of giving up on the turn is just S, and since we have to give up on the river if we bet and are called, we should only bluff the turn if we win enough on that very street to make it worth it. In other words, we basically have a one87

Expert Heads Up No-Limit Hold ’em, Volume 2 street decision that is equivalent to a bluffing decision on the river. Thus, if we plan to give up on the river, we should bluff the turn if and only if Villain calls too little (i.e., less than the unexploitable frequency) on the turn itself. Otherwise, we check-check. This is easy enough – so far, this looks like two single-street decisions in sequence. Now, what if bluffing is best on the river because Villain is too tight there? In this case, our turn decision is between check-check and bet-bet. If he calls too infrequently on the turn also, we can happily bet-bet and expect to make money on both bets, individually and combined. However, if he is too loose on the turn, the tradeoffs are a bit more complicated. Our decision has to involve Villain’s play over two streets. Let’s see which option has the higher EV. We have

where fT and fR are Villain’s fold-to-turn-c-bet and fold-to-river-c-bet statistics, respectively. Bluffing twice is better if and only if the second EV is larger than the first. Setting the two EVs equal and rearranging, we find that we have a break-even decision when

If he folds any more than that, bet-bet is best, and if he folds any less, we should just give up with bluffs on the turn. This formula essentially gives us a decision rule. It can tell us how to play on the turn in the case where Villain folds too much on the river. It’s not terrible – we could conceivably calculate it on the fly – but it is somewhat complicated. We’ll explore it further soon. As usual, this calculation has assumed the second bet is all-in. Incidentally, though, if there is any money left behind after the second bet, it can never come into play. Thus, we can apply all these results to a non-all-in betting situation by ignoring the extra money behind and setting S to the amount of remaining stacks that can actually come into play through two bets. Figure 10.6 summarizes our conclusions so far. There are four cases, depending on whether Villain’s strategy is tighter or looser than GTO play on each street individually. The intuition is that if we can bluff each street profitably in and of itself, we bet-bet. If we cannot bluff either individual 88

Turn Play: Polar Versus Bluff-catchers Redux street profitably, then we have to give up. If we can profitably bluff the turn but not the river, then we do that, and if we can profitably bluff the river in a vacuum but not the turn, we have to evaluate the tradeoffs to see whether our gain on the river makes up for the turn loss.

Figure 10.6: Summary of exploitative bluffing in two-street static PvBC game. In the last case, additional work is necessary to decide between bet-bet and check-check.

Now, we have a decision rule for the last case, but it may not be particularly enlightening in itself. Let us do a quick example to get a feel for the situation. The initial stack sizes in many hyper HU games online are 25BB, and if a min-raise pre-flop and a half-pot c-bet are both called in the first hand, we end up on the turn with a pot of P=8BB and stacks of S=21BB. In this case, the GGOP sizing is 75% of the pot on both the turn and the river, and within the static PvBC situation we have been discussing, the BB’s GTO fold-to-turnc-bet and fold-to-river-c-bet statistics are both 3/7 ≈43%.

Figure 10.7: Exploitative play with air hands in the two-street static PvBC game depends on Villain’s folding frequencies. The numbers correspond to the cases in Figure 10.6.


Expert Heads Up No-Limit Hold ’em, Volume 2 Thus, if Villain’s fT is exactly 3/7, we need his fR to also be at least 3/7 to allow us to bet-bet. As Villain becomes looser, his fT decreases, and he has to fold more and more on the river to make us want to bet-bet. This is illustrated in Figure 10.7, which shows the SB’s maximally exploitative play with his bluffs for all BB frequencies. The equilibrium, where both folding frequencies are 3/7, is located near the center of the graph. The regions labelled “1.check-check”, “2.bet-check”, and “3.bet-bet” correspond to the three simple cases from Figure 10.6 where it is easy to decide on a line by comparing fT and fR to their unexploitable values. In the upper-left region, Villain folds the turn too little and the river too much. The curve passing through the region down to the equilibrium is drawn to indicate where check-check and bet-bet have the same EV. If Villain’s frequencies lie below this curve, he is too loose on both streets. If they lie above it, bet-betting is better. Figure 10.7 is sufficient to choose our exploitative strategy, but to understand what is going on, it may be helpful to think in terms of a 3D plot such as Figure 10.8. Here, each of our three betting lines is associated with a 2D surface in 3D space (like a possibly-warped sheet of paper). The height of each surface is the EV of taking the corresponding line for particular values of Villain’s frequencies. Since we want to take the action with the highest EV, Figure 10.7 is essentially just the 3D plot when viewed from directly above. The dark lines in the first figure show where the surfaces in the second intersect. However, the 3D graph contains additional information about the EVs of each choice and the differences between them. Notice that EV(give up) does not depend on either of Villain’s frequencies, EV(bluff once) depends only on fT, and EV(bluff twice) depends on his turn and river play.


Turn Play: Polar Versus Bluff-catchers Redux

Figure 10.8: Illustration of how the EVs of giving up, bluffing once, and bluffing twice in the two-street static PvBC game depend on Villain’s turn and river folding frequencies.

Let’s return to Figure 10.6 for practical work. Notice that if Villain is folding at least S/(S+B+P)=60% of the time on the river, his turn frequency does not matter – we can bet-bet regardless. Graphically, the curve of indifference is much more horizontal than vertical. This is because it depends more strongly on Villain’s fR than his fT. So, the river frequency is generally more important for our turn decision, at least for the sizings used in this example. This should be intuitive, since a lot more money goes in on the river. Thus, if you are not sure how to weigh the tradeoffs when Villain is too loose on the turn but too tight on the river, focus on Villain’s river play to make your turn decision. In other words, err on the side of bet-betting. If the board is relatively static, we can often plan our play a few decisions in advance. This can help us to cut to the essentials of a situation and make accurate decisions. Instead of evaluating a turn bet-or-check decision with a fuzzy notion of what “bet” means, we know that it means betbet or bet-check. Of course, we may need to re-evaluate the situation because of the new card, to incorporate other information gained, or if Vil91

Expert Heads Up No-Limit Hold ’em, Volume 2 lain takes an unexpected action, but these will be exceptional cases. Thinking ahead can help us make better decisions on later streets as well as on the current one. For example, some players find that they make better bluff-catching decisions before they are actually faced with a bet. In a moment, we will turn to a similar analysis from the point of view of the bluff-catcher, but first, try the following exercise. Within the static PvBC assumptions, these exploitative considerations apply just as well when the BB is relatively polar. Think about your own play in the K♣-7♥-3♦-K♦ hand. What are your turn and river ranges in the SB there? Do you call too much or too little on each street by default? How could Villain best exploit you with his air?

10.5.3 *Hero is the Bluff-catching BB We are in the BB with bluff-catchers. We called a flop c-bet. We now face a second barrel on the turn and could potentially face another on the river. We have three pure strategies in the context of the PvBC game: fold, callfold, and call-call, and we want to choose the best line based on Villain’s turn and river c-betting frequencies, which we will call cT and cR, respectively, as well as the proportion N of his range which is the nuts at the beginning of the turn. If the SB is exploitable, which is best? It is not always immediately clear, and the situation here is a bit more subtle than in the previous section. Start at the bottom of the tree. If we get to the river and face a bet, we should call if the frequency of bluffs in his river betting range is higher than at equilibrium. In this case, we’ll say that he is bluffing too often. We have to be careful, though. This is not the same as saying his river c-betting frequency is higher than at equilibrium, nor does it mean that he too frequently bets any particular air hand that he brings to the river. To use those pieces of information, we would also need to know how much air he brings to the river in the first place. Our river decision depends on the composition of his betting range, but that’s not something we can read off a HUD, nor infer from any one of our convenient single-street statistics. 92

Turn Play: Polar Versus Bluff-catchers Redux We’ll return to this decision in the next section. Once we know the best play when we do make it to the river, we can back up and evaluate our turn choice. In contrast to the river situation, we can determine whether Villain’s betting range on the turn contains too many bluffs by looking at just his turn c-bet frequency, given that we also know N. If it does, we can always call the turn. Why? The key is to remember that it is his single-barrel bluffs – the ones that bet the turn and then give up if called – that make us indifferent between folding the turn and calling once and then folding the river. So, suppose Villain bluffs too much on the turn and then plays the very best he can on the river, given that we’re adjusting to him. That is, he plays equilibrium of the river subgame. But then, he has more bluffs that must give up on the river, so call-fold is certainly better for us than giving up on the turn. Of course, if he plays badly on the river, he can only increase our EV in the subgame, so calling the turn becomes even better for us, whether that is by way of call-fold or call-call. If he bluffs too little on the turn, we have a harder decision. Again, focus on his single-barrel bluffs which, at equilibrium, make us indifferent between folding the turn and calling once and then folding. First, suppose that he plays the equilibrium in river subgame after bluffing too little on the turn. That means we are indifferent on the river, so that our EV(call-fold) equals EV(call-call), but Villain is giving up with too few bluffs on the end, so EV(fold) is better than EV(call-fold) and thus better than EV(call-call) as well. So, if Villain is bluffing too little on the turn and then playing “well” on the river, we have to fold to a turn bet. On the other hand, suppose he begins decreasing his bluffing frequency on the river while keeping his amount of turn bluffs fixed (at some amount less than the equilibrium value). Then, more and more of his air hands must be single-barrel bluffs. This makes call-fold better and better, and at some point, it could become better than folding the turn. So, even if Villain does not bluff enough on the turn, if he also bluffs sufficiently little on the river, we may be able to call-fold. On the other hand, suppose he begins bluffing more than in the subgame equilibrium on the river (and still less than equilibrium on the turn). Then, he’ll have fewer and fewer bluffs that give up on the river, so folding the turn will definitely be preferable to callfold, but call-call will become better and better as Villain is jamming too 93

Expert Heads Up No-Limit Hold ’em, Volume 2 often with air. If his river bluffing frequency becomes too high, call-call could become better than folding the turn. In summary, if Villain bluffs too much on the turn, we can always call the turn, and then we have our usual single-street decision on the river. If he bluffs too little on the turn, we have to look ahead to his river play to make our decision. If he plays “well” there, we have to fold, but if he plays the river sufficiently poorly, it could incentivize us to call. If his river mistake is bluffing too much, we would prefer call-call, and if it is too little, then callfolding is better. Some players find this result counterintuitive. If Villain is bluffing too little on the turn and river, it is natural to think that his range is too strong for us to ever call and that we just have to give him credit and fold when he bets. However, this is not necessarily the case. When considering call-fold, we need to focus on how often he is bluffing the turn and then giving up on the river. So, if Villain is bluffing just slightly too little on the turn and much too little on the river, that means that he is checking down plenty of air on the river after bluffing the turn, and we can happily call-fold. If it is still unclear, it might be helpful to think about an extreme case. Suppose Villain is bluffing only slightly too litle on the turn, but then he is giving up with all his bluffs on the river. Then, calling the turn effectively gets us immediately to showdown – when Villain has air, the river always checks through, and when he has the nuts, check-folding the river is in effect the same for our bluff-catchers as if the river checked through. So, this is essentially a single-street situation. However, Villain is betting only slightly fewer bluffs than would be appropriate for a two-street situation, i.e., much less than is good on the river alone. Thus, we can happily call. Now, let us compare fold to call-call. We face this decision if Villain’s turn betting range has fewer bluffs than at equilibrium, but his river range has more. We have, when facing a turn bet:

Basically, we end up with S+P+B when Villain bluffs once, 2S+P when he bluffs twice, and we go broke when he has the nuts. 94

Turn Play: Polar Versus Bluff-catchers Redux

Figure 10.9: Play with bluff-catchers in the two-street static PvBC game depends on Villain’s bluffing frequencies. Exploitative regions are shown here for the case where S=21BB, P=8BB, B=6BB, and N=25%.

An example and picture will help to make this clear. Suppose we began 25BB deep, and the SB min-raised pre-flop and made a half-pot c-bet. Then, S=21BB, P=8BB, and the GGOP sizing is B=6BB. Say, for example, N=25%. We have plotted the BB’s exploitative play for all of his opponent’s strategies in Figure 10.9. First, notice the axes: turn and river c-bet frequencies. These are the single-street statistics that we actually observe (and which are easily accessible through a HUD). They are not the same variables we used to choose an exploitative strategy – turn c-bet frequency and the fraction of Villain’s river-betting range that is bluffs – but of course they are related. Second, notice that we have labelled a large region in the bottom-left of the graph as “impossible”. Since 25% of Villain’s turn starting range is nuts, and he always bet-bets with these, he cannot c-bet the turn less than 25% of the time. On the river, he also continues betting all of these hands, and if he gave up with some of his air on the turn, then his minimum river c-betting statistic is even greater than 25%. So, the delineated bottom-left portion of the graph just corresponds to frequencies that do not make sense, given that he bet-bets all of his nuts. 95

Expert Heads Up No-Limit Hold ’em, Volume 2 How should we understand this figure? Start in the middle at the equilibrium. This is the only spot on the graph where all three regions meet, i.e., where we are indifferent between all three lines with our bluff-catchers. At equilibrium in this case, the turn c-bet frequency is [(P+2B)/(P+B)]2×N≈51%. We know that if Villain c-bets the turn too often, we can always start with a call. Indeed, everything to the right of cT=0.51 begins with a call. If we move to the left from the equilibrium, we may have to just fold the turn, but if Villain’s river frequency is higher or lower than at equilibrium, it can make us want to call-call or call-fold, respectively. If we move directly up or directly down from the equilibrium, Villain’s turn c-betting frequency, and thus his river starting distribution, stays the same, but he is betting more or less often on the river. When he bets the river more often, we strictly prefer call-call, and when he bets less often we find ourselves in the call-fold region. However, as we move to the right on the graph, Villain starts getting to the river with more air in the first place, so that he has to c-bet the river less and less to make us want to fold. Thus the boundary where we are indifferent between call-call and call-fold moves down as we look to the right from equilibrium. Remember that it is Villain’s river c-bet statistic that controls our fold or call-fold decision on the turn. Thus, we can always call the turn if Villain’s cR is less than the unexploitable value, 70% in this case. Now imagine moving directly to the left from equilibrium along the line between fold and call-fold. Villain’s turn c-bet statistic is decreasing but his river c-bet is constant. So, Villain must be getting to the river with less air in his range proportionally, but since his river c-bet stays constant, he must be betting a lot less air as well, so calling the river is increasingly unprofitable. However, we stay indifferent between fold and call-fold. If we move below this line, Villain is giving up enough on the river that call-fold is better than folding the turn, but if we move above it, folding the turn becomes strictly better than calling just once, and call-call is still a bad choice, so we just fold the turn. If we move directly to the right from equilibrium, Villain c-bets the turn too much but keeps his river frequency the same, so call-call is clearly best, since Villain is betting too many bluffs on both streets. Remember that if


Turn Play: Polar Versus Bluff-catchers Redux Villain uses the equilibrium river c-betting frequency but arrives at the river with too much air in his distribution, then he ends up bluffing too much on the river.

10.6 *Designing Statistics for Effective Decision Making In the previous section, we saw how to use Villain’s turn and river cbetting statistics, cT and cR, to choose an exploitative line for our bluffcatchers. These statistics are known as decision variables. We use them to make our decision, either by plugging them in to an appropriate formula or looking them up on a best response figure. Hopefully, Figure 10.9 now makes sense, and it is clear how a particular combination of Villain’s turn and river c-betting statistics, cT and cR, lead us to a particular exploitative line in theory. However, we still do not have a great way to make decisions at the table. Frankly, it is not easy to decide whether we should call or fold on the turn or on the river from those two numbers. If we face a bet on the river, cR gives us some of the information we need for our call-or-fold decision, but we also have to take cT into account to know how much air Villain brought to the river in the first place. Facing a bet on the turn, we can immediately call if cT is too high, but if it is too low, we have to also consider cR to decide between fold and call-fold. Graphically, these difficulties are indicated by the complicated shapes of the decision regions in the figure. When two action regions are separated by a curved boundary, it means that we must take into account the values of both variables to determine on which side of the boundary our opponent’s strategy falls. Of course, these boundaries move around with different stack sizes, bet sizes, N, etc. If our decision-making process requires us to compute some complicated, non-linear function of multiple statistics, it becomes very difficult to use at the tables. I believe the issue here is even deeper than a matter of computational con97

Expert Heads Up No-Limit Hold ’em, Volume 2 venience. When playing poker, it is very natural to pay attention to certain aspects of Villain’s play, such as how often he double-barrels the turn and triple-barrels the river. Once we start counting how often Villain takes some action, we can decide whether he is doing it too much or too little, and then make one adjustment in one case and another in the other. However, if we try to make a bluff-catching decision in one of these static, two-street PvBC situations according to whether Villain’s turn and river c-bet statistics are too high or low, we are certain to make incorrect adjustments, since that is simply not enough information to make the correct choice. Even if we realize we should take both of his tendencies into account to make a decision, we are unlikely to learn the complicated relationships involved through a trial-and-error process, especially given all the other complications of real play. Finally, even if we do the correct analysis away from the tables, weighing the tradeoffs at the table can use up a lot of brain power that could be put to better use. Thus, even trying to build intuition in terms of those variables about how to adjust to opponents’ play is unlikely to be successful. Thinking in terms of statistics that do not have simple relationships to the actions we want to take will often lead to our making poor decisions. This is something that will happen unconsciously if we do not take explicit measures to correctly develop our exploitative thought processes. There is a real danger in learning to strategize in terms of decision variables that are chosen ad hoc, and there is real value in coming up with something better. So, we need to find decision variables that give us more or less the same information as cT and cR, but which have a simpler relationship to the exploitative response. Ideally, whenever we need to make a decision, we could do so by considering a single statistic describing Villain’s play. In other words, we could look at one number to make our turn bluff-catching decision and another on the river. Graphically, this would mean that the boundaries between action regions were all straight horizontal or vertical lines. For example, in Figure 10.7, straight lines form all the boundaries between decision regions except for the one between bet-bet and checkcheck. In these cases, we can see which side of the boundary we are on by comparing a single statistic to a particular threshold value.


Turn Play: Polar Versus Bluff-catchers Redux Great. We would really like to be able to make each of our decisions exploitatively by considering a single decision variable. Describing Villain’s strategy using a set of decision variables that are decoupled in this way simplifies our decision-making process. Each one of these parameters tells us how Villain is exploitable at one particular point in the game. Just by organizing our thought processes around them, we are likely to come up with more accurate exploitative strategies automatically, since the very terms in which we think about Villain’s play capture just what we need to know to make good decisions. When we get in the habit of thinking in terms of simplified decision variables, we free up brain power to worry about other things. We increase our ability to make mathematically correct plays as well as to build intuition about how to adjust to our opponents. We want these statistics to be quantities that can be estimated from observing Villain’s play, just like cT and cR, so that they can be determined and potentially programmed into a HUD. How might we find them? To choose an exploitative action, we just need to find the choice with the highest EV. Think first about our call-or-fold decision on the river. The river c-bet frequency by itself is insufficient, since the frequency that makes us indifferent between our two options depends on Villain’s river starting distribution, i.e., on his turn c-bet frequency, as well. His turn c-bet frequency alone does not give us the required information either. Here’s one approach that can sometimes help us find one statistic to decide between two actions. Suppose we have two choices, A and B. We start by writing down the EV of each choice in terms of Villain’s strategy. Then, we set EV(A) greater than EV(B). This gives us the condition under which we should take action A rather than B. Then, we do our best to rearrange the inequality so that the numbers that describe the game itself (S, P, etc.) are on the right side of the equation, and the parameters describing Villain’s play are on the left. If all goes well, we find a statistic describing Villain’s play on the left, and on the right, the value of that statistic that makes us indifferent to our choice in terms of the parameters of the game itself. If Villain’s statistic is greater than this value, our exploitative play is A, and otherwise it is B. Let us give it a try for our bluff-catching decision on the river.


Expert Heads Up No-Limit Hold ’em, Volume 2 1. Step 1. Write down the EV of each choice in terms of Villain’s strategy. We should express EV(call−fold) and EV(call−call) in terms of cR and cT so that our result will also be in terms of those numbers. This is important, since it is relatively easy to get those frequencies from observing Villain’s play, as opposed to, say, the fraction of his turn betting range which is air that bluffs twice. So, immediately after we face a bet on the turn, we have

Here, cRcT is the fraction of Villain’s turn starting range that bets the turn and the river, so N/cRcT is the fraction of his river betting range that is value. 2. Step 2. Set one EV greater than the other. The first parts of both EVs are the same, so EV(call−call)>EV(call−fold), and we should call rather than fold to a river bet if and only if

3. Step 3. Move the parameters describing Villain’s strategy to one side and those describing the game itself to the other. We have

Done. On the left is a property of Villain’s strategy that we can estimate by 100

Turn Play: Polar Versus Bluff-catchers Redux observing his play, and on the right is the value of that statistic that makes us indifferent with a bluff-catcher when facing a river bet in terms of some parameters of the game. If Villain’s statistic is any larger than that optimal value, we prefer call-call, and otherwise, we prefer call-fold. Any boundary between the call-call and call-fold decision regions in a plot will be a horizontal or vertical line if we use that statistic as one of the axes. What decision variable did we actually find here? The product of cR and cT is the fraction of Villain’s turn starting range that bets on the river. Combined with the frequency of value hands in his turn starting range, N, this statistic gives us all the information we need to determine the frequency of bluffs in his river betting range. This is, of course, just what we need to know to make our river decision. We can think of this as a two-street statistic, in contrast to one-street frequencies we commonly use. Now, move up the tree. What single number decides our turn call-or-fold choice? Of course, we might need to consider different statistics depending on our plan for river play. In other words, if we will fold to a bet on the river once we get there, then we need to compare fold to call-fold on the turn, and perhaps the best way to do this is different than the best way to compare fold to call-call. Find the statistic necessary to make an exploitative fold or call-fold decision on the turn in the static two-street PvBC game. The normal river c-bet frequency tells us how much of Villain’s turn betting range gives up on the river and is the crucial frequency for a BB with a fold-or-call-once decision on the turn. Now, two statistics are all we need to fully describe Villain’s strategy in this game, and we have two, so let us go ahead and visualize how our best response depends on Villain’s cR and cTcR. This is shown in Figure 10.10. Notice, in the figure, that the fold and call-fold regions are cleanly divided by the cR statistic at cR=0.7. Similarly, the river decision, call-call versus callfold, depends only on whether cTcR is greater or less than about 0.36. Thus, these exploitative decisions are much easier to make than before when we thought in terms of cR and cT. The regions that indicate impossible values 101

Expert Heads Up No-Limit Hold ’em, Volume 2 of the statistics are also easier to understand. First, the fraction of Villain’s turn starting range that bets the river, cRcT, clearly cannot be less than the fraction of his river starting range that bets the river (and they are equal only when Villain bets all of his range on the turn), so the bottom right half of the plot is impossible. And secondly, cRcT cannot be less than N, since he always bets twice with his nuts, so the rectangle on the left where cRcTS and fold otherwise. This solution is not unique. 2


Turn Play: Polar Versus Bluff-catchers Redux call, he needs to consider cTcR, and the threshold value in this case depends on N, the fraction of value hands in Villain’s turn starting distribution. Of course, N depends greatly on the particular board and how it interacts with the SB’s ranges, as well as the SB’s strategy on earlier streets. In practical terms, once we fix the stack-to-pot ratio, all of these situations are the same from the point of view of a polar SB. Of course, subtleties always arise in practice when new cards come, but insofar as we are playing a static PvBC situation and the players know it, the particular board and early street play are unimportant for an SB who is trying to play exploitatively. The BB’s thought processes here, however, must take into account more details of the situation. He needs to estimate N before he can even start to formulate an exploitative strategy for his bluff-catchers. To do that, he must estimate the SB’s range (i.e., what hands the SB brings to the turn) and think about how it connects with the board. Many different combinations of these factors are possible, and the BB must be able to differentiate between them to play well. This is certainly a challenge – it is almost unfair that he must differentiate between many different turn spots while his opponent does not. However, it is also an opportunity to find an edge. To really nail down the BB’s exploitative play we need to give some thought to finding N, the number of value betting hands the SB brings to the turn. In addition to the board and the early-street play, the details of the current board and its texture are non-negligible. Of course, the assumption of staticity can only take us so far – it is almost never entirely correct, and the presence of draws certainly affects turn play even in relatively static spots. In the next chapter, we will address the question of how the players should split their ranges in some more realistic turn spots.

10.7 You Should Now … ♠

Understand the equilibrium of the static, two-street PvBC game, as well as how to play exploitatively from both positions.

Know which turn situations it effectively models. 105

Expert Heads Up No-Limit Hold ’em, Volume 2

Understand the assumptions that led to GGOP bet sizing and when to use it.

Know how to exploit SBs who check back bluff-catchers and then play call-and-re-evaluate facing turn leads.

Be able to design statistics to efficiently choose the better of two actions.


Chapter 11 Nearly Static, Nearly PvBC Turn Play

The beautiful thing about poker is that everybody thinks they can play. – Chris Moneymaker In the previous chapter, we saw several ways the flop action could cause one player’s distribution to become polarized relative to his opponent’s. If the turn card does not interact strongly with either of the players’ ranges nor bring draws, we can find ourselves in a spot that is fairly well described by the static PvBC game. However, this model is rarely perfect. Most boards have some straight draw possibilities. If nothing else, high cards have pair draws, and pairs can improve to two pair or three of a kind. Additionally, the players’ ranges do likely overlap somewhat. For example, after a flop check-call, both are likely to still hold many of their flopped second-pair hands. Loosely speaking, we’ll refer to these realistic situations as nearly static and these distributions as nearly PvBC. In this chapter, we’ll investigate the effect of weak draws and mildly overlapping ranges on turn play. Because hands are not strictly divided into nuts, bluff-catchers, and air, players must decide which hands they should use for value, for bluffs, and to call down. The presence of draws affects these choices as well. We will assume, however, that one player’s range is still particularly bluff-catcher-heavy. This player will not do any of the bet107

Expert Heads Up No-Limit Hold ’em, Volume 2 ting, since he will prefer to keep his few strong hands mixed in with his bluff-catchers. Since we know only one player bets, we can keep using the decision tree from the ideal PvBC game. We will focus on several examples. In addition to providing a realistic portrayal of the turn and river strategies themselves, this will help us understand the initial conditions, e.g., exactly what sort of turn starting distributions we are dealing with after particular flop play. Among other things, this will help us identify the polar player’s nut fraction N, which is necessary to develop an exploitative bluff-catching strategy. Of course, the polar player needs to be aware of his own distribution to implement his unexploitable strategy as well.

11.1 Range Splitting in the Presence of Draws and Mediocre Made Hands

Figure 11.1: Equity distributions at the beginning of the flop and turn when the SB c-bets polar and the BB check-calls (c-c) with bluff-catchers.


Nearly Static: Nearly PvBC Turn Play Suppose the SB opens the top 80% of hands pre-flop, and the BB flat-calls with

This BB range is about the top 60% of hands, less those that are likely to 3bet for value pre-flop. Both ranges are reasonable at short to medium stack depths if the SB is min-raising, and in particular, we’ll assume 25-BB stacks. We get one of several static flops: K♥-Q♠-3♦, K♥-8♠-3♦, A♥-8♠-3♦, or 9♥-9♠-3♦. The SB c-bets a fairly polarized range and is check-called, and the 2♣ comes on the turn. The players’ turn starting distributions after the flop goes bet/call are shown in Figure 11.1. The flop starting distributions are also given for reference. With a few exceptions, we assumed the BB check-calls with his second and third pairs and some high-cards. As for the SB, we have chosen value and bluff portions from his flop starting distribution. Approximately second pair and better are value, and all his no-pair hands without much showdown value are potential bluffs. The precise categorizations for each board are shown in the figure. We’ll assume he c-bets all these value and bluff hands, while checking back with everything in between. With these categorizations, about 30% of his flop starting ranges are value, and his overall flop c-betting frequencies are about 68% on the first board and about 74% on the others. This is in line with the tendencies of many players. With what frequencies does the BB call c-bets given the ranges in Figure 11.1? Are these realistic? Of course, the turn distributions depend on the SB’s pre-flop and flop play. In particular, the SB’s flop bluffing tendencies have a clear effect on the frequency of value hands in his turn starting range, N. If the board is completely static, one more bluff on the flop is necessarily an additional air combination on the turn. The fraction of weak hands can be lower in the turn starting distributions of a player who does not c-bet all of his bluffs, and it can be higher if he has tighter value-betting criteria or slow-plays some good hands on the flop. The effect of his pre-flop raising range on N is a bit more subtle. There is no 109

Expert Heads Up No-Limit Hold ’em, Volume 2 avoiding the fact that the flop changes hand values drastically (recall the discussion surrounding Figure 5.5). We have assumed an 80% SB opening range here. If we give him a 100% range instead, the overall effect is usually to give him more potential bluffs post-flop, but the exact number depends on the details of the particular board. When pre-flop ranges are wide, Villain’s c-betting strategy is usually more important than his preflop opening range when estimating the number of value hands in his range on the turn, since the board cards have such a randomizing effect on relative hand strengths. Compare 60% and 100% pre-flop opening ranges to the 80% case we are working with here. In which particular hand combinations do these ranges differ? For the boards listed in Figure 11.1, do these extra combinations contribute most to the amount of value hands, air hands, or hands in between in the SB’s flop starting range? Does your answer depend strongly on the particular flop? Now, let’s look a bit more closely at the turn starting distributions themselves. Only one card changes between the first and second and between the second and third boards in Figure 11.1. What effect does each change have? The differences are subtle if you focus on the overall shape of the graphs, but the important things to notice about the SB’s distribution are the size of each portion of his range and how much equity each one has. On the K-high boards, most of the SB’s value hands (his pairs of kings) have notably less equity than his top-pair hands on the A-high board. Of course, this is because of the possibility of overcards coming to improve BB’s bluffcatchers to top pair. Take a look at a couple of other effects yourself. The SB’s weak hands have less equity when the second card is an eight than when it is a queen. Why is this? The turn card significantly improved part of the BB’s range on only one of the boards – which board and what part of his range? Of course, the overall shapes of the distributions are also very important. If our reasoning so far has been correct, we should see typical PvBC distributions, and it should be easy to read off N. However, in addition to the fact 110

Nearly Static: Nearly PvBC Turn Play that the SB’s value hands are not the pure nuts and his weak hands are not complete air, he also holds some middling hands. This is partially because of the presence of a few draws, but is largely because both players can hold second-pair hands. Second pair is the top of the BB’s range given our assumptions, so the SB’s best second-pair hands are still almost always ahead, but his worst second pair hands are much more mediocre and are probably not even favorites to win at showdown if they bet again on the turn and get called. In more detail, starting from the top left of each of the SB distributions, we see the strongest value hands quickly followed by a plateau that represents top pairs. Then, an extended downward-sloping portion represents the SB’s second pairs. These hands’ equities depend strongly on the kicker, since they overlap with the BB’s range. Finally, the second half or so of each distribution is composed of low-equity hands. These need to improve to win. The paired board follows a slightly different pattern. The SB’s nines here have somewhat more equity than the top pairs on other boards, but there are fewer of them. Various high-card hands and threes essentially play the role of the second pairs on the other boards. Finally, his air on the paired board has a bit more equity than in the other spots, since the BB’s check-calling range contains more unpaired high cards. In any case, these turn spots are not truly static PvBC situations. Some of the SB’s hands that we labelled value-bets on the flop will continue to bet for value on the turn, but others are not strong enough, and it is not immediately obvious which are which. It is not clear from the distributions what portion, N, of the SB’s turn starting range consists of hands he should bet for value. However, with a bit of logic, we can make a pretty good approximation of the SB’s turn value-betting range. Recall the half street river SB bet-or-check game from Section 7.3.2. Its decision tree is shown on the left in Figure 11.2. With the benefit of our experience, we can quickly see how the SB chooses his value-betting range. It is a two-step process. First, the BB’s calling range is determined by applying the bluffing indifference (see Section 9.4.1) to the SB’s strongest bluffing hand. If the SB’s weak hands are not all equivalent (i.e., not all pure air), we can make an educated guess as to the SB’s strongest bluff. Once we


Expert Heads Up No-Limit Hold ’em, Volume 2 have the BB’s calling range, the SB can value-bet anything that gets called by worse more often than by better. In other words, the SB’s value-betting range is composed of all hands that have at least 50% equity when called.

Figure 11.2: The SB bet-or-check decision tree is on the left, and the twostreet PvBC game is on the right. If he uses the same bet size, then the SB’s value-betting range at his first decision point will be very similar in the two cases.

We can use a similar approach to estimate the SB’s strategy here on the turn. In fact, it turns out that the SB’s value-betting range at his first decision point in the large decision tree in Figure 11.2 is very similar to his value-betting range at his first (and only) decision point in the smaller tree if the initial equity distributions are the same. It is easy to see why. First, in the two-street case, the SB’s turn decision with a borderline value-betting hand will be between check and bet-check – it is hard to imagine a hand that is indifferent between checking down and betting twice for value and strictly prefers both of those options to value-betting once. Now, the SB will make his bet-or-check decision in both cases by comparing the EVs of his two options, checking and betting. So, his value-betting range is the same in these two spots if EV(check) and EV(bet) in the small game are the


Nearly Static: Nearly PvBC Turn Play same as EV(check-check) and EV(bet-check) in the larger one, respectively, for each of the SB’s hands. Why are the EVs of checking down the same in both games? Given the decision trees in Figure 11.2, the BB will never lead the turn or river with a bet. So, the hand always goes to showdown in both cases, and the SB realizes his equity in the pot, no more and no less. Why are the EVs of betting once the same in both games? These EVs depend only on the BB’s calling ranges, and the calling ranges in both cases are determined by the need to keep some of the SB’s weak hands indifferent between giving up and bluffing once. What issues affect this SB indifference? Giving up is the same in both spots – weak hands will just capture their (little) equity – as long as the bluffing hand we’re talking about is the same. Bluffing once turns out the same in both spots if Villain folds, but it can be slightly different if Villain calls, because in the two-street case, bluffs can improve to win on the river. Thus, two issues affect the tradeoffs that determine the SB’s bluffing threshold, and they have opposite effects on the BB’s calling frequency. First, the strongest bluffing hand can be different by virtue of the two-street nature of the game. We have seen that the SB will bluff more often in the twostreet case, and so he will generally have to turn hands with more and more showdown value into bluffs in order to do so. Thus, the equity he would capture were he to check down is a bit greater, and we might expect the BB to have to fold a bit more to enforce the bluffing indifference. Second, the SB’s bluffs can also capture some of the pot after bluffing in the two-street case as well, since they may be able to improve on the river. This effect will make the BB call more to maintain the indifference. Let’s summarize this procedure. We will: 1. Identify the SB’s borderline bluffing hand. 2. Estimate the BB’s calling range needed to keep the SB’s bluff indifferent using the bluffing indifference. 3. Find the range the SB can value-bet against it. To begin thinking about the SB’s bluffs, we will need to talk about draws. In the river SB bet-or-check game, the bluffing cut-off was the SB’s highestequity bluff. Here, the SB’s highest equity air-type hands are draws such as 113

Expert Heads Up No-Limit Hold ’em, Volume 2 5-4s (remember that 3♦ and 2♣ are on board). Does the SB bet the turn with these, or does he prefer to guarantee that they see a river? If he does bluff with draws, does the BB make these indifferent or does his bluffcatching strategy focus on the SB’s “normal” bluffing hands? Once we identify these, we will find the BB calling range needed to make them indifferent. For this purpose, we will need to find how much of the pot the SB’s bluffs can capture after checking and after bluffing. We’ll have to delve into the details of the ranges, the board textures, and some card removal effects to evaluate those issues. Once we have the BB’s calling range, we can find the set of hands the SB can value-bet against it. There are a couple of subtleties in this step, but we expect the result to be something like the case on the river, where the SB should value-bet anything better than at least half of the BB’s calling range. However, we’ll find that his opponent’s draws can make him want to value-bet even wider. We’ll continue with the sizes S=21BB, P=8BB, and B=6BB. For comparison with what follows, if the SB’s strongest bluff were entirely valueless, we would have the following bluffing indifference:

where FT is the BB’s turn folding frequency. Solving, we find FT≈43% in the naive case.

11.2 The Bluffing Range

11.2.1 The Best Bluffs: Draws or Weak Made Hands? Suppose we are on the river, and we want to choose bluffing hands from a distribution that is not strictly polar. That is, we want to find weak hands for which EV(bet)>EV(check). In Chapter 7, we saw that EV(bluff) is the same for all sufficiently-weak hands (neglecting card removal), since they only get called by better and always lose when called. However, 114

Nearly Static: Nearly PvBC Turn Play EV(checking) is less for weaker hands. So, if bluffing one hand is better than showing it down, then bluffing any worse hand must be better than showing down as well. Thus (with some exceptions due to card removal), hands with less showdown equity were strictly better to use as bluffs than hands with more. This line of reasoning no longer holds when draws are possible, i.e., on earlier streets. It may no longer be best to just bluff with our weakest holdings. Unlike weak hands on the river, draws actually maintain much of their equity after they bluff and get called. The EV of betting a good draw will thus usually be higher than that of bluffing a worse one. We will see that this can cause a bettor to give up with his weakest hands while choosing his bluffs from higher-equity parts of his distribution at equilibrium. Our questions in this section are as follows. Which hands should we use to bluff in a nearly static, nearly PvBC turn spot? Is it better to (semi-)bluff with draws or weaker hands? If we don’t just use our lowest-equity holdings, how should we imagine the process of filling out a bluffing range? Finally, which bluffs are indifferent to checking? This last question is particularly important, as we will need to answer it in order to write down the bluffing indifference. We’ll continue to assume our bets do not get raised. We’ll need to relax this assumption later, so try to imagine how our reasoning might change in that case. Now, we need to think about the difference between draws and made hands. A made hand is one that is unlikely to improve in absolute strength. The term “made hand“ does not necessarily imply much value – we could have a made 7-high with no outs. When we speak of bluffing with a weak made hand, we simply mean a weak hand that is unlikely to improve. This is in contrast to draws that are also weak hands (in a nearly static turn spot) but have a significant chance of improving. If a mediocre made hand has the same equity as a draw, it is more or less indistinguishable on an equity-distribution plot. However, there are major, strategically important differences between them (see our discussion of Figure 5.6). First, on future streets, draws tend to turn into hands with either very high equity or very low equity, while mediocre made hands tend to stay mediocre. Second, on the current street, draws tend to have approximately equal equity


Expert Heads Up No-Limit Hold ’em, Volume 2 versus most of the hands in an opponent’s range, while made hands usually have high equity versus some and very low equity versus others. This is why draws can often profitably semi-bluff, while betting is usually counterproductive for mediocre made hands that tend to just get called by better and fold out worse. If we recall why the SB needs bluffs in the first place, it will help us see how he should choose them. If the SB does not bluff enough to make many of the BB’s mediocre hands at least indifferent to calling, then the BB will fold a lot. This will incentivize the SB to begin bluffing a lot, so that the BB begins bluff-catching a lot, so that the SB stops bluffing. Now, imagine the BB slowly decreasing his bluff-catching frequency. For a while, the SB will still prefer to give up with all his bluffs, but as the BB continues to bluff-catch less and less, some of the SB’s weak hands will eventually find themselves able to profitably bet. As we continue to reduce the BB’s calling frequency, the SB will find profitable bluffs with more and more holdings, until the point where the SB is bluffing so much that the BB no longer wants to decrease his calling frequency. At this point, we might say that that SB’s bluffing quota is full – his bluffing range is filled up, starting with the best hands to bluff. We need to explore the EV of bluffing versus checking with mediocre made hands and draws in order to see how much each type of hand “wants to” bluff. Let us get right to it. To figure out which hands the SB actually uses to bluff, we need to determine which hands are first able to bluff profitably. In other words, for which does EV(bluff) first become greater than EV(check) as the BB begins to call with more and more bluff-catchers? To compare bluffing with draws to bluffing with made hands, let us assume that the SB’s draws keep all their equity after a bet (i.e., they have the same chance of winning the hand versus the BB’s turn starting range as versus his turn bet-calling range). We will also suppose the SB’s weak made hands have a bit of equity at the beginning of turn play, but if they bluff and get called, they are always facing a better hand and never improve to win at showdown. In reality, draws tend to have a bit less equity against the stronger range that calls a bet (although the opposite can also be true!), and weak made hands may retain some small chance of improving to win on the river, but these assumptions allow us to focus on the essentials of the situation and see 116

Nearly Static: Nearly PvBC Turn Play the effect of draws’ ability to retain their equity after bluffing. What are the EVs of betting and checking with each type of hand? If we assume the river always checks through, it simplifies the math, and this will still be sufficient to see the effect of draws. In this case, we have for both the draws and made hands, where EQh is the equity of a hand h versus the BB’s turn starting range, since we simply capture our equity in the pot when we check down. Then, if FT is the BB’s folding frequency when he faces a bet, we have for weak made hands: Weak made hands always end up with at least (S−B), and they get their bet back plus the pot whenever the BB folds. For draws, we have (11.1)

This is the same as the EV of bluffing weak made hands, except that draws still win the larger pot of P+2B with frequency EQh when their bluff gets called. Now, suppose that the BB is calling in order to make some made hand indifferent to bluffing. Say that made hand has equity E against the BB’s turn starting range. Then, the bluff-catching indifference tells us that the BB is folding with frequency FT= (B+PE)/(P+B). (Verify this!) Plugging this value of FT into Equation 11.1, we find:

The expression in brackets takes the form of an expected value of a quantity that takes a value of one (1) E of the time and (EQh)(P+2B/P+B) the other (1−E) of the time. In both cases, it is greater than EQh, and so the total quantity in brackets is as well. Thus, the EV of betting the draw is necessarily greater than the EV of checking it, S+P×EQh. Great. We now know that whenever the SB can bluff with a hand that loses its equity when called, he will also bluff with any hand that does not. In


Expert Heads Up No-Limit Hold ’em, Volume 2 other words, draws are at least as good to use as bluffs as weak made hands, and they will be used to fill up the SB’s bluffing range first. Now, if we include contributions to the EVs due to river action, this relationship would still hold. In fact, draws would pull ahead even further. It is easier for the draws to realize their equity on the river on average, when they are more likely to be nuts or air rather than mediocre value hands, and realizing equity in the larger pot translates into more EV. On the other hand, the chance of getting raised can reduce our desire to bluff draws, and this is something we will consider in the future. If we imagine filling up the SB’s bluffing range as described above, the first hands to be added will be his draws. However, in a lot of spots, such as the ones we are considering (don’t forget about Figure 11.1!), the SB does not hold enough real draws to fill out his bluffing range. If he only bluffed with his draws, he would not be bluffing enough to encourage the BB to bluffcatch. He needs to keep adding hands to his bluffing range to reach equilibrium, and he begins including some of his weak made hands that have less of a chance to improve if they get called. Now, which of the weak made hands is best to bluff with – that is, for which of them is EV(bluff)−EV(check) the largest? This decision is essentially the same as on the river: when comparing hands that lose their showdown equity by bluffing, we should prefer to bluff with those with the least to lose and check down the stronger ones. In summary, the best hands to bluff with are those that actually maintain their equity when called. The second-best hands to bluff with are those that lose their equity when called but do not have much to lose. And if the SB still needs more bluffs, he adds more of his weak made hands into his bluffing range, from weakest to strongest, until it is full. Which of the SB’s hands are actually indifferent between bluffing and checking? Of course, it is the last ones to be added into the range in the order we just described. This is simply by construction. We kept adding adding hands to his bluffing range until at last we reached the one for which betting and checking have equal EV. Just beyond that are hands that prefer not to bluff. On static boards where few draws are possible, this last bluff will be a weak made hand. In this case, all his draws and all the weak made hands with less showdown equity than the cut-off hand strictly prefer bluffing. 118

Nearly Static: Nearly PvBC Turn Play In some sense, the BB’s bluff-catching strategy “targets” the SB’s made hands for indifference while letting his semi-bluffs profit by strictly preferring to bluff. Another way to think about this is as follows. The BB has to make the SB’s weak hands indifferent simply because the SB has so many of them. Because he has so many, he can always add more of them to his bluffing range if it is profitable to do so. If the BB is not calling enough to prevent it, the SB can immediately take advantage by betting more weak made hands. (And this will motivate the BB to start calling more.) Contrastingly, the BB is usually not bluff-catching enough to make the SB’s best draws indifferent to bluffing, and indeed, bluffing is strictly more profitable than checking for these hands. However, the SB cannot exploit this by adding additional draws to his betting range, since he holds relatively few of them, and he is already betting all of them. On the other hand, many holdings act like draws, even on fairly static boards. For example, if the cards on board (especially the second-highest one) have relatively low rank, then lots of weak hands will at least have a draw to a pair that beats some of the BB’s turn calling range. The flop and turn cards in our examples, 3♦ and 2♣, were chosen because they do not interact much with the players’ ranges. However, these low cards do not make for the most static of boards. Indeed, we need to keep the effect of draws in mind as we proceed. When the SB does happen to have lots of drawing hands, such that he can bet only these and already be betting “enough” bluffs and have some left over besides, then it will be these hands that the BB has to target for indifference with his calling frequency. The BB will have to bluff-catch much more to keep draws indifferent to betting, as compared to weak made hands. In this case, the SB’s indifferent bluff will be a draw, and his weak made hands will simply give up on the turn. On which of the boards in 11.1 is it most likely that the SB’s indifferent bluffing hand will maintain much of its equity when called? On which is it least likely? If we divide the SB’s equity distribution at the beginning of turn play into betting and checking regions, it will not look like it did on the river where he was strictly polarized, betting his best and worst hands. To build a bluff119

Expert Heads Up No-Limit Hold ’em, Volume 2 ing range here, he begins with some higher-equity holdings (his draws) that are able to keep much of their equity after bluffing, and then if necessary, he starts back with the low-equity holdings that must give up their equity to bluff. This splitting of the SB’s range does not show up clearly on equity distribution plots but is hopefully easy to understand.

11.2.2 The Cut-off Hands We can now estimate the SB’s turn bluffing range in each of the situations in Figure 11.1. Essentially, we’ll estimate how many bluffs he needs, and then, we’ll look closely at his turn starting ranges to understand what sorts of hands are available and pick the best of them to use as bluffs. Once this is done, we’ll also be able to see which hands strictly prefer to bluff and which are likely to be indifferent to bluffing on the turn. We’re going to start with a quick and dirty estimate of the number of bluffs the SB needs to come up with. We will see that large groups of his potential bluffs are equivalent up to card removal effects. So, simply characterizing the weak hands he has to choose from, combined with a rough guess of his bluffing frequency, will be enough to let us identify the sort of hand that is indifferent to bluffing. Once we know that, we’ll be able to proceed, applying the bluffing indifference to estimate the BB’s bluffcatching frequency and finally the SB’s value-betting range. Basically, the BB will call with about half of his turn starting range. (The naive estimate based on the two street PvBC game is P/(P+B)≈57%. He may call more if the SB’s indifferent bluffing hand is a draw that keeps some of its equity after bluffing and getting called, and he may need to call less if the SB’s indifferent bluff is a made hand that gives up its equity to bluff and needs to be compensated by seeing more folds. Fifty percent is a reasonable guess.) Then, the SB will value-bet with anything that beats at least half of this calling range. And finally, to balance those value bets, he needs an amount of bluffs that depends on his sizing. We can estimate this using the result from the two-street, static PvBC situation. For 3/4-potsized turn and river bets, he needs about 1.04 bluffs for each value bet.


Nearly Static: Nearly PvBC Turn Play Decompose the SB’s turn starting range (given in Figure 11.1) into bluffs, value bets, and checks. Assume he value-bets everything that beats at least half of the top 50% of the BB’s turn starting range and bluffs once for each value bet. A computer tool such as the EDVis utility available on this book’s website is very helpful here. Now, we need to characterize the SB’s weak holdings in relation to the BB’s turn starting and turn calling ranges. You just examined the BB’s turn calling range on each board, didn’t you? We’ll now fill out the SB’s bluffing range on each board. Consider the K♥-Q♠-3♦-2♣ spot first. If we take the top half or so of the BB’s turn starting range, we land somewhere in his Q-x hands. So, when facing a bet here, the BB can happily fold all his A-high and threes. Perhaps he will be able to continue profitably with some J-10 as well, but most of his continuing range is pairs of queens. Versus such a calling range, how do the SB’s potential bluffs break down? He has a few straight draws surrounding the 3♦-2♣ (i.e., 4-5, 4-6, and 5-6) but these do not amount to very many hand combinations. The vast majority of the rest of his potential bluffing hands are two cards under the queen. These have some chance to pair up and win if they check, but they have no outs versus a pair of queens, so they give up almost all of their chance of winning at showdown in order to bluff. Thus, in this turn situation, we expect the SB to strictly prefer to bet with his few straight draws, and then the rest of his bluffing range will be filled up with other hands consisting of two cards between the Q and the 3, starting with those whose EV(check) is the least. Are any of these weak made holdings better to use as bluffs than others? When he checks, almost all of them have 6 pair outs versus the threes and aces in the BB’s turn starting range and no outs versus the queens. Thus, the SB’s higher cards (say, J-7) are not any more likely to win at showdown than low cards (say, 7-4), after checking or after betting. Such hands are pretty much all equivalent, so it is left to card-removal effects to determine the weak hands the SB actually uses to bluff here. It turns out that his somewhat higher-card hands block more of the BB’s queens, since the BB does not play the weakest Q-x pre-flop, while not blocking the aces with which the


Expert Heads Up No-Limit Hold ’em, Volume 2 BB will fold on the turn, since the BB preferentially flatted his worst A-x pre-flop. But these are small effects – most importantly, tons of the SB’s potential bluffs fall into the category of two cards between the Q and the 3. All of these have approximately the same chance of improving to win if they check and approximately zero chance of winning a showdown if they bet, and so all these are made nearly indifferent to bluffing the turn. Next, consider the K♥-8♠-3♦-2♣ and A♥-8♠-3♦-2♣ boards. These are similar, so we can take them together. Again, if we take the best half of the BB’s turn starting range, we see that most of his turn calling range will be made up of second-pair hands, his eights. He needs a few more hand combinations, so he probably also calls with some gutshot straight draws, A-4 and A-5 on the K-high board, and K-4 and K-5 on the A-high. (You can verify that these weak draws have more equity versus the SB’s polarized betting range than his pairs of twos and threes!) Many of the SB’s weak hands here include an overcard to second pair. These keep much of their equity if they bet and get called. Thus, in contrast to the K♥-Q♠-3♦-2♣ case, many of the SB’s potential bluffs here are not equivalent. We can fill out his bluffing ranges here as follows, from the best to worst hands to bluff. On the K♥-8♠-3♦-2♣ board, the best are the SB’s 5-4 combinations with 8 outs to a straight. (Of course, 5-4 is a value hand and not a bluff on the A♥-8♠-3♦-2♣.) The best semi-bluffs in the A-high case are the SB’s gutshot-plus-overcard hands, such as Q-4, which have 3 pair outs and 4 straight outs. (This class of hands does not exist on the K-high board). Next on both boards come hands composed of two overcards to the 8, with 6 pair outs. Most of these also have good card-removal benefits since they block a lot of the BB’s calling range (his eights, which tend to have high-card kickers) and do not block high-card, low-kicker hands in his folding range. The next classes of potential bluffs are the SB’s single-overcard hands (e.g., Q-6o) and his low gutshots (e.g., 6-4o). These have 3 and 4 outs, respectively, versus the eights which make up most of the BB’s turn calling range. Now, it might seem that a 4-outer is better to bluff than a 3-outer. However, the gutshots have up to 10 outs versus hands in the BB’s turn starting range, while the overcards only have 6. The gutshots must give up significantly more showdown value in order to turn themselves into a bluff, and


Nearly Static: Nearly PvBC Turn Play thus the SB’s single-overcard hands start bluffing before his gutshots. Of course, the SB has tons of these single-overcard hands, so these top off his bluffing range and are indifferent at equilibrium. Again, relatively minor card-removal effects serve as tiebreakers. This discussion goes to show that careful consideration of the tradeoffs between bluffing and checking certain hands is important. We may have thought that low gutshots are better to bluff than single-overcard hands. However, careful analysis and consideration of BB’s turn starting and turn calling ranges show that the gutshots are something like made hands in that they give up a lot of their equity when they turn themselves into bluffs. The single-overcard hands, however, act more like pure draws and are thus better to bluff, even though they’re weaker draws. Despite the similarities between these two boards, the SB can actually bluff a lot more in the A-high case. The reason is a matter of draws. On the K-high board, the BB’s worst calling hands are A-high with a gutshot. The prospect of a turn call with these holdings is improved by the fact that they have 7 outs to a fairly strong hand, i.e., they have a draw. Thus, they do not need as much encouragement to call the turn as a weaker hand would need. The BB’s weakest calling hands on the A-high board are K-high combinations that have fewer outs to strong hands, and the SB needs to bluff more to make up for it. How many bluffs does the SB need to have for each of his value bets in order to make the BB’s A-4o indifferent to calling the turn on the K♥-8♠-3♦-2♣? How many bluffs per value bet does he need to make K-4o indifferent on the A♥-8♠-3♦-2♣? This question is marked difficult, because a complete answer must take into account the different river scenarios. Lastly, consider the 9♥-9♠-3♦-2♣ board. What does the BB’s turn checkcalling range look like, and how do the SB’s potential bluffs stack up against it? Calling the turn with all his threes does not give the BB much of a calling range, so he has to bluff-catch with almost all of his A-high and perhaps even some other high-card hands. Against this range, the SB’s low straight draws, 4-5 ,5-6, and 4-6, are the best bluffs. These hands maintain nearly all their equity if they bluff and get called. They need to improve to 123

Expert Heads Up No-Limit Hold ’em, Volume 2 win and any improvement almost guarantees a win, versus both the BB’s turn starting range and his turn calling range. However, the SB does not have enough of these holdings. To fill out his bluffing range, he has to turn to his next-best choice: hands with two cards above the 3♦. He has tons of these. For similar reasons as for the straight draws, all of these hands may be thought of as pure draws. They have 6 outs, or about 12 percent equity at the beginning of the turn, and they keep pretty much all of it if they bluff and get called. Paired boards are often descrbed as “dry”, but although this one is quite weak, it is not particularly static. Almost all of the SB’s weak hands here can be thought of as pure draws with a significant number of outs, and drawing effects are a lot more important here than in the previous three cases. There are a couple of subtleties here that distinguish the SB’s potential bluffs. First, consider his 10-x holdings. These block a lot of the high-card hands the BB is check-folding on the turn. Thus, they are somewhat worse bluffs than hands like 7-6o. The SB’s Q-x and J-10 hands are even worse bluffs. We assumed that BB calls the flop with his J-10. This was reasonable, but it easily becomes part of his folding range when he faces a bet on the 2♣ turn. Most of the SB’s bluffs (two cards above the 3) go to the river with 12% equity whether they check or they bet and get called. His queens, however, have around 22% equity at the beginning of the turn, since they are actually ahead of the BB’s J-10. However, if they bluff, they only get called by better and essentially become just another 12% equity hand. So, the SB’s queens and J-10 must actually give up a significant amount of equity in order to turn themselves into bluffs. The SB actually needs to come up with quite a few bluffs here, and it turns out that he eats through all of his straight draws, all of his normal any-two-card type bluffs, and all the way into the hands that have bad card removal effects. Thus, the vast majority of the SB’s weak hands prefer to bluff this turn, and his 10-x hands, such as 10-4o, are made indifferent, while his Q-x, J-10, and some 10-x hands prefer to check back. In summary, on the K♥-Q♠-3♦-2♣ board, many of the SB’s weak hands have some equity if they check, but they give up almost all of it if they bluff and get called. One of the SB’s many hands with two cards between the Q and the 3 (for concreteness, let’s say 9-8o) will be indifferent to bluffing. The BB 124

Nearly Static: Nearly PvBC Turn Play must be calling somewhat less frequently than the naive estimate of 57 percent to encourage it to pass up its showdown equity and bluff. On the K♥8♠-3♦-2♣ and A♥-8♠-3♦-2♣ boards, the deciding factor for the SB’s choice of bluffs was the number of outs to pairs. We decided that hands with one overcard to the 8, such as Q-6o, will be made indifferent to bluffing at the equilibrium. On the 9♥-9♠-3♦-2♣, we saw that almost all of the SB’s potential bluffs were nearly equivalent up to card removal but that card removal was non-negligible. Our representative indifferent bluff here will be 10-4o. NB: we’ll keep these “representative indifferent bluffs,” since it is more convenient to work with specific hands than classes of hands, but none of our work here has uniquely identified these specific hand combinations. In the previous section, we compared the EV on the turn of pure draws that kept all their equity after bluffing and that of pure made-hand bluffs that lost all their equity when called. In reality, many hands lie somewhere in between. In this case, the best hands to bluff are not necessarily those that keep the most equity when called but those that give up the least equity in order to make a bluff. This desire to give up as little equity as possible is the primary consideration for choosing bluffs in this turn situation. Our river bluff-choosing principle can be viewed as a special case of this one. On the river, we favored bluffs with as little showdown equity as possible – since bluffs never won when called, there is no reason to waste a relatively high showdown-value hand while showing down a weaker one. Here, we prefer to bluff with hands that keep their equity, and failing that, we bluff with those that must give up as little equity as possible. As on the river, card-removal effects often effectively serve as tiebreakers. Of course, all other things equal, it will always be best to bluff with holdings that block Villain’s calling range and increase his folding frequency.

11.3 Applying the Bluffing Indifference to Find the BB’s Calling Range We are now well on our way to estimating the players’ turn strategies. We know which of the SB’s hands will be made indifferent to bluffing, so we can now write down the bluffing indifference to find the BB’s calling fre125

Expert Heads Up No-Limit Hold ’em, Volume 2 quency. The naive application gave a frequency of about 57%. What extra contributions to the EVs in the bluffing indifference apply to the cut-off hands we identified in the previous section?

11.3.1 K♥-Q♠-3♦-2♣ Take the K♥-Q♠-3♦-2♣ board first. The 9-8o was representative of the SB’s bluffing cut-off here. It has 6 pair outs versus about half the BB’s turn starting range and no outs versus the rest for a total of about 6% equity if it checks down. We need to account for this in our estimate of EV(check). When he does improve, it will not be to a hand strong enough to value-bet. However, he will have no trouble showing down and capturing that equity whenever he wants, since the BB cannot lead the river. Thus, if he checks the turn, he will just capture his small amount of equity in the pot – no more and no less. We can account for this effect by adding 0.06*P to the SB’s EV(check turn). What about the EV of bluffing with 9-8o? We saw above that this hand gives up essentially all its equity in order to turn itself into a bluff. Thus, the estimate of this hand’s EV(bluff) from the naive estimate is still accurate. If it bluffs, the hand wins the pot the FT of the time it gets a fold, and loses its bet otherwise. It is always effectively air on the river, so it should break even on any river action as well, and no extra contribution is necessary to account for the river play. Applying the bluffing indifference, we see

so that the SB’s turn calling frequency here is 1−FT≈54%. This is a bit less than the naive estimate. As we expected, BB has to fold a bit more to encourage a weak made hand to give up its showdown value and bluff.

11.3.2 K♥-8♠-3♦-2♣ and A♥-8♠-3♦-2♣ The situations on these two boards are again quite similar. Here, hands with one overcard to the eight were indifferent – take Q-6o. This hand has 126

Nearly Static: Nearly PvBC Turn Play approximately 10% equity versus the BB’s turn starting range, so we will add 0.10P to the SB’s EV(check turn). If it bluffs and is called, the BB usually has an eight, so the one overcard is good for about 7% equity. We can estimate the corresponding contribution to EV(bluff turn) by assuming he wins 7% of the larger pot (P+2B) the (1−FT) of the time we get called. Thus, we have

Here, it turns out the extra equity after checking and the smaller amount of extra equity in the larger pot cancel each other out, and we find a calling frequency of (1−FT)=57% as in the naive case. Now, if we hit our overcard on the river, after betting or checking the turn, we have a hand strong enough to value-bet. This is in contrast to the case on the K♥-Q♠-3♦-2♣ board. Here, we effectively assume that we always check down on the river, but in fact, when we improve, the river betting action will increase our EV somewhat. However, this is a relatively small effect, and it applies to both sides of the equation (i.e., after betting and checking the turn), so it cancels out to some degree. We will consider this sort of effect further as we continue exploring the effects of multi-street play.

11.3.3 9♥-9♠-3♦-2♣ In this case, the SB needs to bluff a ton, so much so that his indifferent bluffing hand actually has unfavorable card-removal effects. Take 10-4o, for example. This hand has 6 outs versus much of the BB’s range if it checks down or its bluff is called, for a total of about 10% equity in either case. Making the same sort of adjustments to the bluffing indifference as on the previous boards (and again neglecting the possibility of river action), we find

so that FT=0.4 and the BB’s turn calling frequency is 60%. The bluffing hand here is essentially a pure draw in that it maintains all its equity when 127

Expert Heads Up No-Limit Hold ’em, Volume 2 called. If the BB only called the 57% of the time that the naive estimate would suggest, this hand would strictly prefer to bluff. There is an extra catch here that has to do with the card-removal effects. Keep in mind that the bluffing indifference applies to a particular cut-off hand of the SB’s, and so the folding frequency we solve for is the frequency when the SB holds that particular hand. Here the SB’s cut-off 10-4o was chosen for its negative card-removal properties, since the bluffing range needed to be so wide – it blocks some of the BB’s folding range. Thus, the BB is calling more versus this hand than versus the SB’s average turn bet. That is, the BB needs to call with somewhat less than 60% of his turn starting range in order to end up calling 60% versus the SB’s 10-4o. The opposite case is probably more common. If the bluffing range is relatively tight, the strongest bluffing hand might be chosen for its good card-removal effects, and then the BB will need to fold more of his range on average than he does versus the SB’s cut-off. Anyway, the BB’s average calling frequency here is something close to 57% again.

11.4 Betting for Value and Protection We bootstrapped our estimate of unexploitable turn play using our knowledge of the two-street PvBC game and the single-street SB bet-or-check game. This was enough to estimate the type of bluffing hand the BB needs to be indifferent at equilibrium, and then in the previous section, we used the bluffing indifference to get a more accurate estimate of the BB’s calling frequency. We can go from a calling frequency of C% to a calling range without too much error by simply taking the top C% of the BB’s turn starting range. Now that we have the BB’s calling range, what hands can the SB value-bet? He need never slow-play here, so his value-betting range will be some amount of his strongest hands, and we can describe his whole valuebetting range by simply finding his weakest value bet. So, we want all hands for which the EV of betting is greater than that of checking, and for his weakest value bet, the two EVs will be approximately equal. In particu128

Nearly Static: Nearly PvBC Turn Play lar, we’ll look for the value hand that satisfies EV(bet−check)=EV(check−check), i.e., we’ll assume the weakest turn valuebet checks down on the river. Since we know the BB’s calling range, writing down and comparing these two EVs is easy. We’ll find the value-betting ranges in our particular turn spots, but much of our interest here will be in a general value-betting criterion. We’ll see that it differs from the river case due to the possibility of draws. For comparison, recall the river case that we discussed in Section 7.3.3. Suppose we are deciding between value-betting and checking down on the river, and there is no chance of a bet getting raised. If Villain folds to our value bet, we would have won the pot with a showdown anyway. Against his calling hands, however, we do best by betting as long as we get called by worse more often than by better. Thus, Villain’s folding hands do not factor into our thin value-betting decision, and we can value-bet with any hand that has at least 50% equity versus Villain’s calling range. What part of this logic breaks down on the turn? Will the SB need to have more or less than 50% equity versus the BB’s calling range to make a value bet when draws are possible? For the SB’s weakest value bet, i.e., his strongest bluff, we have


If the SB checks down, he just captures his equity in the pot. If he bets once, he always ends up with at least (S−B), he gets another (B+P) the FT of the time the BB immediately folds, and the other (1−FT) of the time, he wins his equity EQ{vs BB calling range} in the larger pot of (2B+P). Now, the two equities and the folding frequency FT are actually related to each other and to the SB’s equity versus the BB’s folding range. This should make sense intuitively – the BB splits his range when he decides to call or fold, so the SB’s equity versus the BB’s turn starting range is just the weighted average of his equity versus the calling and folding hands. That is, we have (11.3)


Expert Heads Up No-Limit Hold ’em, Volume 2 On the river, this simplifies since EQ{vs folding range}=1, but this is not necessarily so on the turn, since the weak hands that the BB folds on the turn can have outs versus a value-bet. The idea now is to take this expression for EQSB{vs starting range} and plug it into Equation 11.2. We can then solve for EQ{vs calling range}. This will tell us exactly how much equity we need versus the BB’s calling range to find a break-even value bet. Working it out is a helpful exercise, but we will spare you the algebra. Performing these manipulations, we find (11.4)

This is the least equity the SB can have versus the BB’s calling range to have a profitable value-bet. Notice that no equilibrium-based arguments went into finding this. In the context of the decision tree we have assumed, this is the minimum equity versus a calling range needed to make betting once more profitable than checking down. This betting criterion can be used just as well for exploitative play (just like the 50% equity criterion on the river). It is just that we are going to apply this formula to the specific case where the BB is calling with his GTO range, which makes our result a good estimate of the SB’s GTO value-betting range. But first – what is Equation 11.4 telling us? It says that the minimum equity needed to value-bet is at most 1/2 but is somewhat less due to the second term. We call this second term the protection term for reasons that will become clear shortly. If we were on the river, we would have EQ{vs folding range}=1, the second term would go away, and we would need exactly 50% equity to value bet. On the turn, (1−EQ{vs folding range}) is the equity that the BB’s folding range has versus the SB’s value bet, and is nonzero in general. So, unlike on the river, the BB’s folding hands do have a significant effect on our value betting decision! The more equity they have, the weaker the hand the SB needs in order to value-bet. That is, the more successful our bet is in pushing our opponent off his equity, the more we can lower our requirements for value-betting! What is really going on here, of course, is not that the EV of betting is higher when Villain’s weak hands have more outs, but that the EV of checking is lower. The more chance Villain’s folding 130

Nearly Static: Nearly PvBC Turn Play hands have to improve when they see a river, the more we are incentivized to avoid giving them the chance to do so. Betting to avoid giving Villain’s weak hands a free chance to improve is known as betting for protection. Conventional wisdom frowns on protection as a reason for betting, since it is easy to get carried away. Betting middling hands can result in simply folding out worse hands and putting in more money versus better. However the protection effect can certainly lean a borderline case into a clear value-bet, and we will see later that it can sometimes do more than that. How large an effect does the protection term have on our value-betting requirements? Well, for the case of the 3/4 -pot sized bet, the FT/(1−FT) portion of the product will generally be a bit smaller than 1, and the P/B portion is a bit larger than 1, so when these are multiplied, they approximately cancel. In fact, if the BB’s folding frequency FT is chosen to make the SB’s pure air indifferent to bluffing, then FT=B/(P+B) and (1−FT)=P/(P+B), so that FT/(1−FT)=B/P and the last two multiplicands in the protection term become P/B × B/P and cancel exactly! We have seen reasons why the BB might fold more or less than the naive estimate, but those parts often at least nearly cancel, leaving us with the following approximate valuebetting criterion: 11.5

This rule is valuable, since it’s easy enough to use at the tables. Basically, we estimate how much equity Villain’s folding hands have versus our thin value-bet, divide that by two, and subtract it from the baseline of 50% in order to find how much equity we need to value-bet. Let us practice on our four examples. (Perhaps give it a shot yourself before continuing!) Suppose we want to express the SB’s value-betting criterion in terms of his equity versus the BB’s turn starting range. What will it be? Assume the BB’s folding frequency is FT=B/(B+P). For exploitative purposes, when might it be easiest or best to think in terms of our equity versus Villain’s turn starting range, and when might it be best to think in terms of our equity versus his folding and calling ranges?


Expert Heads Up No-Limit Hold ’em, Volume 2 Take first the K♥-Q♠-3♦-2♣ board. Here, the BB’s turn calling range is mostly queens, and his folding range is his twos, threes, and A-high, as well as some J-10 and wheel draws. The SB’s weakest value-bet will lie approximately in the middle of the calling range, i.e., it will be a mediocre queen. The BB’s folding range is composed of A-5o to A-2o, A-9s to A-2s, J3s, and about half the J-10 combinations. Against this range, the SB’s mediocre queens have about 87% equity. That is, the BB’s folding hands have on average 13% equity versus the SB’s thinnest value-bet. So, we can use Equation 11.5 to estimate that the SB needs equity 1/2−(0.13/2)=43.5% versus the BB’s calling range to find a value-bet. The SB’s Q-7 has this but his Q-6 does not. Notice that the protection term has a fairly large effect here, despite the fact that K♥-Q♠-3♦-2♣ is a fairly static board. On K♥-8♠-3♦-2♣, we expect the SB’s weakest value bet to fall around the middle of the BB’s calling range – in his mediocre eights. The BB’s folding range (mostly A-high, deuces, threes, and wheel draws) has about 14% equity versus a pair of eights. Thus, the SB needs something like 1/2−(0.14/2)=43% equity when called to value-bet, and 8-7o is the threshold. Now, in both of the two previous spots, the full protection term in Equation 11.4 is not entirely negligible. The BB’s calling range is chosen to keep the SB’s bluffs indifferent, and the SB’s bluffs do not interact strongly (from a card-removal perspective) with the BB’s calling and folding ranges. Thus, the BB’s overall calling and folding frequencies are about the same as his frequencies when the SB holds a bluff. However, the SB’s thin valuebets and second pairs block a ton of the BB’s calling range. That is, even if the BB’s average folding frequency makes the approximate value-betting criterion look correct, FT is usually much higher when the SB holds a second-pair hand. Examining Equation 11.4, we see that the effect of this is to reduce the amount of equity needed to value-bet even further. If we apply the full criterion, it turns out the SB’s weakest value-betting hands in these spots need something like 41% equity when called, and in effect, this allows the SB to bet his second-pair hands with one lower kicker than we estimated previously. This observation does not depend particularly strongly on the minute details of these two spots. In fact, the SB’s borderline turn value-bets will of-


Nearly Static: Nearly PvBC Turn Play ten block a lot of the BB’s calling range. Thus, our turn value doublebarrelling range should be even wider than suggested by the approximate criterion (and much wider than just those holdings that get called by worse more often than by better). We will make a couple of comments about thin value-betting at the end of this section, but for now, let us finish with the other two boards. On the A♥-8♠-3♦-2♣, the BB’s calling range and the SB’s bluffs look pretty similar to the previous case. The SB’s second-pair hands again push the BB off a range with about 12% equity for an estimated breakeven EV{vs calling range} of 44%, but again, this is an overestimate, since the SB’s thin-value hands reduce the BB’s calling frequency. However, there is an important difference that has to do with how the BB’s pre-flop and flop strategies interact to create his turn starting range. In particular, the BB gets to the turn with many more bare K-high hands on the A♥-8♠-3♦-2♣ board than he has bare A-high on the K♥-8♠-3♦-2♣ board, simply because he flat-calls with fewer aces pre-flop. Since he has more weak hands in this case, he must call a turn bet with weaker hands in order to keep the SB’s bluffs indifferent, and this means calling with his K-4 and K-5 – high-card plus weak drawing hands. This is non-negligible – for example, the SB’s 8-6 has about 10% more equity versus the BB’s calling range on the K-high board than on the A-high board. His weaker eights, 8-5 and 8-4, have more equity than in the K-high case for the same reason, and they have straight draws here as well. Find the equity distribution of the SB’s turn starting range versus the BB’s calling range on the A♥-8♠-3♦-2♣ board. At no point during the game do the players hold these ranges simultaneously, but it is still a good way to visualize how the SB’s potential betting hands stack up versus the BB’s calling range. Verify that all the SB’s 8-x have more than enough equity to bet, while his next weakest hands are all clear checks. Lastly, the 9♥-9♠-3♦-2♣ board also holds a slightly different twist on the standard situation. On the A-high board, the BB had to call with some highcard hands, but here this is taken to the extreme – the BB’s turn calling and folding ranges are mostly just high cards. The SB’s weakest value-bets, his A133

Expert Heads Up No-Limit Hold ’em, Volume 2 high and deuces, have around 81% equity versus the BB’s folding hands. So, we estimate that the SB needs about 40.5% equity against the BB’s calling range in order to value bet – significantly less than in previous cases by virtue of the fact that the BB’s weak hands have significantly higher likelihood of improving to win on the river. Estimating our value-betting criterion more accurately here is unnecessary, because, like on the A-high board but unlike on the K-high boards, the SB has no holdings that fall particularly near the cut-off. All his hands are either clear checks or clear bets. In the perfectly static case, the BB’s folding hands would have zero equity against all the SB’s value bets. In other words, they would have no chance of getting there on the river. However, in these more realistic situations, the SB pushed the BB off ranges with between about 12% and 20% equity versus his thin value-bets. The boards we chose in Figure 11.1 are considered “dry”, but are they really very static? Can you find a turn spot where the BB’s check-folding range has less than 12% equity versus the SB’s thin value-bets? Higher than 20%? How much does this depend on the players’ pre-flop and flop strategies? These hands provide rich examples of the ways that details of the players’ ranges and how they interact with the board can affect our decisionmaking. Poker is a game of small edges that add up, and paying attention to these effects is a good place to find edges. Finally, so we do not get too out of line, it is important to note that the SB’s ability to make thin valuebets on the turn is kept in check by at least three considerations that we did not account for here:

We assumed here that the BB’s flop check-calling range is capped fairly low. However, the more often the SB follows through on the turn, the more the BB is incentivized to slow-play the flop. If he does slow-play, the SB will need a stronger hand to have the equity necessary to value-bet.

As on the river (see Section 7.3.3), the possibility of being raised, especially bluff-raised, certainly affects the SB’s desire to make thin value-bets. If the BB raises for value, his hand is ahead of the SB’s thin value-bet, and the SB would likely have lost the pot anyway.


Nearly Static: Nearly PvBC Turn Play However, he also loses any chance he had to improve on the river. If the BB check-raise bluffs, the SB loses even more relative to showing down. This all said, many players check-raise the turn after check-calling the flop quite rarely, especially on static boards. It is difficult to get to the turn with a distribution that allows them to put in many raises, at least if they play strong hands fast on the flop, and if they had flopped a hand they wanted to bluff-raise, they likely would have done so on the flop.

The SB cannot be certain of a free showdown on the river if he checks back the turn. Even if his turn starting range was nearly polar, that may not be the case on the river after he checks back all his middling hands on the turn. With regards to the last point, what do the river starting distributions look like after the SB checks back each of our four turn spots? Given these distributions, is the BB likely to have a significant river leading frequency? If so, do you think it will motivate the SB to change his turn play so as to get to the river with different distributions?

We will see each of these issues again as we continue our study of multistreet play. Each of these issues will lead the SB to tighten up his turn value range. So, although the ranges we found in this chapter are quite accurate given the decision tree and turn starting ranges we assumed, they’re likely looser than in full HUNL. One thing that can let the SB value-bet even wider, however, is if he can play particularly well on the river after betting the turn. We focused here on the indifference between check-check and bet-check for the SB’s thinnest value-betting hands. However, if it turns out that betbetting is better than bet-checking for these hands on a significant number of rivers, it could make betting the turn even better on average.

11.5 *Wrapping it up: Exploitative Bluffing In this chapter we looked at nearly static, nearly PvBC situations. We fo135

Expert Heads Up No-Limit Hold ’em, Volume 2 cused on play within the larger of the two trees in Figure 11.2. This game tree and our discussion apply to the case where one player takes the betting initiative early in the hand, and by doing so, defines his range as relatively polar. His opponent just plays “check and guess”, and we have argued that this is reasonable given the distributions. Of course, we know that raises and other changes of betting initiative are important parts of real play, and we will begin to consider them in the next chapter. Here we saw how the presence of draws affects players’ range-splitting decisions at equilibrium. Given the decision tree, the structure of the SB’s turn strategy looks a lot like the analogous case on the river, the SB bet-orcheck game. It is just that the betting thresholds are affected by the possibility of changing hand values. The BB’s calling range can be looser or tighter than the naive case depending on the SB’s bluffing cut-off. If the SB’s borderline bluff is a made hand that loses its showdown value by bluffing, the BB calls tighter to make up for it, but if the SB’s indifferent bluff is a draw, then the BB must call the turn more frequently. As for the SB, he value-bets hands corresponding to something like the top half of the BB’s calling range, but on early streets, he can bet somewhat wider for protection-related reasons. Finally, coming full circle, the SB will have to include enough bluffs in his betting range to make the BB’s bluff-catchers indifferent to calling. We saw that naive applications of indifference equations lead to valuable intuition about unexploitable play but can lead to untrustworthy results in practice, and we had to account for some details to obtain accurate strategies. Along the way, we found a rule for choosing our value bets on the turn, and we saw how the presence of draws modifies the rule of thumb from river play. On the river, with some simplifying assumptions, we can valuebet from the SB if we have at least 50% equity versus the BB’s calling range. On the turn, we found that this requirement is lessened because of draws – essentially, we are motivated to bet because we gain by pushing Villain off some of his equity. Can we find an analogous rule for exploitative bluffing as well? How will it compare to the river case? Let us try to find an approximate bluffing criterion for the turn by comparing bluffing to shutting down. We looked at the


Nearly Static: Nearly PvBC Turn Play bluffing indifference above, but now we want to find a quick rule of thumb for evaluating the tradeoff between bluffing and checking, à la Section 10.6. For convenience, let Es, Ec, and Ef be the equity of the SB’s potential bluff versus the BB’s starting, calling, and folding ranges, respectively. Then, assuming we just go to showdown once we make it to the river, we have a profitable single-barrel bluff if

Plugging in Equation 11.3 for Ec and rearranging, we find that a bluff is best if (11.6)

where FT is the BB’s turn folding frequency, as usual. This looks messy, but it is actually easy to understand. First of all, if we are on the river with pure air, both Ef and Ec are 0, and this equation tells us that we need FT>B/(B+P) to have a profitable bluff. No surprise there. If we are on the river with a hand with some showdown value, Ec is still 0, since we only get called by better, but Ef is not, and we find a profitable bluff when (11.7)

What is this fraction on the right? Consider the numerator and denominator separately at first. Since Ec=0, Equation 11.3 tells us that FTEf=Es, and so the numerator here equals FT−Es. FT is the total fraction of hands Villain folds to a bet, and Es is the total fraction of his range we are ahead of at the beginning of the street, so FT−Es is the fraction of his turn starting range consisting of hands better than ours that fold to a bet. And the denominator, 1−Es is just the amount of Villain’s hands that are better than ours at the beginning of the street. So, the entire fraction on the right side of Equation 11.7 is just the proportion of Villain’s hands that are better than ours and that fold to a bet. Thus, if we hold a hand with showdown value on the river, we need Villain 137

Expert Heads Up No-Limit Hold ’em, Volume 2 to fold at least B/(B+P) of hands better than ours to profitably turn our hand into a bluff. As a quick example, suppose we are considering making a potsized bluff in position on the river. If our hand has no showdown value, we have a profitable bluff if Villain folds at least 1/2 of his turn starting range. If our hand has some showdown value, however, he needs to fold at least half of hands that are better than ours so that we can profitably bluff, and this requires him to fold more overall. So, this gives us a good way to think about bluffing on the river that takes into account our loss of showdown value without explicitly comparing EV(check) to EV(bet) 3. Finally, on the turn, both Ec and Ef in our bluffing criterion might be nonzero – Ef because we have some showdown value if we check down, and Ec because we have some chance to improve if we bluff and get called. So we need to consider the full right side of Equation 11.6. The only difference, in comparison to the river case, which has a very simple interpretation, is the bolded term in the denominator. This term depends only on Ec and necessarily makes the right side of the inequality bigger. So, the more equity we have versus a calling range, i.e., the more likely we improve on the river after our bluff gets called, the more we prefer a bluff to a check. No surprise there. In order to see exactly how much bigger, it would be really great if we could write Equation 11.6 in the form of the easily-understood river result plus some extra contribution that depends on Ec, just like we wrote the minimum equity needed to value-bet as the river value ( 1/2 ) minus the protection term. It turns out that we can do that, approximately, using a mathematical technique known as a Taylor expansion. The details of the technique are not important – suffice it to say that, for fairly small values of Ec, we have the approximate bluffing criterion

Again, we see that the more equity retained by our bluffs, the less Villain needs to fold for us to profitably bluff. There are a number of other interesting, practical questions to be answered here, but they are mostly small extensions of ideas we have already 3

Thanks to Owen Gaines for this tip!


Nearly Static: Nearly PvBC Turn Play spent plenty of time on, so we will leave the exploration to you. Consider the following questions.

The extra term here is due to any chance our bluffs have to improve if they get called – how big an effect does this have on our bluffing ranges in real spots?

How does this bluffing criterion look for the ideal made hand (Ec=0) and the ideal drawing hand (Ec=Es=Ef)?

Comparing check-check to bet-check gave us a quick way to get an idea of the effect of draws on our turn bluffing decision. However, to play most exploitatively, we know we need to consider the possibility that bluffing twice is actually best. What is a good rule of thumb for comparing check-check to bet-bet on the turn?

How might the possibility of other river action change our bluffing decision?

11.6 EV Distributions Finally, let’s look at some EV distributions to solidify what we have seen. We will focus on the K♥-8♠-3♦-2♣ case. Consider a game tree like the larger one in Figure 11.2, except that all the rivers are explicitly included. In other words, whenever the hand goes to the river, the tree includes river subtrees corresponding to each of the 48 possible river cards, and the players’ ranges and strategies are potentially different for each of them. We have solved this game computationally, and the slight differences in river play affect EVs of the players’ turn decisions only slightly as compared to our earlier discussion. The primary difference from our earlier estimate is that the SB value-bets a little wider on the turn, since he can win some extra value by playing well on specific rivers. The other issues mentioned at the end of the last section do not come into play because of the fixed starting ranges and decision tree. Figure 11.3 shows EV distributions for both players’ hands on the turn when players use their GTO strategies. As in an equity distribution, we 139

Expert Heads Up No-Limit Hold ’em, Volume 2 have ranked hands from weakest to strongest according to their equity versus the opponent’s range. Unlike in an equity distribution, we show the hands’ EVs instead of equities at different points in the tree. We have drawn these distributions to include all hands with which the players get to the turn, not just those with which they actually take each action. For reference, the corresponding turn starting equity distributions are shown as insets. These are the same plots that were drawn larger in the second row and third column of Figure 11.1. We have indicated some corresponding holdings and numbered them for easy reference.

Figure 11.3: EV distributions for hands in players’ turn starting ranges in the K♥-8♠-3♦-2♣ spot. Equity distributions are shown as insets for comparison.

The plot on the left illustrates the EV of betting and of checking the turn with all hands in SB’s turn starting range. These are his only two options at his first decision point, so of course, his actual play with any given hand corresponds to whichever EV is higher. That is, his actual EV curve at that decision point can be thought of as a third that is the maximum of the two drawn. Starting on the left side of the plot with the SB’s best hands, betting is clearly best. The top of his value region consists of 2-pair and better hands. The EVs here appear a bit erratic due to card-removal effects. For example, K-8o actually has significantly lower EV than K-3o, since it blocks more of the BB’s calling range. The first plateau in his EV (see arrow 1) corresponds to his kings as well as nines through jacks. These are all nearly equivalent, since we assumed the BB’s range is capped at second pair. The 140

Nearly Static: Nearly PvBC Turn Play SB’s eights, however, are certainly not all equivalent. These fall into the downward sloping section of the EV curves indicated by arrow 2, and the SB’s kicker makes a big difference here. The two EV curves meet, so that the SB is indifferent between his options, right around the bottom of his 8-x hands. The small group of hands indicated by arrow 3 is the SB’s deuces. They have significantly more equity than the SB’s weakest hands, but not all that much more EV. They are also more or less indifferent between betting and checking at equilibrium. Arrow 4 points to the SB’s draws. The best of these are low straight draws, but the bulk of them are hands with two overcards to the 8. These all have clearly higher EV(bet) than EV(check) (Why is this?) Finally, much of the rest of the SB’s range (see arrow 5) has one overcard to the 8 and is made indifferent, while his worst hands have no overcard or straight outs and strictly prefer to just give up on the turn. Which parts of the SB’s betting and checking EV distributions should we expect to overlap in the three other turn situations featured in this chapter? Now, how would these EV curves change if the BB were exploitable? Generally speaking, if the BB called too frequently, the EV of checking would stay the same, while the EV of betting would increase for the SB’s better hands and decrease for his worse ones. If he called too little, the opposite trend would generally be true. What effect would these changes have on the SB’s range splitting, i.e., when would it affect which of the EV curves was higher at each point? The second panel in Figure 11.3 shows the BB’s EV distributions, both at the beginning of turn play (solid lines) and after he actually faces a bet (dotted lines). The first pair of curves looks a lot like the second, just shifted upwards and slightly tilted. Why is this? Focus on the BB’s EVs after he actually faces a bet, since that is when he actually has a decision to make. His strongest hands are eights. These have EV(calling) strictly higher than EV(folding) by virtue of the fact that they are actually ahead of some of the SB’s value-betting range. Most of the rest of his range is made indifferent by the SB’s polar betting range.


Expert Heads Up No-Limit Hold ’em, Volume 2 We can get a sense of scale for the EVs here by keeping in mind the stack size at the beginning of turn play, S=21BB, and stack plus pot size, S+P=29BB, that one player would end up with if his opponent just gave up on the turn with his entire range. Notice that SB’s value hands are able to capture a lot of equity above and beyond the pot. For the BB, having a bluff catcher is not much better at all than having air once he faces a bet on the turn, and he usually faces a bet. Compare the EV curves in Figure 11.3 to the corresponding equity distributions. If the players just checked down the turn and river, their EVs would be directly related to their equity. How does the betting affect the players’ ability to capture equity? Which hands end up winning more than the entire pot at the beginning of turn play, on average? Do any expect to end up with less than their turn starting stack? Which hands is it most important to play correctly to maximize overall EV in these spots? For which hands does the player’s choice not significantly affect his EV? In this section, we assumed the SB c-bet a polar range and the BB’s check-calling distribution was capped. If the players play these strategies, we can also find the SB’s flop checking range and the BB’s turn starting range after the SB checks. What do the equity distributions look like at the beginning of the turn after the SB checks the flop? Use the methods of this chapter to estimate the players’ equilibrium turn play, assuming the BB does all the betting.

11.7 You Should Now … ♠

Understand how to choose bluffs on the turn.

Be able to estimate equilibrium turn and river play in realistic SB barreling spots.


Nearly Static: Nearly PvBC Turn Play

Understand how concerns related to hand protection affect exploitative turn value-betting criteria.

Know how to read an EV distribution.

Of course, we covered each of these points under the assumption that one player does all the betting. There are many reasons why this is not always the case in real hands, and we’ll begin worrying about those now.


Chapter 12 Initiative and Less Common Turn Lines

Fall seven times, stand up eight. – Japanese proverb

12.1 Changes of Initiative If a player chose an aggressive action (a bet or a raise, as opposed to a check or a call) in the most recent spot where he had the chance to do so, then we say that he holds the betting initiative. For example, if the SB raises pre-flop, c-bets the flop, and the BB calls, the SB has held the initiative throughout the hand, up to the beginning of turn play. Traditional wisdom says that the player with the initiative either thinks he has a strong hand or is betting to represent such a hand. Players without the initiative may respect that and check to the player with the initiative at the beginning of each street. Continuing with our example, the BB will usually check to the SB on the turn. It would be somewhat unusual if he decided instead to steal the initiative by leading the turn. On the other hand, if he checks but then the SB declines to follow up and also checks, nobody really has the initiative at the beginning of river play.


Initiative and Less Common Turn LInes Betting initiative can be understood in terms of the players’ equity distributions in common spots. We have seen that it often makes sense to play aggressively with strong hands and bluffs but not with intermediate holdings. So, aggressive play tends to polarize the aggressor’s range at the same time as it earns him the betting initiative. Then, by virtue of polarization, he keeps the initiative – we have seen that his bluff-catching opponent will usually play passively. In some sense, betting initiative is illusory. Many players place significant constraints on their strategies to respect initiative, despite the fact that these constraints are not enforced by the rules of the game. However, it does describe a pattern that emerges out of the solutions to very complex games. GTO strategies often respect initiative. The multi-street decision trees we have used so far are ones where the owner of the initiative does not change (i.e., only one player bets). This was justified by early-street action that polarized one player’s distribution. In Chapter 11, we saw how draws and varied hand strengths can affect the range splitting within the context of this decision tree, but the set of lines we allowed was limited to check-check, bet-check, and bet-bet for the polar player, and fold, call once, and call twice, for his opponent. These are the most common lines in real turn and river play because of the initiative-seizing, range-polarizing dynamic we just described. It is fair to think of any hand that breaks outside of this pattern as unusual. If the betting initiative switches from one player to the other, something at least moderately uncommon must have occurred. Perhaps a player slow-played an earlier street and is thus able to find a raise later, or perhaps a card came which significantly affected relative hand strengths. More mundane reasons for initiative switches boil down to hand protection (which also has to do with the possibility of changing relative hand strengths). There are two primary ways for the initiative to change owners, and it is useful to separate them, since they give rise to very different spots. We will refer to these as type P and type M initiative switches. In the type P case, one player seizes the initiative. The BB can do this by leading a street when the SB holds the initiative, and both players can do this by raising a bet. A type M initiative switch is much less dramatic: one player voluntarily gives up the initiative. For example, if the BB 3-bets pre-flop, and his flop c-bet is


Expert Heads Up No-Limit Hold ’em, Volume 2 called, but then he checks on the turn, he has given up the initiative. Note that when the SB gives up his initiative, we see another street immediately, but this is not so for the BB. Find a friend and play a session where you agree that neither of you can steal the initiative on the turn or river. How does this affect your turn and river strategy? How does proper early street play change? Type P and M switches differ in the sorts of equity distributions involved. Suppose the BB holds the initiative at the beginning of turn play (as in the K♣-7♥-3♦-K♦ hand to which we will return shortly). Consider first a type P switch wherein the SB aggressively seizes initiative. Since he is in position, he can only do this with a raise. We saw what this looks like on the river in Section 7.3.3. In particular, Figure 7.19 shows the distributions after a BB bet and SB raise. The BB’s initial betting range was fairly polar, and the SB’s raising range was even more so. Thus, we have an absolutely polar range running into another absolutely polar range. In the model situation, the players essentially swapped places in a nearly-PvBC dynamic. Now, the situation is not always so dramatic on earlier streets. When draws are possible, there are some protection-related reasons to raise without a polarized range, as we will see. And, of course, if a raise is called, new cards can come and they can mix up hand values. In any case, a type P initiative switch involves two absolutely polar distributions running into each other – thus the label, P. A Type M switch involves ranges composed primarily of mediocre hands running into each other, and M stands for mediocre. This tends to happen when one player voluntarily gives up the initiative. For example, suppose the SB checks back the flop and calls a turn lead, and then the BB checks the river. In this case, the originally-bluff-catching SB still has a range of mediocre hands, and the BB’s failure to continue betting indicates that he is also unlikely to have a strong hand. Perhaps the BB holds one of the few medium-strength hands in his range – these might overlap significantly with the SB’s bluff-catchers. Alternately, perhaps the BB just has air that is giving up on its bluff. Thus, the BB’s equity distribution here might look a lot like that in Figure 7.10(c). The top part of it is a lot like the symmetric case, but he has some air in his range as well. 146

Initiative and Less Common Turn LInes This discussion gives us a good framework for organizing the various possible later street lines. First, we have those that involve no initiative switching and are contained in the PvBC tree. Then, there are lines that indicate that something unusual happened and the initiative changes hands. Both types of initiative switches frequently involve relatively similar ranges running into each other: polar vs polar or mediocre vs mediocre. When players hold similar ranges, we’ll tend to see changes of initiative, i.e., raises and reraises, as in the symmetric distribution river games. In nearly-PvBC spots, especially on the river, we can expect one player to do most of the betting. In the next section, we will examine the exact solution to a two-street, nearly-static, nearly-PvBC situation, and we will see that the players do break out of the PvBC game tree with significant frequency. We will need to figure out why the (bluff-catching) SB might bet when checked to on the turn and how the BB should adjust his play in response. We will consider some other “outside the box” lines in this chapter as well. As you read over a description of the computational solution, pay special attention to spots where the initiative changes and how the subsequent action proceeds after each type of switch.

12.2 K♣-7♥-3♦-K♦ Part Deux: The Computational Solution Let’s look at unexploitable turn-onward play in our K♣-7♥-3♦-K♦ example. We first considered this spot in the context of the ideal PvBC game. The players begin with 30-BB effective stacks, the SB opens to 2.5BB pre-flop, and the BB calls. The flop checks through, and we find ourselves at the beginning of turn play with a pot of 5BB and the turn starting distributions given in Section 10.4. What does our approximate game tree look like? In our study of river play, we were able to tune the decision trees by hand to include the options that made the most sense depending on the players’ early-street play, river starting distributions, etc. However, most of our computational models


Expert Heads Up No-Limit Hold ’em, Volume 2 describing play from the turn onward will contain hundreds of different river spots, and hand-tuning each of them is inconvenient. Thus, we will use some general rules for generating decision trees. The following paragraph describes this approach. Our turn approximate games include both turn and river play. Whenever turn play ends without a fold or all-in, 48 different rivers are possible. Each of these is included in our game trees and is dealt with the correct frequency. The bet and raise sizing options are generated as follows. When a player has the chance to bet, he can always go all-in. He can also use each of the following sizes as long as it is at least a min-bet and at most 6/10 of an all-in: 1/2 P, 3/4 P, P, and (3/2 )iP for any positive integer i, where P is the size of the pot. In other words, the over-bet sizings are 1.5P, 2.25P, 3.375P, etc. Additionally, we give the BB the option to block-bet 1/5 P at the beginning of each street and the SB the option to min-raise when facing a bet. Thus, the players have a wide range of strategic options available to them. We will use these bet-sizing options in all the computational calculations we present in this chapter and the next unless otherwise noted. The turn starting distributions in our K♣-7♥-3♦-K♦ hand are shown in Figure 10.3. The BB is fairly polar. He has some near-nut hands (mostly kings), some other high-equity holdings that are ahead of all of the SB’s range but which are not quite as invincible (his sevens). Overall, about a quarter of his range is a value hand. Then, he holds some hands with mediocre equity, some of which are draws, and finally a sizeable portion of his hands need to improve to have much chance of winning. What do you expect equilibrium play to look like here? Let’s see some results. First, consider the BB’s first turn decision point. He bets out about 80% of the time. This is more than we predicted when we applied the ideal two-street PvBC solutions to this spot. He also uses bigger bet sizings. In particular, the BB almost never uses his options to bet the size of the pot or smaller. Instead, he goes 3/2-pot with about threequarters of his betting range and 2.25 times the pot with the rest. Recall that the GGOP sizing, although a bit larger than the pot, is smaller than both of these choices. To see why he bets this way, it is helpful to look at the holdings he plays with each sizing.


Initiative and Less Common Turn LInes The BB’s strongest hands, and many other value hands as well, use the smaller sizing which is closer to the GGOP choice. This is desirable, since GGOP sizing essentially lets the bettor include as many bluffs as possible in his betting range while still keeping bluff-catchers indifferent. On the other hand, his range for using the larger sizing is capped – the strongest hand is a seven. These weaker value hands are also ahead of all of the SB’s range on the turn, but they only have about 90% equity. In other words, there is a small but significant chance they lose on the river. These hands have something of a tradeoff. It benefits them to leave enough behind to make a significant river bet when called in order to bluff as often as possible. However, unlike the BB’s very strong holdings, they also gain from going a bit larger on the turn in order to prevent some of the SB’s hands from realizing their equity. We have seen that GGOP sizing is the multi-street analog to the river over-bet. On the river, nut hands can over-bet jam with impunity, while thin value hands often do better by betting smaller to ensure they get called by worse. Here, protection effects can make more vulnerable hands bet larger for protection, while the nuts take a closer-to-GGOP approach. Why can the BB get away with capping his second value-betting range? On the river, if our value range’s best hand has 90% equity, then 10% of Villain’s range is effectively the nuts and can punish us with re-raises (neglecting card removal and chopping). Here, however, the SB’s 10% comes from the chance to improve on the river, not from any nut holdings that can happily re-raise on the turn. However, his capped betting range does result in his getting pretty well crushed by some rivers. For example, suppose the SB calls a 2.25P turn bet, and then an ace comes on the end. The BB’s previously polarized range becomes about half air and half mediocre showdown value, while the SB now holds almost all pairs, many of which are rivered top pairs. In this case, the BB must almost always check the river, and he very often faces a bet when he does. Nonetheless, this is a relatively rare contingency, and it turns out that it does not happen often enough to motivate the BB to include any stronger hands in his 2.25P turn betting range. The BB’s turned flush draws are almost all included in his smaller-sized betting range. Why does he bet them, and why does he use the smaller sizing to do so?


Expert Heads Up No-Limit Hold ’em, Volume 2 How should the SB react when facing a bet? His considerations here are of the sort we focused on in the last chapter. He has to continue enough to make some of the BB’s low-equity air hands indifferent to checking. When the BB checks one of those hands, it gets to see a free river about half the time, and it improves on about 12% of rivers. So, when these hands try to check and get to showdown, they might expect to capture about 6% of the pot. However, the river action gains them a bit extra, so let us say they expect 8% of the pot when they check. On the other hand, when they bet and get called, they mostly lose that bet, but again they have a bit of equity (in a larger pot) and a bit extra EV due to the river betting where they will be part of a relatively polar range. How often should the SB continue when facing the BB’s 1.5P and 2.25P bets here? 4 Now, back to the BB’s first decision point – his checking range contains some nuts and some air (i.e., weak draws), but the majority of it is made up of those relatively few hands that do not fall into the nuts-or-air category, i.e., his weak showdown value. These include most A-x hands, which are both ahead of and behind significant portions of the SB’s range. The kicker has a big effect on these hands’ equities. The BB’s A-2 has about 20% equity while his A-10 has well over 40%. There are two reasons for this. First, the higher kickers are more likely to improve the BB to a winning hand if they pair up on the river. More significant, however, is the fact that A-x makes up a large portion of the SB’s range as well, and on most rivers, kickers will play. How does the SB respond to a turn check? Of course, it is helpful to know what the distributions look like following the BB’s check – these are shown in Figure 12.1. The SB has about 66% equity on average. He holds no strong value hands nor any air, but some of his holdings are certainly better than others. The SB’s range at the end of flop play was composed of nearly equivalent bluff-catchers, but the BB starts the turn by checking primarily with his own mediocre hands, and this makes the distributions closer to symmetric than they were previously.


Against the smaller sizing, the answer turns out to be right around half of the time.


Initiative and Less Common Turn LInes

Figure 12.1: Equity distributions after the BB checks the turn in our K♣-7♥-3♦-K♦ example.

Now, it turns out that the SB bets a bit over half the time and checks back the rest when facing the BB’s turn check. Most of his hands here play with a mixed strategy – sometimes betting and sometimes checking. However, he does bet a somewhat polar range, considering the constraints of his starting distribution. Hands stronger than A-J and weaker than A-6 tend to bet, while hands from A-8 to A-10 tend to check. We saw that, in a pure static PvBC situation, the bluff-catching player will never bet. After checking the turn, the BB always held air that was giving up on the pot, the players checked to showdown, and the SB won the pot with his bluff-catchers. Here, unlike in the model, the BB’s weak hands do have some chance of improving to win. This motivates the SB to bet to just take down the pot – a plan that works out great if the BB holds only bluffs that plan to check-fold. This is a protection-type effect reminiscent of the one that affects the polar player’s value betting decision as well. Of course, the possibility of a SB bet can tempt the BB to check value hands on the turn. We will discuss this dynamic in more detail in the next section. The SB always uses his smallest sizing option – 1/2 pot – when he bets here. It turns out that if we give him some smaller sizing options, 1/2 pot is still his favorite, although he uses a slightly smaller bet sometimes. If he 151

Expert Heads Up No-Limit Hold ’em, Volume 2 goes much smaller than 1/2 pot, holdings with two overcards to the seven, such as 10-8o, begin to have the correct implied odds to call, which is nonideal. In the next section, we will talk about some of the reasons behind these bets, and we will see why a small sizing works well. Facing a bet, the BB folds over half his checking range, and the rest of his actions are split fairly evenly between calling and re-raising. As you might expect, he folds his worst holdings, re-raises relatively polar, and calls the middling. Why might this result be surprising? If the SB’s bluffs are pure air, a half-pot sized bet would only need to take down the pot 1/3 of the time for bluffing to be better than giving up and losing the pot. Here, however, the BB is folding more than half his range! This BB strategy is unexploitable only because the SB does not get to this turn spot with any real air, or at least not much of it anyway. The BB is calling only enough to keep hands with a reasonable amount of showdown value from wanting to turn themselves into bluffs. This is another example of a spot where the naive application of indifference equations can lead to results that are quite far off the mark.

12.3 The Delayed c-bet Versus the Turn Check-raise In the previous chapter, we saw how slight violations of the assumptions of staticity and PvBC distributions could affect the players’ frequencies. In the last example, however, we saw that the presence of draws could cause play to break out of the PvBC decision tree altogether. In particular, the SB bet the turn when checked to around half the time. This line is known as the delayed c-bet, since he raised pre-flop, declined to c-bet the flop, but did stab the turn when given the option. This can be the result of having been improved by the turn card, but playing flopped mediocre hands this way can be good as well. Let’s first consider some exploitative reasons for the delayed c-bet. Often, if the BB had much of a hand, he would have tried harder to get a bet in before the river. So, betting when checked to twice can be used for value with some fairly weak value holdings – say, A♥-3♠ in our example. Such hands 152

Initiative and Less Common Turn LInes can be called by a few worse made hands as well as turned draws. This move is especially good versus BBs who would lead the turn with any better hands and do not often check-raise the turn after the flop checks through. The SB’s turn bet then gets some value while denying the BB a free chance to improve. It can also help to ensure that we do not face a river lead with our weak value hands (if the BB is the type to respect initiative). Thus, going for a delayed c-bet could be better than making a normal flop c-bet with weaker value hands that might want to put in a bet but do not want to give the BB the opportunity to float out-of-position. Delayed c-betting weak showdown value also makes sense in the context of equilibrium play. In the ideal PvBC solutions, the BB always checked down and lost after checking the turn. There was no motivation for the SB to bet out, since he could check down and win the pot 100% of the time anyway. However, if he had done so and the BB had always folded, it would not have changed the value of the game for either player. From the BB’s point of view, checking with a range of entirely air was unexploitable, since he could play his value hands more profitably by bet-betting. In real play, however, many of the BB’s air hands on the turn will have at least a little chance of improving on the river. In our K♣-7♥-3♦-K♦ hand, almost all of the BB’s holdings have at least two live cards that will pair up to win on about 13 percent of rivers. If the BB’s turn checking range is entirely air (or, we could say, very weak draws), i.e., hands that are planning to fold to a bet, the SB can just bet the turn and take down the pot with his bluff-catchers. This is clearly better than giving a free card. Thus, the presence of draws gives the SB some genuine incentive to bet, and he is essentially betting for protection. In this case, the SB’s bets do not often get called by worse or fold out better. However, if none of the BB’s hands are going to put any more money in the pot except in the rare (but not insignificant) case that they improve on the river, the SB does best to just bet and preemptively take down the pot. At this point, delayed c-betting with weak showdown value should seem like a pretty good idea. It is useful both for exploitative reasons and to simply take down the pot when we know we have the best hand but Villain could have a few outs. Certainly, if the SB knows that the BB is always check-folding after checking the turn, he definitely should go ahead and 153

Expert Heads Up No-Limit Hold ’em, Volume 2 bet to pick up the pot with most if not all of his range. Winning the whole pot all the time is the best he can expect to do on average at equilibrium. However, this isn’t the whole story. If the SB bets all of his range when checked to twice, how might the BB react? The BB’s nut hands want to play so as to cause the SB to put as much money in the pot as possible, on average. If the SB is always stabbing when checked to, the BB’s nuts should always check and let the SB bet. In this way, he gets a bet in versus all of the SB’s bluff-catchers rather than the approximately half of them that would call a lead. He also figures to get more future bets in versus more of these hands, since the pot is inflated. So, checking is definitely best versus a SB who holds bluff-catchers and plans to delayed c-bet with all of them, but should the BB check-call or check-raise? If he check-calls and then checks the river, then we have a genuine static board with mostly PvBC type distributions. The SB is usually just going to show down his mediocre hands, and the BB will miss value. Check-calling the turn and then leading the river might be an option, but we will generally do best to spread our action into as many bets as possible with the nuts. Thus, if the SB is always betting when checked to, the BB should go for the check-raise on the turn. We have essentially just done a bit of equilibration exercise. If the BB is always leading the turn with his value hands as in the ideal PvBC game solutions, the SB’s counter-strategy is to always bet the turn when checked to twice. This incentivizes the BB to begin going for a check-raise on the turn with his nuts. On the other hand, if the BB is always check-raising the turn, the SB will clearly prefer to check back rather than to bet. Failing to get any money in on the turn with his value hands is a poor result for the BB, so this motivates him to switch back to leading the turn for value. Thus, in a nearly static situation where the BB is nearly polar at the beginning of turn play, neither always leading nor always checking with his value hands can be the GTO strategy. Similarly, neither always betting nor always checking can be the SB’s GTO play after the BB checks. The players will employ mixed strategies in these spots. Does similar reasoning suggest that mixed strategies arise at equilibrium when the SB is polar and the BB is bluff-catching on the turn? 154

Initiative and Less Common Turn LInes How can we estimate the players’ unexploitable frequencies? As usual, because of the mixed strategies, we can apply the Indifference Principle. We have, for the SB’s weak made hands, (12.1)

The left side here is the EV when we go straight to the river, so it is essentially the EV of playing a PvBC situation on the river. The right side is the weighted average of the value of winning the whole pot when BB checkfolds and losing the pot and turn bet whenever he raises, assuming the SB is indifferent between calling and folding when facing a raise. (Of course, the SB cannot always bet-fold on the turn when he bets, or else the BB could always raise!) For the BB, we have the indifference (12.2)

Intuitively, the BB must be compensated for his losses when the SB checks behind by his gains when he successfully pulls off the check-raise. Solving these equations to good approximation is certainly possible. However, we will use another approach to build intuition and to develop a tool that can be used in other spots. Solve the above indifference equations to estimate the BB’s check-raise frequency and the SB’s delayed c-betting frequency in the K♣-7♥-3♦-K♦ turn spot.

12.4 Estimating Mixed Equilibria with Matrix Games 12.4.1 Tradeoffs in the Delayed c-betting vs Check-raising Mini-game How will this spot play out? The BB will sometimes bet-bet with his nuts and sometimes go for the check-raise, and the SB will sometimes delayed c-bet and sometimes check behind with his bluff-catchers. The more the SB 155

Expert Heads Up No-Limit Hold ’em, Volume 2 delayed c-bets, the more the BB will be motivated to go for the check-raise, and vice versa. The more the BB tries for the check-raise, the more the SB prefers to check back when given the option, and vice versa. Both players face tradeoffs. They each have two strategies. If we labelled them without regard to their actual strategic content, we might say that SB can choose to play A or B, and BB can choose to play 1 or 2. If the SB plays A, then the BB wants to play 1, but that makes the SB want to play B, which motivates BB to play 2, which makes the SB want to play A again. And so on. This sort of alternating sequence of strategies when players adjust maximally exploitatively is nothing new for us, but we are getting a new way to estimate the equilibrium in these situations. This is essentially the same as Matching Pennies (MP), an often-studied, archetypal situation in game theory. In this simple game, two players simultaneously reveal coins. If the two coins match (i.e., they’re both heads or both tails), then one player wins both coins. If they don’t match, then the other player wins. A more common game, Rock-Paper-Scissors (RPS), has essentially the same structure except that each player has three options. The only difference has to do with the payoffs. In MP and RPS, we generally think of losing being just as bad as winning is good, and we think of losing in one way as being just as bad as losing in any other way. For example, in RPS, losing is losing, whether we threw rock and lost or threw paper and lost. However, if throwing rock and losing were much more painful than throwing paper and losing, then we would tend to throw paper more often. We would not always throw paper (since then Villain would always throw scissors, and then we would be motivated to throw rock regardless of the downside), but we would certainly want to throw paper more often at equilibrium. Which other poker situations have a two-strategy alternating structure like MP? Can you think of any situations where each player essentially has three options and the sequence of maximally exploitative counterstrategies cycles through all of them as in RPS? What are the relative payoffs in our delayed c-betting situation? Consider the situation from the BB’s point of view. He wants to get as much money in the pot as possible with his value hands. If he leads the turn, he is cer156

Initiative and Less Common Turn LInes tain to get two bets in versus a SB who calls down, but if he checks, he gives this up for the chance to get three. However, the smaller the SB’s delayed c-bet, the less benefit gained by the BB when he pulls off a checkraise. In the limit that the delayed c-bet is extremely small, getting to put in a check-raise is pretty much the same for the BB as having led out himself, so going for the check-raise has no upside but has significant downside if the turn checks through. Indeed, we noted that the SB used his 1/2pot sizing to delayed c-bet in our computational solutions. Of course, his bet is only meant to fold out the BB’s air, and only a small bet is needed to discourage these holdings from trying to draw to a pair on the river. Thus, a successful check-raise is usually a significant-but-modest win for the BB, while missing a bet completely on the turn is quite unfortunate. From the SB’s point-of-view, a successful delayed c-bet is only slightly better than checking down, since the BB’s air does not have much chance to improve. On the other hand, delayed c-betting and facing a raise is quite painful. So, what should the equilibrium look like in the end? The SB will be quite happy if both players choose strategies such that the turn goes check-check a lot, but we do not expect the BB to allow this to happen. Thus, we expect the BB to lead quite frequently at equilibrium to make certain that a bet usually goes in on the turn when he holds a value hand. This will allow the SB to delayed c-bet with a fairly high frequency. An easy way to organize our thoughts on this sort of spot is as follows. Ignore for a moment the complexities of the multi-street situation and focus on a single choice of each player: the BB can choose whether or not to check the turn with the nuts and the SB can choose whether or not to delayed cbet with his bluff-catchers when given the chance. We know that the players will adopt mixed strategies at the equilibrium, but consider how the pure strategies stack up against each other. The SB has two pure strategies: delayed-c-bet and check back facing a check, and the BB also has two: bet-turn and check-turn. Notice that the names of the BB’s strategies describe play with his value hands, but the strategies tell us how he plays his air as well. When he plays bet-turn, he bets the turn with enough air to balance his value and checks the rest. When he plays check-turn, he checks the turn with all of his air as well as his value. Overall, there are four pure strategy combinations. How favorable is each of them for each player? 157

Expert Heads Up No-Limit Hold ’em, Volume 2 We want to see how the magnitudes of the players’ tradeoffs lead to the equilibrium frequencies. We’ll start with a quick approximation of the SB’s average EVs for each of the four possible strategy pairs. (This gives us all the necessary information about the BB’s EVs, too, since any gain for one player is a loss for his opponent.) We won’t follow the indifference equation approach to this problem all the way to the end, but we will do enough to build some intuition. Suppose the effective stack and pot sizes are S and P, as usual, and let B be the SB’s delayed c-bet sizing. When the BB plays bet-turn, he always folds to a turn bet after checking, and when he plays check-turn, he checks all of his air as well as his value. Assume the SB is indifferent to folding his bluff-catchers when facing a bet, as will generally be the case at equilibrium. Then, the following are the SB’s EVs at the beginning of turn play:

Check that each of these makes sense. For example, in the first case, the SB is playing delayed-c-bet while the BB plays lead-turn. Here, the BB either leads the turn, and the SB expects S, or he checks the turn, and the SB’s delayed c-bet always takes down the pot. If N is the nut fraction of the BB’s turn starting range, then we can usually think of (BB turn lead frequency), (BB turn c−r frequency), etc., as N times some factor to account for bluffs, e.g., 2N if he bluffs once for each value hand. What can we glean from these? Suppose we know the BB is playing leadturn. Then, comparing the EVs of the SB’s options means comparing the first and third equations. These differ only on the end, and delayed cbetting is clearly the superior choice, as expected. If we know the BB is playing check-turn, then the EVs of the SB’s two options are given by the second and fourth equations. Is it clear why the latter is greater? If not, think about how the SB’s equity versus the BB’s turn starting range compares to the BB’s turn check-raising frequency. 158

Initiative and Less Common Turn LInes Now, intuitively, how good are the best cases for the players, and how bad are the worst cases? How much should each player risk to try to obtain his preferred situation, given the potential downside that occurs when his opponent does not cooperate? For concreteness, consider what we might call the “standard line” – the BB plays bet-turn and the SB plays check-back – and compare it to each of the other three possibilities. If the BB plays bet-turn and the SB plays delayed-c-bet, the BB still gets to play his value hands and a lot of bluffs quite profitably by bet-betting, but when he does check, his air does not get to see a free river. So, this pair of strategies is slightly worse than the standard line for the BB, and thus slightly advantageous for the SB. Secondly, what if the SB plays check-back while the BB plays check-turn? Missing a bet with his value hands is quite bad for the BB and a big win for the SB relative to the standard line. Lastly, if the SB plays delayed-c-bet and the BB plays check-turn, the result is a fairly large loss for the SB, relatively.

12.4.2 Expressing the Tradeoffs as a Matrix Game

Figure 12.2: Matrix for turn PvBC situation where the BB’s air hands have a bit of equity. Each pair of pure strategies is compared to the one in the top left, and the numbers approximate the value of each strategy pair for the SB.


Expert Heads Up No-Limit Hold ’em, Volume 2 This information can be organized into a grid or matrix such as the one shown in Figure 12.2. The SB’s two pure strategies are listed across the top, and the BB’s down the side. In each of the four squares is a description of the outcome if both players play their respective strategies. We have also some specific numbers for payoffs in each case in addition to the qualitative descriptions. These are meant to represent the EV differences for each of the strategy pairs as compared to the standard line. Plus or minus 3 indicate very good or bad results, and +1 indicates a more modest positive outcome. These numbers could have been calculated from the EV equations above, but instead, we simply chose some values that more-or-less indicate the relative values of each strategy pair, given our previous discussion. They are written from the point of view of the SB, but since any money won by one player is money lost by the other, the value for the BB is just the opposite. Note that the absolute magnitude of the numbers is not really important here. If we multiplied all the values by the same number, or added a constant to each of them, it would not change the resulting strategies. As usual, it is the differences between the values, not the absolute values themselves, that are strategically important. How closely do the payoffs indicated in Figure 12.2 agree with the EV equations we sketched out above? (A qualitative answer is fine here.) The matrix in Figure 12.2 actually gives us a complete description of a game. These so-called matrix games are the natural choice for modelling strategic situations where both players make their actions simultaneously, and the payoff to each player depends on both choices. This is in contrast to games such as poker, which are more naturally represented using decision trees, since they involve multiple actions taken in a specific order. As with decision trees, there is a fairly straightforward mathematical process for solving matrix games. If our matrix is 2-by-2 like the one here, and we know the equilibrium is mixed, then we can essentially just parameterize the players’ strategies and apply the Indifference Principle as usual. We will not take that approach here. We would have to consider the full multistreet, multi-action situation to get exact payoffs anyway, so we will stick with our previous methods when we want to chug through math to find 160

Initiative and Less Common Turn LInes exact results. The matrix approach will aid in visualization and help us build intuition. So, what intuition is there to be gained? As usual, the SB’s indifference will give us information about the BB’s strategy, and vice versa. The indifference relationships are the same as we had above (Equations 12.1 and 12.2), but having organized the possibilities into a matrix can help us see their consequences. The basic recipe for thinking about these spots is as follows. Suppose Hero’s first option makes Villain prefer one of his actions by an amount X, and Hero’s second option makes Villain prefer his other action by an amount Y. Then, Hero can make Villain indifferent by playing his first option Y times for every X times he plays his second. An example will help to make this clear. Suppose the BB wants to make the SB indifferent to delayed c-betting. If he plays bet-turn, then we are in the second column of Figure 12.2, and the SB prefers to delayed-c-bet by an amount 1−0=1. If the BB plays check-turn, then we compare the two numbers in the first column of the figure and see that the SB prefers check-back by 3−(−3)=6. So, to make the SB indifferent, the BB must play bet-turn 6 times for every 1 check-turn. That is, he leads on the turn with his nut hands 6/7 of the time. Intuitively, the SB loses a lot more from delayed c-betting when the BB goes for the checkraise than he wins when he plays delayed-c-bet and the BB plays bet-turn. Thus, the BB can go for the check-raise only very rarely if the SB is to be indifferent between his options. He has to be check-folding a lot when he checks to make the SB want to delayed-c-bet at all. On the other hand, suppose the SB wants to make the BB indifferent. If he plays check-back, the BB can achieve a relative profit of 0 with bet-turn and -3 with check-turn. (Remember that the BB’s EVs are the negative of the SB EVs shown in the figure.) Thus, the BB prefers betting the turn by an amount 0−(−3)=3. If the SB plays delayed-c-bet, checking gains the BB +3 while leading gains him -1 for a difference of 3−(−1)=4. Thus, the SB can keep the BB indifferent between his options by playing check-back 4 times for every 3 times he delayed-c-bets. That is, he checks back 4/7 of the times he is given the chance. Here, the BB gains slightly more when he plays check-turn and the SB delayed-c-bets than he gains by playing bet-turn


Expert Heads Up No-Limit Hold ’em, Volume 2 when SB plays check-back. So, the SB has to check back a bit more than he c-bets to make the BB indifferent. This spot is one where fuzzy logic can create trouble for a player who is not used to thinking strategically. Suppose Hero is in the SB and is more or less ignorant of the BB’s strategy. If the BB is more or less mixing it up evenly between his options, the SB should very strongly prefer to always check back on the turn since he loses so much more when BB plays check-turn than he gains when BB plays bet-turn. With such considerations, Hero might decide that check-back is best versus an unknown opponent, to be safe. However, the SB in fact delayed c-bets nearly half of the time at the equilibrium. In other words, his actual best response given uncertainty about the BB’s strategy is not the obvious choice. The way to identify your GTO play in a situation like this is not by thinking about your own payoffs but rather those of your opponent. The benefit the BB gains from leading the turn when we play check-back (3) is about the same as the benefit he gains from check-raising the turn when we played delayed-c-bet (4). So, we have to take both actions with about the same frequency to keep him indifferent. On the other hand, the benefit we gain from checking back when Villain goes for the check-raise (6) is much larger than the benefit we gain from betting if Villain is leading with all his value (1). So, the BB must be leading a lot to make us want to start delayed cbetting. In the SB, we fear getting check-raised, so it might seem like we should not often c-bet and give the BB the chance to raise. But in fact, we have to be betting quite frequently before the BB can profit by going for the turn check-raise, because he loses so much when we check back.

12.4.3 Reasoning Using the Matrix Game Finally, let us think about how our results might depend on a couple of variables: the SB’s delayed c-bet sizing and the equity of the BB’s bluffing hands. These are not necessarily separate issues: if the BB’s air (i.e., weak drawing) hands have more equity, a larger bet sizing might be necessary on the part of the SB to prevent the BB from calling to draw. That said, we will consider the two issues separately for now. 162

Initiative and Less Common Turn LInes Try to figure out the effect on the equilibrium of changing these two variables with either logic or math before you continue. If the SB increases his bet sizing, which of the payoffs in Figure 12.2 are affected? Of course, if SB plays check-back, the sizing is irrelevant, so none of the payoffs in the left column change. Also, when the BB plays lead-turn, he always holds air when he checks, and so the delayed c-bet sizing does not matter as long as it is large enough to keep the BB from trying to draw to a pair. Thus, none of the payoffs in the top row change. Assuming the SB bets large enough to fold out the BB’s air, the only payoff that changes is that in the bottom-right square, which corresponds to the case when the BB goes for the check-raise on the turn and the SB delayed c-bets. How does this payoff depend on the SB’s sizing? On one hand, when the SB bets larger, it reduces the effective stack size, thus reducing the amount of bluffs the BB can play to balance his value hands. In other words, when the SB bets larger, he gets check-raised less often. This is a small win for the SB. That said, the primary effect comes from the fact that he loses more when he does get check-raised. So, overall, the effect of increasing the SB’s bet size is to decrease his payoff in the bottom-right square of the matrix. Thus, the only effect of increasing his sizing is to decrease his payoff in one case, and there is no corresponding upside. Thus, the SB should bet as small as possible in this game, as long as it is large enough to make the BB check-fold air. By the way, how should the BB play here if he checks and the SB delayed cbets with too small a sizing? He should follow through with his plan to check-raise any value hands he slow-played along with bluffs to balance. Other air hands (i.e., weak draws) can flat-call if they have odds to do so. On the river, some of the BB’s air will improve to become value hands, and the rest will still be air, while SB’s range can still be thought of as mostly bluff-catchers, maybe with a few rivered stronger hands as well. That is, the BB will be the polar player in a PvBC or PvBC-plus-traps river situation, and so he should lead out with his rivered value hands and some bluffs for balance (thus stealing the initiative). Of course, we know that hands have significantly more EV than just their equity in the pot when they are part of a polar distribution, so the BB can draw here more frequently than a


Expert Heads Up No-Limit Hold ’em, Volume 2 simple pot-odds analysis would suggest. Recall that in the K♣-7♥-3♦-K♦ example, the SB could not get away with betting much less than half-pot, since then the BB could start chasing weak draws. We now know that the SB should delayed c-bet for protection as small as possible, but no smaller. How would the equilibrium change if his bet sizing were fixed to a larger value? Since he loses even more now when he plays delayed-c-bet and the BB correctly counters by going for the check-raise, the BB must play check-turn even less often than before to keep the SB indifferent between his options. On the other hand, the BB wins more than before when he plays check-turn and the SB delayed c-bets, so the SB must bet less often in order to keep the BB indifferent. That is, the bigger the SB’s sizing, the more that both players take the “standard” line at equilibrium. Now – what about the case when the BB’s air has more equity? We saw in the previous chapter that this depends strongly on the board and how it interacts with the players’ ranges. In particular, the ranks of the cards on board factor significantly into whether pair draws are worth much. Anyhow, if the BB’s air has more equity, it turns out that his payoffs increase in all four cases since they all involve the BB getting to the river with “air” some of the time, and a larger chance of that air improving to a value hand is better than a smaller chance. (Even when the SB is delayed c-betting, some of the BB’s air gets to the river by virtue of having been used to bluff.) However, the biggest benefit to the BB comes when the SB plays checkback. In this case, all of the BB’s weak hands get a shot at improving, while, when the SB plays delayed-c-bet, many of them do not. For concreteness, suppose that both of the SB’s payoffs on the left column in Figure 12.2, which correspond to the case when he plays check-back, decrease by 1. Then the top-left and bottom-left entries change to -1 and +2, respectively. How does this change the equilibrium? The SB’s GTO strategy is the one that keeps the BB indifferent, and the tradeoffs facing the BB are effectively the same. The BB’s bet-turn option still does better by 3 when SB plays check-back than when he plays delayed-c-bet, and the BB’s other option does better by 4 when SB plays his other option. So, the SB still has to play bet-turn 4 times for each 3 he plays delayed-c-bet to enforce the BB’s indifference. That is, the SB’s strategy


Initiative and Less Common Turn LInes does not change! So basically, if one of the SB’s options gets uniformly worse regardless of the BB’s play, then the overall situation for the BB is improved, but his strategic situation is not affected, so the SB strategy needed to keep him indifferent does not change. Suppose we continued uniformly decreasing both of the SB’s payoffs in the left column of Figure 12.2. How far could we go before indifference breaks down? What about BB’s new strategy? Previously, when the BB went for the checkraise with his value, the SB did significantly worse by delayed-c-betting than checking, while if the BB led the turn for value, the SB came out only slightly worse by checking than by betting. Thus, the BB had to lead the turn very frequently to keep the EVs of the SB’s two options equal. Now, the gap between SB’s possible payoffs when BB bets the turn has grown from 1 to 2 – there is more of an advantage for the SB in trying to counter BB’s bet-turn choice by playing delayed-c-bet, since he wins more by denying the BB’s air a free river card. At the same time, the difference between the SB’s payoffs when BB plays check-turn has shrunk from 6 to 5 – there is less of an advantage for the SB to try to counter the BB’s check-turn choice by playing checkback, since checking back is now less attractive. Thus, the BB now needs to play check-turn more than he did before to keep the SB indifferent. In particular, he only needs to play his bet-turn option 5 times for every 2 times he plays his other. In effect, the BB checks the turn with his nuts over twice as often as he did before! The SB has more reason to want to delayed c-bet and less reason to want to check back, so the BB has to go for the turn check-raise more often to keep the SB from preferring to delayed-c-bet. Does this agree with your intuition – why or why not? In this section, we got a new tool for estimating GTO play. Expressing tradeoffs in matrix form is very useful for estimating equilibrium frequencies when we know there will be a mixed strategy played at equilibrium. Thus, it is very useful for coming up with unexploitable strategies, especially when used in tandem with the equilibration exercise. It was the equilibration exercise that let us see that each player would adopt a mixed strategy at equilibrium in the first place. Only simple logic was necessary to see that maximally exploitative adaptation would never converge to a 165

Expert Heads Up No-Limit Hold ’em, Volume 2 pure-strategy equilibrium. If the SB always checks back, then BB should lead with his nuts, but then the SB should always delayed c-bet his bluffcatchers, but then BB should check-raise his nuts, and so on. This sort of pattern lets us know that mixed strategies will be played at equilibrium but does not give much hint as to the actual unexploitable frequencies. Of course indifference equations can be used to find these frequencies. However, we can gain more intuition by thinking about the tradeoffs shown in the games’ matrix representations – if Hero’s first option makes Villain prefer one of his by an amount X, and Hero’s second option makes Villain prefer his other by an amount Y, then, Villain will be indifferent to his choice when Hero plays his first option Y times for every X times he plays his second. Note that this rule applies to the case of a 2-by-2 matrix where a mixed strategy is played at equilibrium. The situation can be a bit more complicated if players have more than two strategic options each, or if indifference breaks down. In the simple river PvBC game, the polar player can bluff or check with his air, and his opponent can call or fold with his bluff-catchers. Why does this situation have the same structure as MP? Describe this situation as a matrix game and solve it. If you chose the correct payoffs, your result should be identical to the one we have found previously. Before we move on – a couple notes on hand selection if Hero is in the BB and wants to mix his turn check-raising range. There are often reasons to use our strongest hands to go for the check-raise and just bet-bet weaker value hands. Our nutted value is not scared to see a river, while nonnutted hands benefit from protection. Practically, if Villain sees us checkraise the turn and continue on the river, and then we show down secondpair-good-kicker, he may find it much more noteworthy and thus adjust more quickly than if we had shown up with a stronger hand.

12.5 River Leads in Checked-down Pots Suppose the SB opens pre-flop and the flop checks through, so that the BB 166

Initiative and Less Common Turn LInes is relatively polar at the beginning of turn play. The turn checks through as well, and then the BB leads the river. In Section 10.2, we referred to this as a weird spot, since it never comes up at equilibrium in the PvBC game. Without much idea of what the BB’s leading range looked like, it was hard to say exactly how the SB should react. However, since then, we’ve seen a few ways the BB can make it to the river this way with hands other than just air, even if he is playing well. We have treated river situations in a vacuum already, but the issue here is: what are the river starting ranges? The SB looks quite weak after checking twice, and the BB can certainly have plenty of hands with very little chance of winning at showdown – perhaps we are just seeing a desperate bluff on the BB’s part. However, many players do not often bluff this way in static spots. They are a lot more likely to bet the turn with hands they want to turn into bluffs. That is how they likely play their value hands, after all, and that way they can apply more pressure by threatening the river bet. Additionally, the fact that SB has twice passed up the opportunity to stab at the pot means he likely has showdown value and is likely to call once. Why else might the BB check-bet? Perhaps he did indeed check the turn to give up with air but managed to improve on the river and is now betting for value. On any river that improves some of the BB’s air to value hands, he can bet a few bluffs as well for balance. This is a lot more likely on some rivers than others. If we see, for example, a 10♥ river after the K♣-7♥-3♦-K♦ board, any rivered pairs can happily bet for value. On, say, a 2♥ river, however, almost none of the BB’s range should fall into this category. On many boards, the set of hands that improved to something valuable on the river is a small and well-defined group, and this is generally a rare possibility that cannot account for too high a leading frequency. Lastly, given our discussion of the delayed c-bet versus check-raise dynamic, the BB could have been going for a check-raise on the turn. Depending on the details of the situation, this could contribute a significant number of value hands to his river starting range. Any way you slice it, the BB’s river starting distribution looks fairly polar here. He holds air, air that improved to value on the river, and value that missed a turn check-raise. Thus, having a leading range makes a lot of sense – it is just that he should not be doing it


Expert Heads Up No-Limit Hold ’em, Volume 2 very frequently, since he should not get to this spot with too many value hands. If the BB does have a check-betting range versus missed flop c-bets on static boards, there are two main possibilities. Either he is bluffing too much on the river, or he is slow-playing too much on the turn. If you notice that an opponent has too high a check-bet frequency on the turn and river in the BB, how would you distinguish between the two reasons for it, and how would you adjust to each of them? Suppose you are in the SB playing our K♣-7♥-3♦-K♦ hand. The turn goes check-check. The river comes either the 10♥ or the 2♥, and Villain checks to you. You put in a bet, and he checkraises! What might the distributions look like in each case, and how do you split your range facing the raise? In the ideal PvBC game, position was unimportant. Does position matter in a nearly-static, nearly-PvBC scenario? Compare the situation described in this section to that where the SB is polar at the beginning of turn play, checks back the turn, and then bets the river.

12.6 Lessons so Far: The c-bet Polar Dynamic We have spent a lot of effort analyzing turn play when the SB checks back only mediocre made hands. Why have we done this? It is not only because it leads to the PvBC situation that is so useful and tractable. The dynamic initiated by polarized c-betting is right at the heart of modern HUNL strategy. Understanding and attacking a player’s flop c-betting strategy is very important, and of course, you must understand the situation from the SB’s perspective as well. We will discuss these issues further in Chapter 14, but let’s take a first pass at it now. Think a bit more about the SB’s motivation for c-betting polar. How does it


Initiative and Less Common Turn LInes help him play the various parts of his range profitably on later streets? We have seen over and over that a polarized range is almost always more profitable to play than a range of mediocre hands with the same average equity. Generally, it is also easier to play. One does not tend to face many hard decisions when holding the nuts, and air is relatively simple to play as well. To be slightly more quantitative, recall the EV distributions in Figure 11.3. One of the surrounding exercises involved comparing the EV distributions to the equity distributions in that K♥-8♠-3♦-2♣ turn spot. Of course, if there were no betting and both players just checked down, the EV and equity distributions would map directly onto one another, since all the hands of both players would simply realize their equity in the pot. So, in comparing the two distributions, you saw which hands benefit from the betting action. And of course, it is a zero-sum game, so if some hands benefit, then others must lose. When you performed the comparison, you saw that value hands tend to earn much more than their equity in the pot, and the EV of the nuts was significantly more than the pot itself. You also saw that air tended to capture more or less just its equity in the pot (i.e., nearly zero). So, the extra EV gained by the value hands comes at the expense of the bluff-catchers. A hand’s playability relates to how easy it is to realize its equity in the pot. It is very easy to realize all the equity of air, since it has none. We simply give up the hand or put in a bluff when it has at least zero expected profit. It is also easy to capture all our equity in the pot with the nuts – it’s hard to do too badly as long as we don’t fold. However, it is often very hard to effectively contest the pot with our middling hands. These generally do worse than if they could just check down. Thus, we often say that particularly strong and particularly weak hands (and pre-flop holdings that tend to make particularly strong or weak hands post-flop, such as suited connectors) have good playability, while middling hands (and pre-flop holdings that tend to make middling hands post-flop, like K-3o) do not. The primary message here is that weak hands do not really care if there is betting action, while strong holdings tend to gain value at the expense of mediocre made hands when there is betting. This is one of the primary benefits of being in position, at least on the river. When the SB faces a bet-


Expert Heads Up No-Limit Hold ’em, Volume 2 or-check decision there, he can bet a polarized range and check back his middling holdings. This lets him capture all of his equity with hands that just want to see a showdown while still getting a bet in with all those hands that benefit from it. The idea behind c-betting a polarized range on the flop is more or less the same. The SB bets particularly strong and particularly weak hands, but his mediocre holdings would prefer to just go directly to showdown. Unfortunately for him, in contrast to the river case, checking back the flop does not get the SB directly to showdown. We have seen over the past couple of chapters that after the SB defines his range as bluff-catchers, the BB gets the best of the situation, even if the SB plays the later streets perfectly. We have also seen that the multi-street PvBC situation is in some sense even worse for the bluff-catching player than the river-only case. Additionally, in the extreme limit represented by the ideal PvBC situation, position does not matter, so when a SB checks back only bluff-catchers, he is giving up much of his positional advantage. So, when a SB c-bets polar and narrowly defines his checking-back range, he puts himself in a potentially poor spot. However, there is an upside: His c-betting range is polarized and thus easily playable. Thus, this common approach to c-betting works by sacrificing the playability of the SB’s checking-back range in order to enhance the playability of his betting distribution. It makes him vulnerable after he checks back the flop but works quite well against BBs who do not take advantage. When we find ourselves facing such a SB, it is hard to combat him by attacking his c-bets. That part of Villain’s strategy is artificially reinforced. Instead, we have to apply pressure after he checks, because that is where he is exposed. Characterizing and attacking the SB’s checking-back range is the key to OOP play versus many opponents. Once we begin attacking his checkbacks on the turn and river, Villain will need to begin balancing that range. We will gain immediately when he begins missing bets with his value hands. Additionally, if he c-bets less value, he will have to c-bet fewer bluffs as well. Many players adjust incorrectly by focusing on play versus the SB’s c-bets while approaching his checks relatively passively. Perhaps this is because the SB checks back less often, so it seems less important, or it is


Initiative and Less Common Turn LInes harder to get good data about his range there. However, this is playing right into the SB’s plan! He wants to play large pots with his c-betting range, while taking his checking range to showdown. Perhaps BBs play passively since they figure the SB has at least some showdown value so he is likely to call at least once and maybe twice, but we saw in Section 10.4.2 how to play against the call-and-re-evaluate approach, and we make even more versus these opponents on average than we do versus someone with a more balanced strategy. Now, we are getting a bit ahead of ourselves. Flop play has its own chapter and we will return to these points there. For now, I just wanted to give a bit of context to show where our previous results fit into the overall picture of post-flop play. It will be important in the future once we take another step up the decision tree and consider play on the flop. Remember that good early-street play depends on a solid understanding of later-street situations. For now, however, try to characterize the SB checking-back ranges of opponents in your games. Play around with some larger turn and river bet sizings in these spots. At many stack sizes, choosing a standard bet size of around the full pot would not be unreasonable, and making it significantly smaller without a good exploitative reason could cost you a lot of money. How does board texture affect the dynamic we have discussed in this section? Consider a particularly volatile flop. Do many SBs still have a “polarized” c-betting range here? If so, is the BB’s best counter facing a check still to lead the turn aggressively? On which turn cards is this the case and on which not? Then, how will SB’s readjustment work? Will he begin to slowplay the flop more often? With which hands?

12.7 Example: The BB Check-calls Bluffcatchers and Some Traps on 9♥-2♠-9♦-A♠ Let us turn our attention to a spot that arises after the SB c-bets the flop. Most of the BB’s range here will just be trying to get to showdown after he 171

Expert Heads Up No-Limit Hold ’em, Volume 2 check-calls, but he will have some slow-played strong value hands as well. In addition to seeing the effect of the slow-plays, we will be able to compare turn play when the SB has the initiative to the case when the BB does, which we have focused on so far in this chapter. We discussed SB’s turn play with the initiative in the previous chapter but neglected the possibility of raises. The discussion here will lead us to investigate the turn protection raise. The following hand was Example 1 in Chapter 7. There, we made some assumptions about the players’ pre-flop, flop, and turn play, and solved for the players’ strategies on the 6♥ river. Here, we will use slightly modified assumptions regarding the pre-flop and flop ranges, and we will look at strategies from the beginning of turn play onward. The action goes: Effective stacks: 25BB Pre-flop: SB raises to 2BB, BB calls 1BB Flop: 9♥-2♠-9♦ (4BB) BB checks, SB bets 2BB, BB calls Turn: A♠ (8BB)

For simplicity, assume the SB min-raises pre-flop and c-bets this board with 100% of hands. Are these good assumptions? At 25-BB deep, minraising 100% is probably exploitably loose, and certainly many players fold or limp some hands pre-flop. The SB’s open-folding range can significantly affect how many pairs of deuces he makes on this flop. Some players tend to limp middling hands with good post-flop playability, such as Q-9, J-9, 10-9, and 9-8. This tendency could significantly affect the amount of nines in the SB’s flop starting range here. Also suspect is the assumption that the SB always continues on the flop. Certainly, if one is looking to choose some checking hands, high cards are good candidates. If the SB does not bet his A-high hands on the flop, the A♠ turn has a much different effect on the turn starting distributions. So, be aware that an analysis of this situation could be much different than the one that follows, and remember that it is important to characterize an opponent’s flop checking-back range. As for the BB, we will assume he arrives at the turn with a variety of K-high, some slow-played nines, and some overcards with back-door draw potential: 172

Initiative and Less Common Turn LInes K2s+, QTs+, J9s+, T9s, KTo, K8o-K5o, Qto+, JTo We have assumed he holds no aces (or pocket pairs), presumably because he would have 3-bet them pre-flop. This is reasonable play from the BB, especially versus a SB opening 100% of hands. Twenty-five-BB deep is probably around the deepest at which this is a good assumption in most player pools. If the stacks were 5 BB shorter, we could be more confident. The turn starting distributions are shown in Figure 12.3. As expected, the BB is quite bluff-catcher-heavy. Almost all of his hands have around 50% equity, although he has a few high-equity, slow-played nines. This turn card was strong for the SB but probably not nearly as strong as many players assume. Notice that the BB’s average equity did not actually decrease much on the A♠ turn. The reason is that all of his high-card bluff-catchers were already losing to the SB’s A-high, so it did not really change too much. It is true that the BB’s overcards lost their pair outs versus the SB’s A-highs, but its arrival makes it less likely that the SB holds an ace in the first place. Additionally, this turn could not have improved any of the SB’s hands to winners that were not already ahead, as any random middle-to-low card would have done. This turn card makes the SB somewhat more polar, but it does not affect the average equity of the BB’s bluff-catchers too much overall, nor does it cap his range. Perhaps the ace is not quite as bad a card for the BB as one might initially think.

Figure 12.3: SB and BB flop ending and turn starting ranges on 9♥-2♠-9♦-A♠.


Expert Heads Up No-Limit Hold ’em, Volume 2 Now, how might play go at equilibrium? In the pure PvBC case with the SB polar, the BB will check the turn and check-call sometimes. When he calls, he will again check-and-guess on the river. The SB will bet-bet (with GGOP sizing) with all of his value and enough bluffs to balance. How does this situation differ from the ideal PvBC case? There are several ways. First of all, our BB holds some slow-played three-of-a-kind. We expect to find that he plays these the same way as his bluff-catchers until the last possible moment. One might say that this is for information-hiding reasons. However, there is nothing fundamental about information hiding. Certainly, hiding information is not a good reason to take a particular line in and of itself. Rather, since the SB figures to put in plenty of action versus a range of mostly bluff-catchers, representing a weak hand is the way to get the most money in the pot. When the BB faces a river bet, he can call with any bluff-catchers he wants to show down, ending the action, and check-raise polar. In this way, the SB will not be able to take advantage of any information he gains when the BB finally splits his range. Second, the SB’s air does have a chance to improve on the river. When the BB was polar, we found that the SB was motivated to bet his bluff-catchers for protection when checked to on the turn, and the BB was thus incentivized to sometimes go for a turn check-raise. However, no analogous dynamic arises when the positions are reversed. In the ideal PvBC game, the action is checked to the player with initiative who then bets or checks, and when he checks, he has defined his range as complete air. When the BB was polar and checked the turn, the SB had the chance to act and attack the BB’s weak range. Here, however, if the SB checks the turn, he gets to see a river immediately. The BB has no chance to attack the SB’s weak distribution and prevent his air hands from seeing a river card. The SB’s weak hands thus always get their chance to improve. This is yet another benefit of position. Additionally, the previous dynamic meant that the BB could get to the river with value hands that missed a turn check-raise. Here, the SB does not have the same incentive to slow-play on the turn. If the SB indeed checks back only weak hands, what do the river situations look like? On low-to-medium card rivers, most of the SB’s checked back air hands will still be effectively air


Initiative and Less Common Turn LInes while the few that paired up on the river will beat everything except the BB’s nines. Thus, the BB will essentially have bluff-catchers plus traps, and SB will be fairly polar. (The river PvBC-plus-traps situation was covered in Section 7.2.5. How does it go again?) High-card rivers, on the other hand, improve significant portions of the BB’s range. These rivered pairs are ahead of almost all the SB’s hands as long as he did not slow-play on the turn, and so in this case, the BB will lead the river with a significant frequency. We will identify a couple other interesting things in this spot’s solution as well. The decision tree generation scheme we described in Section 12.2 is still in force, so we can jump to the results. First, the BB always starts with a check, as expected. The SB then bets over 90% of his range. This is a lot, but he has a lot of value hands here, and he needs more bluffs to balance when there are multiple streets of betting. GGOP sizing is 6BB, but he uses both half- and ¾-pot-sized bets, that is 4BB and 6BB bets, in about a 45:55 ratio. Most of his hands are bet both ways, i.e., played with mixed strategies, although there are a few exceptions. Most notably, his vulnerable 2-x hands all strictly prefer the larger sizing for protection-related reasons. However, unlike the K♣-7♥-3♦-K♦ case, where the BB’s larger-sized betting range was capped at 7-x, here the SB finds it valuable to balance his larger betting range with many strong hands as well. We have seen how the presence of draws can make a bettor go larger than the GGOP sizing for protection, but why might the SB want to bet smaller here? The reason has to do with the fact that the river card can have a significant effect on the distributions. In particular, a king (or to a lesser degree another high card) is quite strong for the BB. When the SB makes a bet on the turn, he might regret it if certain cards come on the end. By virtue of his position and the fact that he is fairly polar, the SB is in a better spot to make use of the river card information and will be able to play more profitably on average if he leaves a larger stack behind to play after seeing the river. Incidentally, the BB’s slow-plays are not the reason for the SB’s smaller bets here. As we saw in the river PvBC-plus-traps game, if the number of traps is relatively small, the polar player’s betting strategy continues to focus on his opponent’s bluff-catchers. In this hand, the SB’s river bet sizing is always all-in, so a smaller turn bet does not help him lose less versus the BB’s slow-plays. 175

Expert Heads Up No-Limit Hold ’em, Volume 2 All of the SB’s A-x hands have approximately the same equity regardless of the kicker. However, the SB tends to use the smaller sizing with higher A-x hands and the larger sizing when he has a lower kicker. Why might this be? What does the SB check behind on the turn? Interestingly, the SB’s Q-Q to 10-10 bet using the half-pot sizing, while his K-K checks back. In fact, K-K is significantly less valuable than the lower pocket pairs. Because K-high makes up a large fraction of the BB’s bluff-catching range, the BB is much more likely to hold a slow-play when the SB has K-K. The SB also checks back some weak showdown value holdings such as Q-J and K-x, and some flush draws and a few weak, unpaired holdings are indifferent to bluffing and sometimes check back. Now, the fact that some of the SB’s flush draws are indifferent to bluffing should come as a surprise given the discussion in Section 11.2 where we saw that, under certain conditions, draws are the first hands that begin to bluff. We will return to this point shortly. What happens after the SB bets? The situations following bets with the two sizings are quite similar, so we will focus on play after the 6-BB bet for the sake of discussion. The BB will fold enough to make the SB indifferent between checking and bluffing with some of his air hands. In particular, the SB is indifferent with a number of hands. Here are a few examples:

Q♦-7♦ with 24% equity after checking and 15% if its bluff gets called

10♦-8♦ with 14% equity after checking and 13% if its bluff gets called

7♦-3♦ with 15% equity after checking and 14% if its bluff gets called

7♠-3♠ with 37% in both cases

When it comes to contesting the pot after checking, low cards are actually better than some high ones, since they have clean outs. For example, the 7♦-3♦ captures more of the pot (about 22%) on the river after checking than the Q♦-7♦ (20%). All of these hands can be thought of as draws. They need to improve to win, and since so much of the BB’s calling range is unpaired, any improvement gives them a winning hand most of the time. The SB’s weak hands barely lose any equity at all if they bluff and get called. 176

Initiative and Less Common Turn LInes In the naive approximation, the SB is bluffing 6 BB to win 8 BB, so the indifference implies that his bet has to work 6/(6+8)≈43% of the time to break even. However, in the actual solution, the BB’s folding frequency is closer to 45%. This is somewhat unexpected. First, we expect the BB to need to call more than the naive frequency to keep draws indifferent to bluffing. Second, we expect draws of a single strength to be indifferent, while stronger ones strictly prefer to bet. We can chalk minor violations up to the subtleties of the river strategies where the BB has adjusted to keep the SB from being able to find an edge with any of these hands. However, there is another issue here. The BB can check-raise, and assuming the SB’s draws must fold to the raise, a raising range will have a much more punishing effect on stronger draws. Check-raises will incentivize draws to check back on the turn while having little effect on weak madehand bluffs. In fact, the BB uses his option to check-raise here. Facing the 6BB bet, the BB raises all-in with a bit over an eighth of his range, calls about 45% of the time, and folds the rest. He continues check-calling with all of his slow-plays, and his check-raising range is composed entirely of bluffcatchers. This unusual line is what we are going to look at next.

12.8 The Turn Protection Raise In the previous example, the BB check-called the flop on a 9♥-2♠-9♦-A♠ board. We gave the BB a starting range of bluff-catchers and some traps, and made the SB mostly polar. The BB checked his whole range on the turn, and facing a bet, he called, jammed, and folded with non-zero frequencies. His jamming range was composed of bluff-catchers, not slow-played value hands. Why might he do this? If the hand reaches the river, the SB’s semibluffs have a chance to improve. Jamming prevents weak draws from seeing a river and decreases the SB’s EV of bluffing. Essentially, this is a protection raise. Of course, raising for protection does not come for free – the downside for the BB’s bluff-catchers is that they lose the maximum when the SB shows up with a value hand, and they prevent the SB from putting in any more money 177

Expert Heads Up No-Limit Hold ’em, Volume 2 with bluffs. In the context of the simple PvBC situation, we can see that jamming is worse than calling with our bluff-catchers by looking at things from the bettor’s point of view. His bluffs have the same expectation (S−B) when we call as when we raise, but his value hands clearly prefer us to raise all-in. Conventional wisdom frowns upon betting or raising a hand when our opponent will call with all his better holdings and fold all his worse ones. Indeed, this is more or less what happens at equilibrium in the example above. Facing a jam, the SB’s hands break down into those with much more equity than is necessary to call and those with significantly less. Few of the SB’s holdings are made anywhere near indifferent by the BB’s check-raise. Given all these concerns, it is not clear how protection raising is at least as profitable as the BB’s other choices, and the fact that he does raise with significant frequency at equilibrium needs some explanation. We will attack this situation from several directions, starting with the equilibration exercise to explore the players’ tradeoffs in this situation. Hopefully, we will see when this move is actually good in practice. This discussion will also give us the opportunity to explore the matrix game concept further and to introduce an important new concept, the capture factor.

12.8.1 Why Jam? A Matrix Model There are a couple of important effects here that arise from changing hand values. First, as we already mentioned, there is a protection effect in play – jamming allows the BB to fold out a lot of the SB’s equity. Raising essentially protects the BB’s decent but vulnerable hands from both the chance that the SB’s draws will improve and the possibility that the SB can take advantage of this in the river betting round. This was an especially important factor in the previous example, because all of the SB’s bluffs had a lot of equity, since much of the BB’s calling range was unpaired. Second, if the BB’s check-raising bluff-catchers have a few outs when called, that helps a lot as well. Either of these things by itself can make protection raising a viable strategy, but when combined, they make it pretty common. Let us look at the first effect with a simple model. That is to say, we will essentially work with the PvBC game, but we will take into account the fact 178

Initiative and Less Common Turn LInes that the BB’s bluffs are almost all draws that keep their equity if called. The first steps here represent only small tweaks to things we have seen before, so we will move quickly. The exercise will also help us learn more about using matrix games to understand equilibriums. Let S, P, and B be the stack, pot, and SB bet sizes as usual. Let C, J, and F be the BB’s calling, jamming, and folding frequencies when facing a bet. So, C+J+F=1. Suppose the SB’s draws have equity E, and they keep all of it if they bluff and get called. Then, neglecting the river action for simplicity, we can approximate the SB’s EV of betting his value hands and of checking and betting his draws (i.e., bluffs). There is no need to worry about his EV of checking with value, since he never will do that. We have since the SB’s value always wins the pot, it gets another B when BB flats a bet, and gains another S when the BB jams. Also, and When the SB checks, he just realizes his equity (since we’re neglecting the river action), and when he bets, he wins the pot when the BB folds, loses his bet when the BB jams, and realizes his equity in the larger pot when the BB calls. Note that we are assuming the SB’s weak draws must fold to a jam. Now, we can organize this information using a matrix. We need to identify each of the combinations of pure strategies the players can choose, and write down the EVs for each of them. The BB has three pure strategies – he can call, jam, or fold with his bluff-catchers. We will take it as given that the SB always bets his value hands, so he has two pure strategies to choose from: bet-draws and check-draws. There are six combinations, and we need to find the players’ average EVs on the turn for each of them. Of course, this depends on what fraction of the SB’s range consists of value hands and what fraction constitutes draws. Let’s say V of his turn starting range is value, and the remaining 1−V is draws. Then, his average EVs for his two strategies are just the weighted averages of his EVs with his individual types of hands. We have 179

Expert Heads Up No-Limit Hold ’em, Volume 2

We can now plug in numbers to find all six EVs. We have S=21, B=6, and P=8, and let’s say E=0.25 and V=0.2. By the way, you can verify that with these numbers, the SB does not have the odds to call all-in when he faces a turn check-raise with his draws, so our assumption that he folds to the jam is good. To evaluate the SB’s EVs when the BB plays call, we let C=1, J=0, and F=0, and similarly when the BB plays fold or jam. Filling out the table is then just arithmetic, and the result is shown in Figure 12.4.

Figure 12.4: The SB’s payoffs for each of the six pure strategy combinations in a simple model game. The SB can choose which column to play, and wants to pick the one that achieves the largest value. The BB chooses which row to play and prefers small SB values.

The first point here is that when the SB plays bet-draws, he clearly prefers that the BB calls rather than jams. If we had set E=0, so that the SB’s bluffs were all completely valueless holdings, we would have found that each number in the first row of the matrix was smaller than the one directly below it. That is, we would have found that calling beats jamming for the BB, no matter what strategy the SB plays. (We would say that call dominates jam.) That is not the case here. Jamming is sometimes better than calling for the BB’s bluff-catchers. Now, organizing the pure strategy EVs in a table like this makes the equilibration exercise easy. This can help us guess the indifferences that might arise at equilibrium, although as we will see, it is important to be careful. Consider the following. Any square in the matrix gives the SB’s EV for some particular combination of the players’ pure strategies. If the BB chooses his pure strategy, then that fixes the row of the matrix we are playing in, but the SB can still choose a column (i.e., move left or right in the matrix) to find the strategy that maximizes his EV. On the other hand, if the SB’s strategy is fixed, then that fixes the column we are playing in, but the BB can still move play up and down in the matrix to find his best strategy – he 180

Initiative and Less Common Turn LInes wants the one that gives the smallest SB EV. Let’s see how the sequence of maximally exploitative adjustments will play out. Follow along in Figure 12.4. Start off with the SB playing bet, i.e. choosing the first column. The BB’s best response is to choose the row that gives the smallest number in the first column. The number is 22 BB, corresponding to the BB strategy jam. Can the SB adjust to improve his EV? Yes, he can move to the right to improve from 22 BB to 28.4 BB. Can the BB profitably readjust? The smallest number in the right column is 23.8 BB, so the BB begins playing fold. And so on. So, starting from the case where the SB is playing bet, if we label the pair of pure strategies as an ordered pair containing the SB’s play and then the BB’s play, the sequence of strategies goes 1. The BB switches to (bet-draws,jam) for a SB EV of 22 BB 2. The SB switches to (check-draws,jam) for a SB EV of 28 BB 3. The BB switches to (check-draws,fold) for a SB EV of 23.8 BB 4. The SB switches to (bet-draws,fold) for a SB EV of 28.6 BB 5. The BB switches to (bet-draws,jam) for a SB EV of 22 BB At the end, we are back where we started, and the process repeats. The BB switches between jamming and folding his bluff-catchers, and the SB switches between checking and betting. This cycle goes through four of the six boxes, and we can check the other two as well to see that there is always some incentive for at least one player to deviate from any pair of pure strategies. Thus, the equilibrium here must be mixed. Which strategies will actually be used at equilibrium? Consider the SB first. We know he is playing a mixed strategy, and he only has two pure strategies to choose from, so his only option is to mix between betting and checking at the equilibrium. As for the BB, it is tempting to hypothesize that he is mixing between jam and fold but is never calling, since those are the two pure strategies he alternated between above. Suppose this is the case. If he is playing both of those strategies at equilibrium, he must be indifferent between them. When the SB plays bet-draws, the BB prefers fold by 28.6−22=6.6 BB, and when the SB plays check-draws, the BB prefers jam by 28.4−23.8=4.6 BB. (Notice how we can use the rule for 2-by-2 matri-


Expert Heads Up No-Limit Hold ’em, Volume 2 ces after we have narrowed down the options to two pure strategies each.) So, the BB is indifferent when the SB plays bet-draws 4.6 times for every 6.6 times he plays check-draws. In this case, the SB’s average payoff is [(22×4.6)+(28.4×6.6)]/(4.6+6.6) =25.77 BB. This is clearly a bad result for the BB, since the SB is making more than he could possibly expect if the BB simply played call! Thus, even if the BB plays as well as possible using just his jamming and folding options, he will be incentivized to begin calling. The situation here is as follows. The BB’s maximally exploitative responses to the SB’s two pure strategies are jamming and folding. If the SB never (semi-)bluffs, then the BB should always fold to a bet, and if the SB bets all of his draws, the BB does best to jam for protection. However, that does not necessarily mean that one of the two must be the BB’s best response when the SB is mixing between jamming and folding. The BB’s third option, call, turns out to be best against some mixed strategies here. Suppose the BB is constrained to playing jam-or-fold facing a bet on the turn. How do the equilibrium frequencies compare to the naive PvBC solutions? Why is this? Unfortunately, I know of no easy trick to simply read off mixed-strategy solutions to matrix games that are larger than 2-by-2. It turns out that the primary hang-up is in determining exactly which strategies the players are mixing between. In this case, we know the BB is not mixing between jamming and folding his bluff-catchers, because we explicitly checked, and it did not work out. Intuitively, we know he must sometimes be folding. Thus, he must be mixing between calling and folding. (The SB’s strategy only has one degree of freedom, so he can’t generally make the BB indifferent between all three of his options.) The fact that the BB is indifferent between calling and folding at the equilibrium must tell us something about the SB’s strategy. When the SB plays bet-draws, the BB prefers call by 28.6−23=5.6, and when he plays check-draws, the BB prefers fold by 25.4−23.8=1.6. So, the SB must be playing check-draws 5.6 times for every 1.6 times he plays bet-draws. In other words, whenever he finds himself holding a draw on the turn, the SB’s unexploitable play is to turn it into a bluff 1.6/(1.6+5.6)=2/9≈0.22 of the time. We will not give more general methods for dealing with large matrix 182

Initiative and Less Common Turn LInes games in this book. It is not particularly difficult, but it would be something of a digression, and plenty of good resources are already available on the topic.5 Here, we will simply give the solutions to matrix games when necessary. Of course, you do not need to take our word for them – once you have the strategies, it is easy to verify that they are equilibria by checking that neither player can improve his EV by deviating. We know the SB is mixing between checking and betting at equilibrium given the matrix game in Figure 12.4. What does this tell us about the BB’s GTO play?

12.8.2 Exploitative Protection Jamming and the Capture Factor The previous model showed us how jamming could be reasonable under certain conditions but not how it could coexist with both calling and folding at equilibrium. To see that, we need a model that more accurately captures certain aspects of the game. Now, we could work harder to write down the exact EVs of various holdings, and we will. However, the crucial element turns out to be that the SB has multiple different kinds of bluffing hands – say, pair draws and flush draws as in the previous example, or draws and weak made hands, as we’ll consider here. By folding some and then picking the right balance between jamming and calling when he does continue, the BB can make both types of SB bluffs indifferent. Thus, our work here will not only have consequences for the BB’s protection raising, but also for the SB’s bluff selection. But we’re getting ahead of ourselves. In this section, we’ll look to make more accurate estimates of the relevant EVs and use them to look at the BB’s exploitative raise-vs-call decision. In the next section, we will take a more equilibrium-oriented approach. When exactly does he prefer jamming to calling? It depends on the SB’s The book The Compleat Strategyst by J. D. Williams gives an elementary introduction to two-player, zero-sum matrix games. Recordings of a Yale University course entitled Game Theory by Professor Ben Polak are highly recommended for anyone who prefers video to text. Both of these high-quality resources are freely available online. 5


Expert Heads Up No-Limit Hold ’em, Volume 2 betting range, which we will assume is composed of draws, weak made hands, and value hands. We will suppose that draws keep all of their equity and weak made hands lose all their equity after bluffing. This last assumption is not great for the previous hand example, where the SB’s weak, unpaired holdings also act like draws. It fares much better on most other boards, where the BB’s ranges will not contain nearly as many unpaired hands, and the SB’s weakest holdings cannot count on pair outs after their flop and turn bluffs get called. As far as the EVs, here are a few issues the previous model ignored: 1. The BB’s bluff-catchers have a significant chance to improve on the river, even if they go all-in and get called. This makes jamming and calling both better as compared to folding. 2. If there is money behind at the end of turn play, there is potential river action, and it matters. We have seen that bluff-catchers are not generally able to realize much of their equity in the pot when there is another round of betting. This makes calling significantly worse than in the previous model. Both of these effects make jamming better for the BB as compared to his other options. We need to write down the BB’s EVs on the turn. Let S, P, and B be the stack, pot, and bet sizes, as usual, and let D, W, V be the fractions of the SB’s betting range that are draws, weak made hands, and value hands, respectively. In order to account for point 1 above, let EQBB{vs value} be the BB’s equity with a bluff-catcher versus the SB’s jam-calling range (i.e., his value hands and a few strong draws). Then, once he faces a turn bet, the BB’s EV of protection check-raising all-in is Whenever the SB has a bluff, he folds to the jam, leaving the BB a stack of (S+P+B), but when he has value, he calls, and the BB gets his equity in the pot. On the other hand, if the BB calls, his EV is:


Initiative and Less Common Turn LInes This deserves a bit of explanation, since we stuck something new in here. Generally, we can write the BB’s EV of calling as the amount of chips he has left after the call, (S−B), plus the size of the pot after he calls, (2B+P), times the fraction of that pot that he ends up capturing, which we have called R. Now, if the bet were all-in or if the river were guaranteed to check through, then R would just be the BB’s equity. However, due to the future street of betting, the BB’s bluff-catchers will not be able to realize all of their equity in the pot. It turns out that R≈0.30 – that is, he is able to capture about 30% of the pot at equilibrium. This is somewhat less than the 48.4% equity in the pot that he would win if the players were forced to check down on the river. We will call R the capture factor, because it describes the amount of the pot a player is able to capture. Now, this number might be bigger if the BB can play exploitatively on the river. Also, in general, it will not just be a single number – it will depend on the hand, just like equity depends on the hand – although for now, we can get away with a single number, since the BB holds mostly nearly-equivalent bluff-catchers here. Generally, a first good guess at a hand’s capture factor is its equity, and then we can adjust that estimate according to the type of hand and expected future action. We know that air hands tend to more or less break-even with respect to just giving up, so they will usually capture none of the pot on average – that is, they have R≈0. Nut hands will have R>1, since they win more than the whole pot if there is money left behind to bet. Bluff-catchers will generally capture somewhat less than their equity, as is the case here (here, the BB’s bluff-catchers realize about 0.30/0.484≈62% of their equity). Notice that the chance that the SB’s drawing hands improve on the river has been subsumed into the capture factor. That is, R is not only a function of equity. It also depends on things like playability of all the hands in Villain’s range. In fact, it accounts for lots of effects. In some sense, all of the multi-street effects that are difficult to account for are represented in that one number, so we have not really solved any problems – we’ve just shifted all the hard work into the problem of finding R. That said, there is a lot of practical utility here. More on this later. The EVs we wrote above did not rely on any equilibrium-based arguments. We can see if jamming is better than calling by determining whether 185

Expert Heads Up No-Limit Hold ’em, Volume 2 EV(jam)>EV(call). The result does not simplify to anything especially nice, so we will not reproduce it here. Anyhow, it expresses a trade-off between the amount of equity the SB’s draws are folded off of by a jam and the fraction of the pot the BB’s bluff-catchers expect after calling. The more high-equity the draws that the SB folds, the more the BB likes a jam. The more of the pot he captures after calling, the more he likes to call, of course. The result also depends on the composition of the SB’s betting range. The higher his value-hands-to-draws ratio, the more equity the BB needs to be folding out to find a profitable shove. On the other hand, the more equity the BB has when his jams do get called, the less equity he has to fold out to make shoving good. In our 9♥-2♠-9♦-A♠ example, the SB’s draws need only have modest equity before the BB begins to prefer jamming to calling. Of course, if the SB is bluffing with fewer draws and more weak made hands, the draws he is bluffing with will need to have more equity to make up for it. What is the general requirement for EV(jam)>EV(call), given the above EVs? It involves a number of parameters. Can you think of a more useful way to express it than as a constraint on the equity of the SB’s draws?

12.8.3 Jamming and Calling: Equilibrium Play Facing a bet on the turn, the BB can call, raise, or fold. The bluffing indifference, essentially, tells us that the BB’s bluff-catchers must sometimes be folding and sometimes not-folding at equilibrium. However, he can make the SB’s bluffs indifferent through either of his not-folding actions. When exploring equilibrium play in SB barreling spots previously, we assumed not-folding meant calling. However, under certain conditions, the BB prefers jamming to calling with his bluff-catchers facing a turn bet. If the BB stops calling on the turn, and jams instead, what consequences does it have for the SB’s strategy? Certainly his value hands still do well to bet. And, at equilibrium, he will still bluff. However, his choice of bluffing hands changes. If the BB is playing jam-or-fold versus a bet, then the SB’s EV of bluff-folding is the same with draws and weak made hands – he 186

Initiative and Less Common Turn LInes never sees a showdown, so his particular holding does not matter. The EV of checking, however, is better for draws, since they will be worth more on the river. Thus, versus a BB who is not flatting bets, the SB should fill out his bluffing range starting with those that capture the least of the pot on the river. This is different than what we found when the betting initiative was not allowed to change hands. So, we can think of the turn-protection raise like this – generally, the polar SB will bet the turn with value and bluffs. He will fill out his bluffs starting with his draws, and the BB will play check-and-guess. However, if conditions are right for re-raising – stacks are short enough, bluffs have equity they can be pushed off of, the BB’s bluff-catchers have some outs versus the SB’s value, etc. – then the dynamic changes. The BB prefers to play jamor-fold versus a turn bet. But, the SB will then be motivated to bluff with weak hands instead of draws. Then the BB’s jams lose their protection effect. The SB is no longer pushed off draws, and the BB is incentivized to return to playing call-or-fold versus a bet. And so on. Thus, nearly-polar, nearly-static turn play will often proceed as before, without any initiative switches. However, if the SB’s using draws to fill his bluffing range makes the BB want to jam rather than call the turn, then the nature of play will change. The BB will be motivated to raise for protection. (We can check when the BB might want to incorporate any jamming into his strategy using EV equations such as the ones in the previous section.) However, he can’t play strictly jam-or-fold, or else the SB’s response will motivate him to call rather than jam. Thus, at equilibrium, he will be indifferent between all three of his choices. Similarly, at equilibrium, both the SB’s draws and his weak made bluffs will be indifferent to bluffing. He cannot bluff all draws, since the BB’s counter-adjustment (playing jam-orfold) will incentivize him to deviate, and similarly he cannot bluff only weak made hands. Now that we have a few indifferences whose origin we understand, we can solve them to find some GTO frequencies. Because BB calling and jamming have the same effect on the SB’s weak made hands, while folding has the same effect on SB’s draws and weak hands, we can break down the process of finding the BB’s frequencies into two steps. First, he can make the weak


Expert Heads Up No-Limit Hold ’em, Volume 2 made hands indifferent by choosing an appropriate folding frequency. Then, within his remaining not-folding hands, he can adjust the ratio of calls to jams to make the draws indifferent as well. If we are in the BB, we can choose our folding frequency to make the SB’s air indifferent to bluffing and checking. However, his EV of bluffing draws is equal to his EV of bluffing weak made holdings only when all of our not-folds are jams, i.e., when we do not call any. How can we still make his draws indifferent to bluffing when some of our not-folds are calls? Now, we can solve for the BB’s GTO frequencies, and we will use numbers corresponding to our 9♥-2♠-9♦-A♠ example. In that hand, the SB has essentially two types of bluffs. He has weak, unpaired hands with pair draws, and he has flush draws that have significantly higher equity. The BB will call, fold, and raise so as to make both types of draws indifferent to bluffing. The SB EVs here will be similar to those at the beginning of this section, just with a more detailed accounting of the various possibilities corresponding to the deficiencies in the first model, which we listed above. A number of equities are necessary to write down the EVs here. Let:

EQSBcheckdraw=0.34 be the equity of the SB’s flush draws if they check back the turn (i.e., their equity versus the BB’s turn starting range).

EQSBsemibluff=0.30 be the equity of the SB’s flush draws if they bet and get called (i.e., their equity versus the BB’s check-calling range).

EQSBdrawallin=0.32 be the equity of the SB’s flush draws if they bet the turn and then call an all-in.

EQSBcheckweak=0.16 be equity of the SB’s weaker bluffs if they check back the turn.

EQSBbluffweak=0.14 be equity of the SB’s weaker bluffs if they bluff and get called.

EQSBjam=0.96 be the equity of the SB’s value hands after they bet and call a jam.

EQSBcall=0.88 be the equity of the SB’s value hands if they bet and get flat-called.


Initiative and Less Common Turn LInes The first two equities, EQSBcheckdraw and EQSBsemibluff, are equal in the simple case where the SB’s draws keep all their equity after bluffing. The fifth, EQSBbluffweak, would be 0 if the SB’s weak made hand bluffs lost all their equity after bluffing. Here, however, they keep most of it. The last two equities correspond to the SB’s value hands. They should be close to 1 but are somewhat less due to the BB’s chance to improve on the river. The SB has various value hands, and his draws are not all perfectly equivalent either, but these numbers are chosen to be representative. Notice that the SB’s value hands have more equity when facing a jam than a call, since the BB check-calls all of his slow-played three-of-a-kind. Now for the EVs. How should we account for the river action? To good approximation, we can write the EVs of any SB hands that are air on the end as if there were no river action, since they will be indifferent to bluffing. The SB’s value hands, however, gain from the river betting, so we need to find their capture factor. Strong value hands will expect to win the whole pot plus another bet whenever the BB calls a river bet. If the river bet is of size b into a pot of p, we know the BB’s calling frequency will be something like p/(b+p). So, a value bet will capture approximately p+[bp/(b+p)]. (Previously, in neglecting the river action, we just awarded him the pot p.) Dividing by p, we find that the fraction of the river pot captured by the SB’s value hands is


Note that this has the same numerical value whether the turn checks through and the bet is 6BB into 8BB on the river or whether a bet goes in on the turn leaving 15BB into a pot of 20BB – these two cases could have been different if we were not using GGOP sizings. So, the SB’s value hands win more than the whole pot on average, as expected. The closer the situation is to the true static PvBC case, the better approximation this will be. In reality, at equilibrium here, the SB’s highest-value hand, A-A, has a capture factor (CF) of about 1.42 after the BB calls a turn bet. K-9o, a very strong hand that is still vulnerable to the flush draw, has a CF of 1.27. Finally, the SB’s A-x hands have a CF of only about 1.13, since they are vulnerable to some draws and lose to the BB’s slow-plays. So, our approximation looks to 189

Expert Heads Up No-Limit Hold ’em, Volume 2 be right on the money for the pure nuts, but the CFs of non-nut value hands fall off fairly quickly. As for the SB’s flush draws, we will assume that when they improve on the river, they will get 1.43 times the pot, just like the SB’s other value hands, and we will assume their equity is the chance they improve on the river. The same goes for the SB’s weak hands – it is just that they improve to value less frequently on the end. With this in mind, we can write the SB’s EVs. We have

In these two, the capture factor accounts for the value the SB gets when he improves on the river. Now, his EV of value-betting is just a case analysis – the amount of chips he ends up with after each of the BB’s possible responses, weighted by the frequencies with which the BB takes each action. The additional value the SB achieves with a river bet is again accounted for by Rvalue here, and so is the fact that the SB does not always win the hand if he gets called or jammed. Finally,

We assume here that bluffs fold if they face a turn all-in while semi-bluffs have the odds to call it off. Notice that we have neglected the BB’s slowplays in writing down these EVs. We have seen before that if there are few of these and stacks are sufficiently shallow, they will affect profits but not players’ strategies. We have accounted for each of the deficiencies in the original model listed above. The fact that the BB’s bluff-catchers have some chance to win on the river is incorporated into the SB’s various equities. The SB’s two different types of bluffing hands are modelled explicitly. Finally, we have accounted for river action, as well, using the capture factor. Now, we want to find the BB strategy that makes the SB indifferent to bluffing with each of his holdings. That is, we want to find C, J, and F such


Initiative and Less Common Turn LInes that

We have three equations and three unknown variables, so we can solve. Plugging in for the equities and sizings in this hand, we find J=0.14, C=0.42, and F=0.44. These numbers agree with the computationally-generated equilibrium to within a percent. The result is quite sensitive to the amount of equity had by the SB’s draws facing an all-in, EQSBdrawallin. The higher the equity, the more the BB needs to jam to keep draws indifferent to semibluffing the turn. In summary, the BB’s primary turn plays after check-calling the flop with bluff-catchers are fold, call-call, and call-fold. Generally, he is indifferent between all of these at equilibrium: when he checks the turn and faces a bet, they all have an EV of S. However, even in the ideal PvBC game, as stacks get shorter and shorter, his EV of jamming the turn gets closer and closer to his EV of calling. Several additional factors can push it over the edge: the equity of the SB’s bluffs, the BB’s equity versus the SB’s value, and poor playability of bluff-catchers on the river. Remember that in multistreet situations, a polar player can include many bluffs in his betting range. In the example above, he bet more than one bluff for each value hand on the turn, even with relatively modest bet sizing. If a jam makes the SB fold all of these bluffs, each of which had a significant amount of equity, then the result can be a large win for the BB. Generally, a turn-jamming BB will motivate the SB to re-compose his bluffing range and begin using his weakest hands to bluff rather than his draws. Another option is for him to change his bet sizing. We have seen that GGOP sizing lets him include as many bluffs as possible in his betting range. By deviating and, say, going larger on the turn and smaller on the river, he can decrease his number of bluffs and thus make the BB’s jams less profitable. This is part of the reason we saw the polar player bet larger than GGOP with much of his range in the solution to the K♣-7♥-3♦-K♦ spot.


Expert Heads Up No-Limit Hold ’em, Volume 2 The BB’s turn jamming range is composed of his best bluffcatchers, generally speaking: e.g., K-Qo in this section’s hand example. His slow-played nines do much better at equilibrium by continuing to call down on the turn. Why are his best bluffcatchers better than mediocre ones for raising, and why do the actual value hands prefer to call, given the BB’s EVs we wrote down above? Suppose you are building a double-barrelling range in position. How would you choose your turn bluffs if the BB tends to play call-or-fold? What if he frequently check-raises? This discussion foreshadows strategies we will encounter on earlier streets and in larger model games. We’ve seen that GTO strategies will often make Villain indifferent to certain decisions. We will often be able to do this more effectively when we allow ourselves more strategic options. We saw in the previous chapter that in SB barrelling situations where the BB can only call or fold the turn, he can generally only make one sort of SB hand indifferent to bluffing. Here, however, since the BB has two independent frequencies to adjust (his calling and jamming frequencies), he is able to make two sorts of holdings, weak hands and draws, indifferent to bluffing. Now consider the real game, or at least a larger model. Suppose we’re making a flop c-bet. Not only can Villain adjust calling and raising frequencies on the flop, he can also adjust tons of subtle details of his strategy on all later streets. Maybe he can make our bluffs indifferent by folding enough to the c-bet, while making our draws indifferent by changing how much he pays off on run-outs when draws come in, while making strong made hands indifferent by playing aggressively enough versus missed c-bets, etc. Thus, we can expect to see lots of indifferent hands (and thus very highly-mixed strategies) in early-street play.

12.9 You Should Now … ♠


Understand the connection between PvBC distributions and the betting initiative.

Initiative and Less Common Turn LInes

Understand how hand protection issues can motivate players to seize the betting initiative with mediocre hands, even in nearlystatic, nearly-PvBC spots. We focused in particular on the SB’s delayed c-bet the BB’s turn check-raise and we now know how these moves are used at equilibrium and exploitatively.

Estimate equilibrium frequencies in spots where each player is mixing between two pure strategies.

Understand how the possibility of getting raised affects the SB’s double-barrel bluff hand selection.

Know what a capture factor is.


Chapter 13 Turn Play: Volatile Boards and Capture Factors

Empty your mind, be formless, shapeless – like water. Now you put water into a cup, it becomes the cup, you put water into a bottle, it becomes the bottle, you put it in a teapot, it becomes the teapot. Now water can flow or it can crash. Be water, my friend. – Bruce Lee The primary purpose of this chapter is to gain a more practical understanding of GTO turn (and river) play. We have covered most of the analytically-tractable models we’ll see in this book, and we should now have enough background to understand what we see in some real results. We also want to build a few more practical tricks and rules of thumb for real-time play. In particular, we’ll focus on using the capture factor to make decisions quickly. It turns out that we can use many of the singlestreet exploitative decision rules in multi-street situations, simply by replacing the equity with the capture factor. The capture factor gives us a way to incorporate multi-street effects, approximately, in a way that is easy enough to use at the tables. In this sense, substituting the capture factor for equity is like substituting implied odds for pot odds. Hopefully, however, we will see that the capture factor can be applied to a wider variety of decisions than implied odds.


Turn Play: Volatile Boards and Capture Factors Let’s start off with a hand example. We will start with a bit of hand reading to choose the turn starting distributions and then consider the computationally-generated equilibrium play. The decision trees used in this chapter are generated according to the rules described in Section 12.2.

13.1 Example: Turn Play after a Flop c-bet on K♣-Q♠-8♥-5♥ 100BB deep

13.1.1 The Setup Consider the following hand Effective stacks: 100BB Pre-flop: SB raises to 3BB, BB calls 2BB Flop: K♣-Q♠-8♥ (6BB) BB checks, SB bets 4BB, BB calls Turn: 5♥ (14BB)

This is Example 3 from the River Play (see Section 7.1.3). Here, we will step back a street and consider both the turn and river strategies. Previously, we characterized this flop as static and strong for the BB and mentioned that this is a common spot for the SB to c-bet once and then give up if called. Considering that, it would be reasonable for the BB to look for excuses to out-of-position float here. Of course, it is easy to think of counteradjustments for the SB. Previously, we considered river play after the turn checked through. If the BB never leads the turn after check-calling the flop, his range at the beginning of turn play is the same as that at the beginning of the river. We’ve no real reason to expect a BB to want to lead often on the 5♥, and any leading here will be very player-dependent, practically speaking. Thus, we will as195

Expert Heads Up No-Limit Hold ’em, Volume 2 sume that the BB starts the turn with the river starting range we justified previously, and later, we can check our solution to see if our assumption about his turn-leading range is true.

This is all the queens and eights in his pre-flop defending range, weaker kings, and some draws and slow-played two-pair hands as well. As for the SB, we will assume he is c-betting frequently but prefers to check back bluff-catchers. In particular, we will give him a pre-flop opening range of the top 85% of hands – somewhat conservative since he opens 3BB – and then we will assume that his c-betting strategy effectively removes from his range 50% of each hand combination with absolute strength A-high through J-J. That is, we will assume he c-bets each of those hands half the time. In this way, we do not make too many assumptions about the SB’s play: we make some nod towards common tendencies, but we assume he can get to the turn with lots of hands. The turn starting distributions for both players are shown in Figure 13.1. It is clear that BB’s range is relatively strong. The SB holds most of the same hands the BB does – he just holds a lot of air too. On the other hand, the BB’s range is capped high, since we assumed he cannot have any of the sets or top two pair. Notably, the SB also holds a lot of turned flush draws that BB folded on the flop. Stacks are deep and the ranges overlap a lot, so it will be interesting to see how things play out.

13.1.2 The Solution Turn play begins with 14 BB in the pot and 93 BB behind. Consider the BB’s initial decision point. He mostly checks, but he makes 1/5− and 1/2−pot bets with single-digit frequencies. His ranges are highly mixed, and his betting ranges have nearly the same composition as his checking range. However, he has a slight preference towards leading with strong draws (e.g., J♥-9♥) and good pairs.


Turn Play: Volatile Boards and Capture Factors

Figure 13.1: Turn starting distributions for our example on the K♣-Q♠-8♥-5♥ board.

The fact that so many hands are played with mixed strategies, despite his checking the vast majority of the time, means that almost every hand in the BB’s range is indifferent between betting and checking at equilibrium. The SB’s strategy must be finely tuned to make this true, and small deviations in the SB’s play will have a large effect on the BB’s best response. Knowing all this, it would be reasonable for a human player to just check his whole range for simplicity (in the absence of exploitative reads). However, these results indicate that leading is also reasonable here, contrary to standard play. Does the standard approach exploit population tendencies, or is there another reason that leading (with small sizings) is rare here? Let us look at how exactly the SB’s strategy makes one of the BB’s hands indifferent to leading. This will let us see how the SB responds to a lead generally and when the BB might be able to play exploitatively. Consider the BB’s K♥-8♦. With about 93% equity at the beginning of the turn, it is the strongest made hand in his range. He will very rarely be looking to fold it, and if the EVs of taking two actions with this hand are equal, then the SB figures to be putting approximately the same amount of money in the pot on average against both. Now, the BB actually takes a number of lines on the turn with this hand. He does not just bet or check, but he bet-calls, bet-3-bets, check-calls, check-raises, etc. Let’s investigate a couple of ex197

Expert Heads Up No-Limit Hold ’em, Volume 2 treme choices: the check-call and the bet-3-bet all-in. First, the check-call. The BB checks about 2/3 of his K-8 combinations on the turn. Facing the check, the SB frequently checks back. When he bets, he uses his 3/4-pot and full-pot options – he almost never bets half-pot or over the pot. More specifically, when the BB checks K♥-8♦:

The SB checks back about 72.6% of the time, in which case we get to the river with a 14BB pot and 93BB behind. The BB expects to capture about 1.15 times this pot on the river for an EV of about 93+(1.15×14)=109 BB.

The SB bets 3/4-pot about 4.8% of the time, and we get to the river with a pot of 35BB and stacks of 82.5BB. At equilibrium, the BB’s capture factor here is about 1.20, for an EV of about 82.5+(1.20×35)=124.5 BB.

The SB bets full pot about 22.6% of the time, in which case we get to the river with a pot of 42 BB and 79 BB behind. At equilibrium, the BB captures 1.07 times that for an EV of about 79+(1.07×42)=124 BB.

Clearly, the BB hopes to see a bet after he checks the turn with this strong hand, and when he does, he almost always just calls, keeping his strongest hand combined with other hands he plays passively. Regardless of whether a bet goes in on the turn, it expects to win a bit over the size of the pot when it sees a river. Overall, his average EV when he goes for the check-call is approximately Now, what happens when the BB goes for the bet-3-bet all-in with his K-8 on the turn? Actually, he uses two bet sizes, but focus on the 1/2-pot lead for now. Facing this, the SB:

Folds 49.0% of the time, so that the BB ends up with 107 BB.

Calls 40.0% of the time, in which case the BB captures 1.17 times the 28-BB pot in addition to his 86-BB stack for an expectation of 118.9 BB.

Raises to 21 BB 1.6% of the time. Facing BB’s subsequent jam, he


Turn Play: Volatile Boards and Capture Factors folds around 2/3 of the time (the BB’s K♥ blocks a lot of his calling range) and calls the rest, leaving our BB an average equity of 123.4 BB here.

Raises to 28 BB 9.4% of the time. Facing the BB’s jam, he folds about 57% (his wider range for getting here is less affected by card removal) giving the BB an average EV of jamming of about 120.6 BB.

These two raise sizes correspond to half-pot and 3/4-pot raises. The SB does not use his min-raise or larger betting options when facing the lead. The BB’s average EV when he goes for the bet-3-bet is thus This is the same as his EV when he goes for the check-call, as expected. So, the BB leads occasionally, but he mostly checks. This makes sense, judging by the distributions. His average equity is quite high, simply because he always has “something” after calling the flop, but he has few very strong hands. We briefly mentioned the SB’s frequencies after the BB checks K♥-8♦, but now let’s take a closer look at his play facing the check. His threequarters and pot-sized bets are much smaller than the GGOP choice, since the distributions are nowhere near PvBC, but are still very large. His betting ranges are quite polar. They contain more or less no hands with equities between 25% and 75% versus the BB’s checking range. His value betting cutoff (i.e., his weakest value bet) falls among his weak kings, and he has some tendency to use the larger sizing with stronger hands. He bets everything with above about 80% equity, and as a consequence, his checking range is capped. The river can strongly affect hand values on this board, so the fact that his checking range is capped does not necessarily mean his river starting distribution is, and he simply needs to get value with his strong hands. Certainly, the BB leads the river a significant amount of the time after the SB checks back the turn, especially on “blank” rivers. Facing a bet, the BB’s strategy includes calling, folding, and jamming. It keeps both the SB’s bluffs and various semi-bluffs more or less indifferent to betting the turn à la Section 12.8. His range for check-raising seems fairly standard: some two-pair hands and some semi-bluffs for balance. 199

Expert Heads Up No-Limit Hold ’em, Volume 2 This contrasts with the solution in Section 12.8, where the BB check-raised for protection while continuing to trap with stronger hands. It is important to understand the reasons the BB splits his ranges so differently in these two hands. Previously, the SB’s nearly-polar distribution allowed him to barrel very frequently. In fact, he jammed the river with almost all hands that would have called a turn raise and with some bluffs on top of that, so of course nut hands maximized their value by continuing to slow-play. Here, the SB does not bet the river nearly as often, and he holds many more thin value bets that we can put in a tough spot and get extra value from by raising. These raises are sufficient to keep the SB’s barrelling frequencies and distribution contained so that the BB no longer finds profitable raises with bluff-catchers. Lastly, out of curiosity, how do the equilibrium ranges here compare to those we assumed when we considered river-only play of this hand after a turn check/check? First, how do the “standard” starting distributions we assumed previously stack up versus the ones here? If they are the same, then the subsequent river play is likely to be more or less the same as well. We have already said that the BB checks the turn with most of his starting range, so the BB strategy matches up. As for the SB, the main effect of a check on the turn is to remove the top of his range. Indeed, the SB ranges we assumed, which included most air hands but were capped at weak kings, are actually very similar to the ranges found here. So – chalk one up for standard play. On the other hand, an equilibrium for a turn-and-river-only situation may still not necessarily be representative of GTO play of the full game. In particular, the fact that the SB is forced to give up with so much air here may very well mean that he should not have c-bet the flop so frequently in the first place. And if the SB c-bet this turn more often, perhaps the BB would no longer need a leading range to get value with his strong hands. Appropriate flop c-betting frequencies will certainly be a theme of our discussion of flop play.


Turn Play: Volatile Boards and Capture Factors

13.2 A Philosophical Note on the Program of Solving Subgames Sometimes, we start with the assumption that Villain is a poor player, and we make some strong assumptions about his pre-flop and flop play in order to choose the turn starting ranges. Then, we try to find the equilibrium turn and river play given those ranges. At least two objections might come to mind. First, if Villain is bad, and we have enough reads to confidently choose his turn starting ranges, perhaps we should not be worried about GTO play at all and should just try to play exploitatively. Second, if the turn starting ranges we have assigned are unlikely to closely resemble the equilibrium ranges of the full game, perhaps the results we find here are unlikely to be representative of GTO play for the full game or otherwise very useful. However, we will often find ourselves with some preliminary information or reads on Villain’s early-street play but little to no idea about his turn and river tendencies. It is much easier to make reads on the early streets. We generally get more data on opponents’ early-street play, and it is much easier to make sense of that data, since there are far fewer distinct earlystreet situations. Additionally, populations of players tend to converge on one style of play in common spots but not in less common ones. So, for example, we’ll find it useful to study flop play under the assumption that the SB open-raises all of his playable hands, regardless of whether we think that unexploitable play involves lots of limping. When we find ourselves with actionable intel on Villain’s early-street play but no knowledge of his later-street strategy, a reasonable exploitative approach is to use our reads to the best of our ability early in the hand and then fall back on the equilibrium of the subgame, given the known earlystreet ranges to choose our later-street strategy. This gives us the benefits in the subgame we have come to expect from equilibrium play, including the provision of a baseline to work off of once we gain reads and want to adjust. Essentially, we play exploitatively while we can, and adopt equilib201

Expert Heads Up No-Limit Hold ’em, Volume 2 rium play in the subgame since that’s the best we can do there, given our ignorance of Villain’s strategy. Sometimes, we’ll decide that we need to arrive at the spot with a different range in the first place, but we have to understand the values of hands in subgames before we can make that call. Lastly, when we solve subgames that involve assumed starting ranges, take note of the starting distributions. Decide whether you agree with them, for what sorts of players are they appropriate, etc. If they are different in your match, the resulting strategies will be, too. On the other hand, aspects of many spots are similar, and the equity distributions can help us see the essentials of situations without worrying about the particular cards involved. In the next section, we’ll look at a spot where we may be able to narrow down Villain’s flop range, but we are unlikely to be able to say much about his tendencies on later streets.

13.3 Example: Turn Play 145BB deep on A♣-4♥-2♥-7♦

13.3.1 The Setup Consider the following hand. Hero is in the BB here. Effective stacks: 145BB Pre-flop: SB raises to 3BB, BB calls 2BB Flop: A♣-4♥-2♥ (6BB) BB checks, SB bets 6BB, BB calls Turn: 7♦ (18BB)

Stacks are deep, and the board is moderately volatile. In addition to the heart draw, many rivers will make (more) straights possible. On the other hand, there is no danger to top-pair hands of being beaten by rivered single 202

Turn Play: Volatile Boards and Capture Factors pairs. How should we choose the turn starting ranges here? We try to pick ranges as if both players are solid, mid-stakes types, but that is difficult here, since most such players do not make pot-sized c-bets with any hands. It’s possible that a good player has made this play (it’s not necessarily bad, just unusual). However, if all the information we have about our opponent is his pre-flop and flop play in this hand, the most likely explanation for this bet is that Villain is a particularly inexperienced player. Thus, even if this was our first hand against an unknown opponent, by the beginning of turn play, we should know it is very likely that the SB is a poor player. If it’s not our first hand with Villain, we should have some idea of whether full-pot is his standard flop c-bet sizing. If so, he likely has a wide range. If he does it rarely, then his range is probably a lot tighter. It can still be hard to say exactly how he chooses this range, but here we can speculate. What hands might Villain have a special reason to bet big? Keep in mind that a recreational player might not interpret a board or think about the game in the same way we do. For instance, this is not really a flop on which the SB should feel a lot of pressure to “protect” a decent hand like top pair with a good kicker. No overcards can come on future streets. There are some stronger draws possible, but the straight draws require low cards, most of which can be heavily discounted from the BB’s range, since he called the raise to 3BB pre-flop. But – we can’t assume Villain knows all this. Anyhow, when a weak player makes a large c-bet on this sort of board, and he usually bets smaller, I’m prone to start building his range with a lot of strong value hands – say, A-Q or better here. We can probably discount 5-3 and 4-2 (at least the off-suit versions) due to the raise to 3 BB pre-flop, especially if we have seen Villain fold (or even limp or min-raise) a button previously. Some players might be capped here, preferring to play their strongest hands less aggressively. Bluffs probably do not make up much of Villain’s range if he does not use this sizing often. For concreteness, we will give Villain all hands with absolute strength A-Q or better, except for 5-3o and 4-2o. We will also throw in Q-5o, K-3o, and 5-4s, to account for the possibility of weird plays, as well as some hearts: K♥-J♥, Q♥-T♥, 8♥-9♥. How about Hero’s play from the BB? If he has the same thought process we just did regarding the SB’s strong range, he will proceed cautiously. That


Expert Heads Up No-Limit Hold ’em, Volume 2 said, there are good reasons to put in a check-raise on the flop with legitimately strong hands to try to get value and build a pot before scare cards have the chance to arrive and kill the action. Villain appears to want to put money in on the flop, so slow-playing could go terribly wrong. Stacks are deep, so we would need to gain a ton of extra bets from bluffs to make up for losing the opportunity to stack him once or twice. So, we will not include flopped two pair or better in Hero’s flop check-calling range. On the other hand, Hero probably will check-call the flop with draws such as hearts. Raising these works best if Villain has a significant amount of weak hands to immediately fold and if he is the sort to just call with strong holdings. Then, a check-raise semi-bluff can fold out hands with good equity while still getting a chance to improve when Villain has a strong holding. Here, however, we assumed that Villain had something after making a 6-BB c-bet, so he is unlikely to fold, and the unusually large c-bet sizing could indicate that he is looking to get money in the pot quickly, so we may have to worry about a 3-bet. An opponent with a thought process that leads him to full-pot the flop when he likes his hand is unlikely to decide to slow-play facing a raise. On the other hand, we can expect to get good value if we make our draw, so we certainly want to continue, and thus we check-call. Lastly, Hero does not have too many deuces or fours, given the pre-flop play, but how does he play these facing the full-pot c-bet? Although it’s a bit conservative, I think we can simply check-fold these on the flop. If pot-sized cbets are rare, then either Villain has a range like the one we gave him above, strongly weighted towards good hands, or he has otherwise decided to play this hand aggressively for some reason. In the first case, we are pretty well crushed by his range. In the second, we are not, but it does not matter too much, since we will find ourselves check-folding to a turn bet either way. So, perhaps Hero gets to the turn this way with all pairs of aces, flush draws, and other combo draws (e.g., 5-4s) in his pre-flop calling range. We could give him a less precise range just to be safe if he were an unknown opponent, but here we identify ourselves with the BB and assume we know that is his strategy. Thus, his turn starting range here is


Turn Play: Volatile Boards and Capture Factors The SB has the hands we assigned to him earlier: The players’ starting distributions, given these ranges, are shown in Figure 13.2. Now, these ranges are not appropriate every time we find ourselves in this turn spot. Villain’s range has been quite restricted – it’s less than 7% of all hands, and the bluffing combinations are rather arbitrary. However, the ranges here are likely representative of the situation when a weak opponent unexpectedly goes full-pot on this flop. If nothing else, hopefully this discussion provides some indication of the things we need to consider in this sort of spot. We’ll take a quick look at the turn and river play strategies for good measure, but the details should only be taken seriously insofar as we think the turn starting ranges are appropriate. How might Hero’s turn starting range be different if the SB is a weak opponent who bets the full pot with all of his c-betting range?

Figure 13.2: Turn starting distributions for our example on the A♣-4♥-2♥-7♦ board.


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13.3.2 The Solution At the beginning of turn play, there are 18 BB in the pot and 136 BB behind – a stack-to-pot ratio of about 7.6. The BB’s range is very bluff-catcherheavy. He holds no real value hands except for A-7 combinations that turned two pair, and even those are decidedly non-nutted given the strong value range we assigned to the SB. Thus, it should be no surprise that the BB starts out by checking his whole range on the turn. Facing the check, the SB bets essentially his whole range. He uses his threequarters and full pot-sized betting options. He makes the larger bet about 4 times as often as the smaller, although the two betting ranges have approximately the same composition. We gave the SB enough strong value hands here that he does not need to bet too large to bet all his weak hands. He would likely bet larger if he had more air that needed to bluff. The BB’s responses to the two bet sizings are similar, with slight differences in the calling thresholds to obtain appropriate frequencies. He calls, more or less, all of his aces that have hearts or a heart blocker and all his draws that paired up or picked up a straight draw on the turn. He actually check-folds higher flush draws while calling with lower ones, since the lower ones interact more strongly with the turn card. He also folds aces without anything else (such as single heart blockers) going for them. How much of the pot does the BB expect to capture after check-calling the pot-sized bet? At this point, we get to the river with 54 BB in the pot and 118 BB behind. If we call the BB’s capture factor at the beginning of river play RH, then the BB’s EV of calling a pot-sized turn bet is (118+54RH) BB. The subscript H here indicates that the capture factor depends on the BB’s hand, H. Of course, different hands will expect to capture different amounts of the pot. He should compare this to his EV of folding, 136BB for all hands, to make a decision. If the BB holds J♥-9♥ here, he wins about 27.8% of the pot on the river on average when facing the SB’s equilibrium strategy. This is somewhat better than its equity, 23.6%, which is to be expected for a draw. However, this


Turn Play: Volatile Boards and Capture Factors leads to an EV of 118+(0.278×54)=133 BB, which is not enough to call a bet. On the other hand, the BB’s 7♥-6♥ has 51.4% equity. Part of that comes from its weak made hand, and this part will not be realized in full on the river, but he will expect to do better than simply win the pot when he makes his draw. It turns out that his average capture factor here is about 40.9%, and thus calling achieves an EV of 118+(0.409×54)=140BB – better than folding. Lastly, if the BB check-calls with A-7o, he has about 75% equity and a capture factor of about 71.9% for an EV(calling) of 118+(0.719×54)=157BB at equilibrium. This fairly strong hand is able to realize most of its equity in the pot. With A-7, the BB does sometimes check-raise when facing a bet, using his half-pot sizing. In fact, he check-raises A♥-7♥ almost always, and he frequently raises with the A-7o combination with no hearts as well. However, those with one heart prefer to call. Why might this be? (Consider that most of his check-raise semi-bluffs for balance are heart draws.) Facing the raise, the SB flats with most of his holdings, folds his bluffs (Q-5, K-3), and is indifferent to calling, jamming, and folding his A-Q and A-K. Now let us look at some of the SB’s capture factors. Consider the situation after he makes a pot-sized bet on the turn and gets called (i.e., the situation in which we looked at some BB capture factors above). The SB’s

K♥-J♥ has 19.9% equity and RH=0.326, for an EV of 118+(0.326×54)=136BB

A♦-Q♠ has equity 0.75 and RH=0.654, for an EV of 118+(0.654×54)=152BB.

5♥-3♥ has equity 0.943 and RH=1.13, for an EV of 177 BB.

How do these hands compare to those we looked at in the BB’s case, and what do the differences tell us about positions’ and distributions’ influence on hands’ ability to contest the pot on the river? We will look into these issues more deeply now.


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13.4 Where the Money Comes From: Turn Play with River Capture Factors In the last couple of examples, we noted players’ capture factors (CFs) in various spots. Hopefully, this has given you some idea of their magnitude for various hands and you can imagine how they are useful for decisionmaking. Here we will make these things more explicit. We can also solve for CFs at equilibrium in some model games, but we will see that CFs are not only a tool for thinking about GTO play. They can be used exploitatively, and with that in mind, we will look at a couple of decision-making rules in a few spots and consider how our play there depends on Villain’s tendencies.

13.4.1 The Capture Factor, Equity, and EV By definition, a CF tells us the fraction of the pot a player will capture with a particular hand in a certain spot, on average. The average is generally over all lines he could end up taking, cards that could come, hands in his opponent’s range, etc. Most importantly, it depends on the players’ strategies. We’ll use CFs to help us make multi-street decisions, so we will primarily be interested in CFs at the end of a street. Let S and P be the stack and pot sizes there, and let R be our hand’s capture factor. Then, our EV is (13.1)

This should look very familiar. If both players are forced to check down, then our EV is just S+P×EQ. The difference is that the first equation is meant to account for the possibility that the hand is not (necessarily) just checked down. In other words, all the complexity of multi-street play is swept into one number, the CF. It would be easy to make early-street decisions based on raw equity if we were forced to check down later streets. As these two EV equations suggest, we will be able to use more or less the same approach in the multi-street case by simply substituting capture factors for equities. 208

Turn Play: Volatile Boards and Capture Factors A picture can help us to further understand the connection between CFs and EVs. Of course, different hands have different CFs, since the fraction of the pot we are able to capture depends on our hand. We can plot CFs for all our hands just like we plot equities in an equity distribution. Let’s take an example from the first situation in this chapter (see Section 13.1). Consider the spot after the turn goes check-check. (This was the most common turn line, you may recall). We will plot the BB’s capture factor for each of his hands at exactly the point after the SB checks back but before the river card is dealt. In other words, we will find the amount of the pot the BB captures averaged over all river cards rather than any specific one – this is, of course, the information that can be available to him during his turn decision-making.

Figure 13.3: BB equities, EVs, and CFs after the turn checks through in the example on the K♣-Q♠-8♥-5♥ board from Section 13.1. Notice that the CF gives us essentially the same information as the EV.

The BB’s CFs at the end of turn play after the street checks through are shown in Figure 13.3. For comparison, we have plotted the equity distribution on the same axes as well. (Hands are sorted by their equity as in normal equity distributions, although we could just as well have sorted hands by their EVs or some other measure.) The equity distribution is the solid curve, 209

Expert Heads Up No-Limit Hold ’em, Volume 2 and each point indicates the CF of one of the BB’s hands. The EVs of each of his hands are shown as well. In fact, the EVs are essentially the same as his CFs, just scaled by a constant (P) and shifted up by another (S), as in Equation 13.1. Thus, we can use the same set of points to represent the CFs and the EVs if we re-scale the vertical axis. Here, the left vertical axis labels the hands according to their CFs, and the right indicates the EVs. Hopefully it is now clear that the CFs give us exactly the same information as EVs. Thus, any decision we can make by comparing EVs, we can make using CFs. CFs are just scaled so they can be more easily predicted and manipulated. Now, what fraction of the pot does the BB capture with his various holdings? This is easiest to understand in comparison to his hands’ equity. Focusing on the middle of the plot, we see that most of the BB’s middling hands capture somewhat less of the pot than their equity would suggest. He does not hold any real air here – his weakest hands have equities and CFs of around 50 percent. Thus, the worst hands in his range are still rather good. The made hands that fall to the far right of the plot capture less than their full equity, and those points that fall right on the equity distribution curve are actually mediocre draws, e.g., J-10. The BB’s strongest holdings capture significantly more of the pot than if the hand was just checked down. Notice that the EVs of his best holdings increase quickly in steps corresponding to rather minute changes in equity. The best holdings benefit greatly from cooler situations, essentially. From the left, the first plateau is K-high two-pair hands. The second is the BB’s Q-high two-pairs combined with his one-pair K♥-X♥, which have slightly less equity but slightly higher EV. The third and fourth plateaus correspond to his pairs of kings and queens, respectively. The fifth, composed of hands that capture significantly less than their equity, corresponds to the BB’s eights. Notice that there are a few points that stand out because they have much higher CF than equity. These are of course draws. The BB does not hold very many here, but for example, the three points on the right that lie significantly above the curve correspond to A♥-10♥, J♥-10♥, and J♥-9♥. These outperform their equity due to playability – on most particular rivers, they are either very strong or very weak.


Turn Play: Volatile Boards and Capture Factors

Figure 13.4: SB equities, EVs, and CFs after the turn checks through in the example on the K♣-Q♠-8♥-5♥ board from Section 13.1.

The SB’s CFs, EVs, and equities at the same point in the hand are shown in Figure 13.4. The SB’s range, you may recall, contains many hands and types of hands, but it is capped, since he does not check the turn with any hands with more than 80% equity. Thus, the strongest hands plotted here are decidedly non-nutted. Other than that, the story here is about the same as in the BB’s case – things are just a bit messier since the SB’s range is less defined. The SB’s made hands have CFs that track his equity distribution pretty well – the weakest ones capture approximately their equity, the mediocre ones, somewhat less, and the strongest ones, somewhat more. His strongest hands are the small amount of Q-5o he checks back, but the second, much larger cluster of value holdings is pairs of kings. The hands with CFs significantly above their equity are draws – the SB holds many draws here of various strengths. Great – these figures along with the previous examples give us some idea what CFs look like in real spots. However, seeing how they arise in some model river games will provide additional insight into their origin. We will begin by looking at CFs on particular rivers (i.e., after the river card is dealt, hand values are fixed, and no draws are possible) in the context of a particular model river game. Then, we will imagine averaging these over all rivers to find the CFs at the end of turn play, before the river card has been dealt. 211

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13.4.2 Single River CFs with Model Games Turn play depends on the value of having hands in various river spots. We can describe those values in a useful way using capture factors, and we have seen a few examples. We can also solve for them in some model situations to better understand their origin. In this section, we will look at the amount of the pot players capture in some single river spots, i.e., after the river card has fallen and hand strengths are fixed. In the next section, we will average over all rivers to estimate CFs at the end of turn play. The PvBC river case is easy, and in fact, we have already done it. Assume Hero is polar and Villain is bluff-catching for convenience. Then, Equation 12.3 in Section 12.8.2 gives the CF of Hero’s nut hands as

where the pot and bet sizes are P and B, respectively. This formula has a natural interpretation. The 1 comes from the fact that Hero wins the full pot, and the second term is associated with the “extra” value Hero gains from the river betting. This extra value increases with B, reminding us that Hero should bet as large as possible in this spot. For example, for a potsized bet, we find Rvalue=1.5, and for a half-pot bet, Rvalue=1.33. Of course, our CF with air hands is 0. (This all assumes the usual case where Villain can bluff-catch enough to make us indifferent to bluffing.) Villain’s bluff-catchers capture whatever fraction of the pot Hero does not. So, if a fraction N of Hero’s river starting range is the nuts, Hero captures N(1+ B/(B+P)) of the pot on average with his whole range. Villain’s bluffcatchers get the rest:

In other words, Villain’s bluff-catchers capture something like their equity (1−N), but when Hero has the nuts, they lose the same extra value that Hero’s nuts gain. So, in the simple PvBC situation, we see again that value 212

Turn Play: Volatile Boards and Capture Factors hands gain from the river betting, relative to their equity, bluff-catchers lose, and air hands break even, since they did not have any equity to lose. Set Rbluff−catcher=0 and solve for N to find the nut fraction that prevents Villain’s bluff-catchers from capturing any of the pot. If N increases further, can Rbluff−catcher become negative? What strategic situation does this correspond to? Do things play out similarly in a more nuanced river situation? This is a good opportunity to leverage a river model we skipped in Chapter 7: the one-bet-behind [0,1] river game. This game’s decision tree was shown in Figure 9.1. As the name indicates, we get to the river with just one bet behind so that any bet is all-in. Thus, the BB can jam, check-call, or check-fold and, facing the check, the SB can jam or check back. The indifference equations and closed-form solutions for the symmetric distributions case are given in Appendix 13.10. Here, we will focus on the solutions themselves when there is one full pot-sized bet behind. This is a useful sizing (and game!) to consider, since many players like to size their early-street bets to set up an approximately pot-sized jam on the river. Symmetric distributions are convenient but less realistic.

Figure 13.5: solution structure for the one-bet-behind river game (decision tree shown in Figure 9.1) drawn for the case of symmetric distributions and one pot-sized bet behind.

The structures of players’ unexploitable strategies in this game are shown in Figure 13.5. It is drawn to scale for the case of a pot-sized bet. For reasons described in Section 9.4.3, the equilibrium parameterization of the 213

Expert Heads Up No-Limit Hold ’em, Volume 2 BB’s strategy shown here is not unique. He is actually indifferent between check-calling and jamming with his nut hands. This implies that the SB should be putting in the same amount of money, on average, versus both BB lines. Does this appear to be the case from the figure? The EVs of each of the BB’s hands at equilibrium are shown in Figure 13.6. Actually, we show the EV of taking every line with every hand. Of course, the choice the BB makes in-game is the one that gives him the highest EV. For example, the far left of the plot gives the EVs of the BB’s strongest hands. We see that these capture half the pot if they play check-fold and 1.5 times the pot if they jam or check-call. What is the BB’s most profitable line with each of his hands? Do these match up with the solution structures in Figure 13.5? Additionally, we have plotted the BB’s equity on the same axes. This is just a straight line because of the symmetric distributions assumption, and it represents the fraction of the pot he would win if the players were forced to check down on the river. We see that strong value hands expect to capture quite a bit more than 100% of the pot, and that extra value comes at the expense of weaker holdings. The BB’s strongest value hand expects 1.5 times the pot on the end. This is the same result we found in the pot-sizedbet PvBC case. This makes sense, because his value-jam gets called with the same frequency in both games – the frequency necessary to make complete air indifferent to bluffing. Weaker value hands, however, do worse, since they sometimes run into better hands and lose the pot. This describes the general trend for CFs at the top of our range when we bet with them on the river (i.e., when we are in position or our distribution lets us lead from the BB). Our nuts will win the pot, plus an extra bet with whatever frequency they get called. (At equilibrium, the calling frequency is chosen to make some bluff indifferent – when is this larger and when is it smaller than in the naive case, if ever?) Our weaker value hands, however, do somewhat worse. This explains the CFs we saw in Section 12.8.3 very nicely. Now, do our nuts do better or worse when we’re in the BB and our distribution is such that we cannot lead the river? Perhaps counterintuitively, our nuts often do better in this spot. Of course they could lead if they wanted to, along with some weaker hands for balance, and get called a lot. They only check at equilibrium because they do even better that way.


Turn Play: Volatile Boards and Capture Factors

Figure 13.6: CFs, EVs, and equities for the BB’s hands at the equilibrium of the one pot-sized-bet-behind river game with symmetric distributions. CFs and equities are indicated on the left vertical axis, and EVs are indicated on the right.

Figure 13.7: CFs, EVs, and equities for the SB’s hands at the equilibrium of the one pot-sized-bet-behind river game with symmetric distributions. EV/CF curves are drawn for each of the SB’s four strategy combinations.


Expert Heads Up No-Limit Hold ’em, Volume 2 A similar plot gives the SB’s EVs, CFs, and equities at equilibrium for each line he can take in this game. It is shown in Figure 13.7. This is a bit more complicated than in the BB case. The SB can call or fold facing a jam, and he can check or jam when facing a check. There are four combinations, so there are four pure strategies the SB can choose from with each hand. Take, for example, his strongest hands, whose values fall on the far left side of the graph. He does best if he plays call-or-jam, i.e., if he calls when facing a shove and jams when facing a check. The trend of CFs for SB’s hands looks generally the same as for the BB. In fact, the SB’s EV distribution here is exactly the same as the BB’s except that it is shifted upwards a bit. Every single hand, air through the nuts, has an EV P/12 higher when the SB plays it instead of the BB. In other words, the CF of every hand is 1/12≈0.083 higher for the SB here. For the SB’s bluffs, this comes from the fact that the BB is folding enough to make the strongest bluff, hsb, rather than pure air, indifferent to bluffing. Where might we say the extra value comes from for the SB’s near-nut holdings? What about those hands in between? We have seen how hands fare on the river in several model games. We have been able to find exact expressions for the CFs of some holdings and otherwise see how they depend on hand value and position. If the board is entirely static on the turn, our EV at the end of turn play is the same as our EV after a particular river card is dealt. Generally, however, that will not be the case. Before we move up the tree and consider turn-ending CFs, let’s think about how stack depth and Villain tendencies affect our ability to contest the pot on the river. If you’re feeling ambitious, find the river CFs for each player in the two-bets-behind game and in the asymmetric [0,1] game situations discussed in Chapter 7.6 Dependence on Stack Size How does the stack size affect hands’ CFs? First, it is of course the stack-to6 The CF of the nuts in the two-bets-behind game with symmetric distributions where the bet and raise are both pot-sized is 1.8.


Turn Play: Volatile Boards and Capture Factors pot ratio (SPR), and not the stack size itself, that sets the scale of the game. Consider some extreme cases. First, as the SPR becomes very small (i.e., close to 0), the CF becomes close to the equity. In fact, when the SPR is exactly 0, it means we are all-in, so we simply capture our equity in the pot. When the SPR is nearly zero, we are nearly all-in, we will nearly always see a showdown, and we capture nearly our equity in the pot. At the other extreme, how do hands do as the stacks become very large relative to the pot? The exact solutions from the PvBC game can give us a feel for the trend. We saw previously that the polar player’s nuts have CF 1+[B/(B+P)]. As B becomes large (i.e., as the stacks get deep), this becomes close to 2. This result holds for large B in the one-bet-behind game as well. Finally, bluff-catchers lose value of N times that which the nuts gain: Rbluff−catcher=(1−N)−N[B/(B+P)]. This becomes close to 1−2N when stacks are deep. Exploitative Play: Dependence on Villain’s Tendencies We can make exploitative decisions by adjusting our estimates of the amount of the pot we will capture on the river with particular hands, based on Villain’s tendencies. I don’t really like using classifiers such as loose, tight, aggressive, and passive to describe opponents’ overall games (see Section 17.1.1). They’re too vague. However, it is safer to think in these terms in specific spots. Tight and loose primarily refer to Villain’s play with his middling hands – tight generally means he is likely to fold these facing action, and loose refers to the opposite. When facing a player who is too tight, our bluffs usually win more than at equilibrium, but our value wins less. Effectively, the graph of CFs flattens out as the left side becomes lower and the right side raises up. Versus a loose opponent, the opposite is more or less true: our value does better, but any hands that gained by bluffing previously can no longer do so. When we say Villain is aggressive, we are primarily referring to his play with particularly strong and particularly weak holdings. If he is aggressive, he tends to bet (or raise) more often, perhaps in a way that still makes our bluff-catchers indifferent to calling. In this case, all of our hands that are 217

Expert Heads Up No-Limit Hold ’em, Volume 2 made indifferent or strictly prefer to fold to a bet perform worse, since they face a bet more often. However, some slightly stronger hands that were indifferent at equilibrium can now strictly profit with a call, since they’re ahead of some of Villain’s (wider) value-betting range, and our actual strong hands get more value as well. In effect, the right side of our EV distribution flattens out towards zero, but the left side becomes greater. A passive player can value-bet and bluff less frequently and potentially still make our middling hands indifferent to continuing. When Villain is aggressive, it tends to increase the curvature of the CF curve – value hands do somewhat better while middling hands do worse. When Villain is passive, the result is not quite the opposite. Our intermediate hands do better, but we do not necessarily lose (compared to equilibrium play) with our value hands, as long as we make sure to bet them ourselves. Thus, the primary effect of passive opponent play on our CF curve is to increase the EV of our middling hands. In all cases, the average height of the graph will be higher – we will capture more of the pot than we would at equilibrium, on average, by virtue of exploiting Villain, whatever his tendencies. Later in this chapter, we will consider how to make decisions using CFs. To use these decision rules exploitatively, we just need to tweak our estimates of the amount of the pot we will win on the river with various hands, given our opponent’s tendencies. More quantitative results are best gained by playing around with a model such as the one-bet-behind game. How do the qualitative descriptors loose, tight, passive, and aggressive correspond to the thresholds between players’ action regions (i.e., the cut-off hands shown in Figure 13.5)? How do the EVs of all the hands in Hero’s range change as we move these thresholds?

13.4.3 Draws and CFs at the End of Turn Play Now that we know what CFs look like once hand values are static on the 218

Turn Play: Volatile Boards and Capture Factors river, we can back up and suppose we are at the end of turn play. The strongest hands on the turn might be devalued if draws come in on the river. Of course, draws themselves will expect to capture very different amounts of the pot on different river cards. The CF of a hand here is just its average value over all possible rivers, weighted by the probabilities that each river is dealt. First, consider draws themselves at the end of turn play. In the simplest case, drawing hands are essentially air on some fraction of rivers and the nuts on the others. If this is true, then in fact the proportion of rivers on which they make their hand is just their current equity; call it EQdraw. They will achieve the CF of the nuts on EQdraw of rivers and the CF of air on the others for an average CF of So, if we know how the nuts and air perform on particular rivers, we can find the CF of a draw as well. In particular, if CFair=0, then draws can expect to capture about CFdraw=EQdrawCFnuts. We have seen that if there is a pot-sized bet left on the river, and Villain is solid, CFnuts often turns out to be a bit less than 1.5, so the turn-ending CF of a draw to the nuts is about 1.5EQdraw. In other words, we can just scale up a draw’s equity by the CF of the nuts to get the draw’s CF. It may be a bit pessimistic to assume that CFair=0, but we must also account for times when our draw is to a non-nut hand. Do these approximations agree with the computational results we saw in Figures 13.4 and 13.3? The BB draws we pointed out previously (the three standout points on the right of the graph) correspond to A♥-10♥, J♥-10♥, and J♥-9♥. These each have about 60% equity. Certainly these are all strong draws on the K♣-Q♠-8♥-5♥ board, but a good portion of their equity comes from pair outs and from simply having the best hand with their high card. This portion of their equity does not correspond to a chance to improve to the nuts on the river, so their CF is somewhat less than 1.5×60%=0.90. By contrast, the SB’s draws, those holdings with CF notably above the equity curve, do appear to capture something like 1.5 times their equity. How can holdings that are the nuts on the turn expect to fare on the river when a draw is present? When draws miss, they are still the nuts and capture CFnuts. When draws get there, the previous-nuts become merely strong 219

Expert Heads Up No-Limit Hold ’em, Volume 2 made hands (in most cases). Thus, if we call their capture factor CFstrong when draws come in, we have the expected CF of nuts on the turn: So, at the end of a turn when draws are possible, nut hands can expect to capture as much as they would if they were guaranteed nuts on the river, minus the difference between having the nuts and having a strong hand whenever the draws get there. Now, what about mediocre made hands – hands that do not figure to improve on the river? These may have a bit more or a bit less equity on various rivers. However, if they capture approximately the same fraction of their equity on each one, then it turns out that they will capture that same fraction of their average equity over all rivers, which is simply their equity on the turn. So, for example, if we expect a weak made hand to capture 75% of its equity on the river, we can also approximate its average CF by 75% of its equity at the end of turn play. Thus, a weak made hand with little chance to improve can think of its turn ending CF as just its turn ending equity scaled down by the same amount we saw in the single river situations. Of course, we’ve seen that many weak draws (draws to pairs, two pair, three-of-a-kind, etc.) are non-negligible. Thus, the nuts at the end of turn play have CF somewhat less than the nuts at the beginning of the river when the board is not static. The CF of a pure draw is approximately that of the nuts on the river times the chance that it makes the nuts on the river (its equity). Weak made hands, insofar as they tend to stay weak made hands on the river, can expect to capture the same fraction of the pot at the end of turn play as they do in the single-river case. Our comments about the BB’s drawing hands in Figure 13.3 hinted at an approach to dealing with hands that have more than one quality. For example, the BB’s A♥-10♥ will be the nuts on some rivers and a weak made hand on the rest of them. In this case, it captures a portion of the pot corresponding to the nuts whenever its draw comes in and a portion corresponding to a weak made hand otherwise. Clearly, more careful calculations could consider exactly how a hand does on all the possible rivers, but here, we’ve seen how made hands and draws, weak, mediocre, and strong, tend to fare. 220

Turn Play: Volatile Boards and Capture Factors

13.4.4 Decision-making with CFs Let’s run over some common decision-making rules using CFs. These will be decision rules in the sense of Section 10.6 – we’ll write down EVs of Hero’s two choices in some archetypal spots, set one larger than the other, and rearrange until properties of the game are on one side and some Villain statistic is on the other. We’ll treat our CF as a property of the game, although in reality it depends somewhat on Villain’s play as well. These will look similar to things we have seen before. In any spot where we previously said, “assume we just check down on later streets,” we can substitute CFs for the equities, and we’re good to go. So, we’ll quickly address a representative sample of decision rules, rather than re-doing everything. These rules apply to flop play as well, although we have yet to look at CFs on the flop. Drawing Imagine we face a bet with a draw and wish to evaluate a call-or-fold decision. If the bet is size B, and the stack and pot sizes were S and P, respectively, before the bet, then we can draw if

where R is our CF after we call, i.e., at the end of the street. Thus, we should call if R>B/(P+2B). As we foreshadowed, this is the same as the usual pot odds equation with equity replaced by CF. For example, in the real hand on which the A♣-4♥-2♥-7♦ example was based, Hero was in the BB with 8♥-6♥. He checked the turn and faced another pot-sized bet. He has about 30% equity. We saw that pure draws to the nuts might capture about 150% of their equity, but here, Hero holds a non-nut draw, and some of that equity comes from the possibility of hitting a pair as well. So, in this spot, Hero’s CF turns out to be about 37%. This is larger than B/(P+2B)=1/3 so we have a call. Notice that a straight pot-odds calculation would indicate a fold, and we’d have to fold here as well if we had not picked up the straight draw on the turn. 221

Expert Heads Up No-Limit Hold ’em, Volume 2 Bluffing Consider the ideal weak made hand that can either check back and realize a little equity in the pot or bet and be guaranteed to lose if it gets called. Let F be Villain’s folding frequency. Then, checking achieves an EV of S+PR, and betting gets us S−B+{(B+P)F}, so we should bluff if

R here is our CF after we check – that is, at the end of the street if we’re in position, or immediately before the Villain’s action if we’re not. How does the critical F here compare to the Villain’s equilibrium folding frequency in the naive PvBC game? Semi-bluffing Suppose we have a choice between seeing a river and semi-bluffing with a hand that keeps all its equity after the bet. Let F and C be Villain’s folding and calling frequencies. Checking gets us S+PR. Betting gets us S−B+(P+B)F+(P+2B)RC. Thus, we have a break-even bluff when

This becomes simpler if Villain plays call-or-fold so that F+C=1, but in general, he can raise as well. This equation is not in a very useful form, unfortunately. Value-betting Consider a thin value-betting decision with bet size B and stack and pot sizes S and P before the bet. If we’re in position so that checking goes to the river, EV(check) = S + P R1 EV(bet) = S − B + (P + B)F + (2B + P )R2 C

where R1 and R2 are the fractions of the pot we capture if we check back and if we bet and get called, respectively. Presumably R2
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