Game Theory Critical Concepts
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Contents Articles Introduction Game theory Nash equilibrium
Definitions
1 1 21 33
Cooperative game
33
Information set
40
Preference
42
Normal-form game
43
Extensive-form game
46
Succinct game
52
Equilibrium Concepts
58
Trembling hand perfect equilibrium
58
Proper equilibrium
61
Evolutionarily stable strategy
63
Risk dominance
69
Self-confirming equilibrium
72
Strategies
73
Dominance
73
Strategy
76
Tit for tat
78
Grim trigger
82
Collusion
83
Backward induction
86
Markov strategy
88
Game Classes
89
Symmetric game
89
Perfect information
91
Simultaneous game
92
Sequential game
92
Repeated game
92
Signaling games
94
Cheap talk
98
Zero-sum
99
Mechanism design
102
Bargaining Problem
110
Stochastic game
113
Large poisson game
115
Nontransitive game
116
Global game
117
Games
118
Prisoner's dilemma
118
Traveler's dilemma
128
Coordination game
130
Chicken
133
Centipede game
141
Volunteer's dilemma
144
Dollar auction
145
Battle of the sexes
146
Stag hunt
150
Matching pennies
152
Ultimatum game
154
Rock-paper-scissors
160
Pirate game
171
Dictator game
172
Public goods game
174
Blotto games
177
War of attrition
178
El Farol Bar problem
180
Fair division
182
Cournot competition
187
Deadlock
193
Unscrupulous diner's dilemma
194
Guess 2/3 of the average
195
Kuhn poker
197
Nash bargaining game
198
Screening game
201
Princess and monster game
201
Theorems
203
Minimax
203
Purification theorem
208
Folk theorem
210
Revelation principle
211
Arrow's impossibility theorem
212
Additional Reading
222
Tragedy of the commons
222
Tyranny of small decisions
232
All-pay auction
236
List of games in game theory
237
References Article Sources and Contributors
240
Image Sources, Licenses and Contributors
245
Article Licenses License
247
1
Introduction Game theory Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others. More formally, it is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers."[1] An alternative term suggested "as a more descriptive name for the discipline" is interactive decision theory.[2] Game theory is mainly used in economics, political science, and psychology, and other, more prescribed sciences, like logic or biology. The subject first addressed zero-sum games, such that one person's gains exactly equal net losses of the other participant(s). Today, however, game theory applies to a wide range of class relations, and has developed into an umbrella term for the logical side of science, to include both human and non-humans, like computers. Classic uses include a sense of balance in numerous games, where each person has found or developed a tactic that cannot successfully better his results, given the other approach. Mathematical game theory had beginnings with some publications by Émile Borel, which led to his book Applications aux Jeux de Hasard. However, his results were limited, and the theory regarding the non-existence of blended-strategy equilibrium in two-player games was incorrect. Modern game theory began with the idea regarding the existence of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann. Von Neumann's original proof used Brouwer's fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by his 1944 book Theory of Games and Economic Behavior, with Oskar Morgenstern, which considered cooperative games of several players. The second edition of this book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty. This theory was developed extensively in the 1950s by many scholars. Game theory was later explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields. Eight game-theorists have won the Nobel Memorial Prize in Economic Sciences, and John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology.
Game theory
History Early discussions of examples of two-person games occurred long before the rise of modern, mathematical game theory. The first known discussion of game theory occurred in a letter written by James Waldegrave in 1713. In this letter, Waldegrave provides a minimax mixed strategy solution to a two-person version of the card game le Her. James Madison made what we now recognize as a game-theoretic analysis of the ways states can be expected to behave under different systems of taxation.[3] [4] In his 1838 Recherches sur les principes mathématiques de la théorie des richesses (Researches into the Mathematical Principles of the Theory of Wealth), Antoine Augustin Cournot considered a duopoly and presents a solution that is a restricted version of the Nash equilibrium. The Danish mathematician Zeuthen proved that a mathematical model has a winning strategy by using Brouwer's fixed point theorem. In his 1938 book Applications aux Jeux de Hasard and earlier notes, Émile John von Neumann Borel proved a minimax theorem for two-person zero-sum matrix games only when the pay-off matrix was symmetric. Borel conjectured that non-existence of a mixed-strategy equilibria in two-person zero-sum games would occur, a conjecture that was proved false. Game theory did not really exist as a unique field until John von Neumann published a paper in 1928.[5] Von Neumann's original proof used Brouwer's fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by his 1944 book Theory of Games and Economic Behavior, with Oskar Morgenstern, which considered cooperative games of several players. The second edition of this book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty. Von Neumann's work in game theory culminated in the 1944 book Theory of Games and Economic Behavior by von Neumann and Oskar Morgenstern. This foundational work contains the method for finding mutually consistent solutions for two-person zero-sum games. During this time period, work on game theory was primarily focused on cooperative game theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies.[6] In 1950, the first discussion of the prisoner's dilemma appeared, and an experiment was undertaken on this game at the RAND corporation. Around this same time, John Nash developed a criterion for mutual consistency of players' strategies, known as Nash equilibrium, applicable to a wider variety of games than the criterion proposed by von Neumann and Morgenstern. This equilibrium is sufficiently general to allow for the analysis of non-cooperative games in addition to cooperative ones. Game theory experienced a flurry of activity in the 1950s, during which time the concepts of the core, the extensive form game, fictitious play, repeated games, and the Shapley value were developed. In addition, the first applications of Game theory to philosophy and political science occurred during this time. In 1965, Reinhard Selten introduced his solution concept of subgame perfect equilibria, which further refined the Nash equilibrium (later he would introduce trembling hand perfection as well). In 1967, John Harsanyi developed the concepts of complete information and Bayesian games. Nash, Selten and Harsanyi became Economics Nobel Laureates in 1994 for their contributions to economic game theory. In the 1970s, game theory was extensively applied in biology, largely as a result of the work of John Maynard Smith and his evolutionarily stable strategy. In addition, the concepts of correlated equilibrium, trembling hand perfection, and common knowledge[7] were introduced and analyzed.
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Game theory
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In 2005, game theorists Thomas Schelling and Robert Aumann followed Nash, Selten and Harsanyi as Nobel Laureates. Schelling worked on dynamic models, early examples of evolutionary game theory. Aumann contributed more to the equilibrium school, introducing an equilibrium coarsening, correlated equilibrium, and developing an extensive formal analysis of the assumption of common knowledge and of its consequences. In 2007, Leonid Hurwicz, together with Eric Maskin and Roger Myerson, was awarded the Nobel Prize in Economics "for having laid the foundations of mechanism design theory." Myerson's contributions include the notion of proper equilibrium, and an important graduate text: Game Theory, Analysis of Conflict (Myerson 1997). Hurwicz introduced and formalized the concept of incentive compatibility.
Representation of games The games studied in game theory are well-defined mathematical objects. A game consists of a set of players, a set of moves (or strategies) available to those players, and a specification of payoffs for each combination of strategies. Most cooperative games are presented in the characteristic function form, while the extensive and the normal forms are used to define noncooperative games.
Extensive form
An extensive form game
The extensive form can be used to formalize games with a time sequencing of moves. Games here are played on trees (as pictured to the left). Here each vertex (or node) represents a point of choice for a player. The player is specified by a number listed by the vertex. The lines out of the vertex represent a possible action for that player. The payoffs are specified at the bottom of the tree. The extensive form can be viewed as a multi-player generalization of a decision tree. (Fudenberg & Tirole 1991, p. 67)
In the game pictured to the left, there are two players. Player 1 moves first and chooses either F or U. Player 2 sees Player 1's move and then chooses A or R. Suppose that Player 1 chooses U and then Player 2 chooses A, then Player 1 gets 8 and Player 2 gets 2. The extensive form can also capture simultaneous-move games and games with imperfect information. To represent it, either a dotted line connects different vertices to represent them as being part of the same information set (i.e., the players do not know at which point they are), or a closed line is drawn around them. (See example in the imperfect information section.)
Normal form Player 2 chooses Left
Player 2 chooses Right
Player 1 chooses Up
4, 3
–1, –1
Player 1 chooses Down
0, 0
3, 4
Normal form or payoff matrix of a 2-player, 2-strategy game
The normal (or strategic form) game is usually represented by a matrix which shows the players, strategies, and payoffs (see the example to the right). More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions. In the accompanying example there are two players; one
Game theory chooses the row and the other chooses the column. Each player has two strategies, which are specified by the number of rows and the number of columns. The payoffs are provided in the interior. The first number is the payoff received by the row player (Player 1 in our example); the second is the payoff for the column player (Player 2 in our example). Suppose that Player 1 plays Up and that Player 2 plays Left. Then Player 1 gets a payoff of 4, and Player 2 gets 3. When a game is presented in normal form, it is presumed that each player acts simultaneously or, at least, without knowing the actions of the other. If players have some information about the choices of other players, the game is usually presented in extensive form. Every extensive-form game has an equivalent normal-form game, however the transformation to normal form may result in an exponential blowup in the size of the representation, making it computationally impractical. (Leyton-Brown & Shoham 2008, p. 35)
Characteristic function form In games that possess removable utility separate rewards are not given; rather, the characteristic function decides the payoff of each unity. The idea is that the unity that is 'empty', so to speak, does not receive a reward at all. The origin of this form is to be found in John von Neumann and Oskar Morgenstern's book; when looking at these instances, they guessed that when a union C appears, it works against the fraction (N/C) as if two individuals were playing a normal game. The balanced payoff of C is a basic function. Although there are differing examples that help determine coalitional amounts from normal games, not all appear that in their function form can be derived from such. Formally, a characteristic function is seen as: (N,v), where N represents the group of people and v:2^N-->R is a normal utility. Such characteristic functions have expanded to describe games where there is no removable utility.
Partition function form The characteristic function form ignores the possible externalities of coalition formation. In the partition function form the payoff of a coalition depends not only on its members, but also on the way the rest of the players are partitioned (Thrall & Lucas 1963).
General and applied uses As a method of applied mathematics, game theory has been used to study a wide variety of human and animal behaviors. It was initially developed in economics to understand a large collection of economic behaviors, including behaviors of firms, markets, and consumers. The use of game theory in the social sciences has expanded, and game theory has been applied to political, sociological, and psychological behaviors as well. Game-theoretic analysis was initially used to study animal behavior by Ronald Fisher in the 1930s (although even Charles Darwin makes a few informal game-theoretic statements). This work predates the name "game theory", but it shares many important features with this field. The developments in economics were later applied to biology largely by John Maynard Smith in his book Evolution and the Theory of Games. In addition to being used to describe, predict, and explain behavior, game theory has also been used to develop theories of ethical or normative behavior and to prescribe such behavior.[8] In economics and philosophy, scholars have applied game theory to help in the understanding of good or proper behavior. Game-theoretic arguments of this type can be found as far back as Plato.[9]
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Game theory
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Description and modeling The first known use is to describe and model how human populations behave. Some scholars believe that by finding the equilibria of games they can predict how actual human populations will behave when confronted with situations analogous to the game being studied. This particular view of A three stage Centipede Game game theory has come under recent criticism. First, it is criticized because the assumptions made by game theorists are often violated. Game theorists may assume players always act in a way to directly maximize their wins (the Homo economicus model), but in practice, human behavior often deviates from this model. Explanations of this phenomenon are many; irrationality, new models of deliberation, or even different motives (like that of altruism). Game theorists respond by comparing their assumptions to those used in physics. Thus while their assumptions do not always hold, they can treat game theory as a reasonable scientific ideal akin to the models used by physicists. However, additional criticism of this use of game theory has been levied because some experiments have demonstrated that individuals do not play equilibrium strategies. For instance, in the centipede game, guess 2/3 of the average game, and the dictator game, people regularly do not play Nash equilibria. There is an ongoing debate regarding the importance of these experiments.[10] Alternatively, some authors claim that Nash equilibria do not provide predictions for human populations, but rather provide an explanation for why populations that play Nash equilibria remain in that state. However, the question of how populations reach those points remains open. Some game theorists have turned to evolutionary game theory in order to resolve these issues. These models presume either no rationality or bounded rationality on the part of players. Despite the name, evolutionary game theory does not necessarily presume natural selection in the biological sense. Evolutionary game theory includes both biological as well as cultural evolution and also models of individual learning (for example, fictitious play dynamics).
Prescriptive or normative analysis Cooperate Defect Cooperate
Defect
-1, -1 -10, 0 0, -10 -5, -5
The Prisoner's Dilemma
On the other hand, some scholars see game theory not as a predictive tool for the behavior of human beings, but as a suggestion for how people ought to behave. Since a Nash equilibrium of a game constitutes one's best response to the actions of the other players, playing a strategy that is part of a Nash equilibrium seems appropriate. However, this use for game theory has also come under criticism. First, in some cases it is appropriate to play a non-equilibrium strategy if one expects others to play non-equilibrium strategies as well. For an example, see Guess 2/3 of the average. Second, the Prisoner's dilemma presents another potential counterexample. In the Prisoner's Dilemma, each player pursuing his own self-interest leads both players to be worse off than had they not pursued their own self-interests.
Game theory
Economics and business Game theory is a major method used in mathematical economics and business for modeling competing behaviors of interacting agents.[11] Applications include a wide array of economic phenomena and approaches, such as auctions, bargaining, fair division, duopolies, oligopolies, social network formation, agent-based computational economics,[12] general equilibrium, mechanism design,[13] and voting systems,[14] and across such broad areas as experimental economics,[15] behavioral economics,[16] information economics,[17] industrial organization,[18] and political economy.[19] [20] This research usually focuses on particular sets of strategies known as equilibria in games. These "solution concepts" are usually based on what is required by norms of rationality. In non-cooperative games, the most famous of these is the Nash equilibrium. A set of strategies is a Nash equilibrium if each represents a best response to the other strategies. So, if all the players are playing the strategies in a Nash equilibrium, they have no unilateral incentive to deviate, since their strategy is the best they can do given what others are doing. The payoffs of the game are generally taken to represent the utility of individual players. Often in modeling situations the payoffs represent money, which presumably corresponds to an individual's utility. This assumption, however, can be faulty. A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of some particular economic situation. One or more solution concepts are chosen, and the author demonstrates which strategy sets in the presented game are equilibria of the appropriate type. Naturally one might wonder to what use should this information be put. Economists and business professors suggest two primary uses (noted above): descriptive and prescriptive.[8]
Political science The application of game theory to political science is focused in the overlapping areas of fair division, political economy, public choice, war bargaining, positive political theory, and social choice theory. In each of these areas, researchers have developed game-theoretic models in which the players are often voters, states, special interest groups, and politicians. For early examples of game theory applied to political science, see the work of Anthony Downs. In his book An Economic Theory of Democracy (Downs 1957), he applies the Hotelling firm location model to the political process. In the Downsian model, political candidates commit to ideologies on a one-dimensional policy space. The theorist shows how the political candidates will converge to the ideology preferred by the median voter. A game-theoretic explanation for democratic peace is that public and open debate in democracies send clear and reliable information regarding their intentions to other states. In contrast, it is difficult to know the intentions of nondemocratic leaders, what effect concessions will have, and if promises will be kept. Thus there will be mistrust and unwillingness to make concessions if at least one of the parties in a dispute is a non-democracy (Levy & Razin 2003).
Biology
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Game theory
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Hawk Dove Hawk
Dove
20, 20 40, 80
80, 40 60, 60
The hawk-dove game
Unlike economics, the payoffs for games in biology are often interpreted as corresponding to fitness. In addition, the focus has been less on equilibria that correspond to a notion of rationality, but rather on ones that would be maintained by evolutionary forces. The best known equilibrium in biology is known as the evolutionarily stable strategy (or ESS), and was first introduced in (Smith & Price 1973). Although its initial motivation did not involve any of the mental requirements of the Nash equilibrium, every ESS is a Nash equilibrium. In biology, game theory has been used to understand many different phenomena. It was first used to explain the evolution (and stability) of the approximate 1:1 sex ratios. (Fisher 1930) suggested that the 1:1 sex ratios are a result of evolutionary forces acting on individuals who could be seen as trying to maximize their number of grandchildren. Additionally, biologists have used evolutionary game theory and the ESS to explain the emergence of animal communication (Harper & Maynard Smith 2003). The analysis of signaling games and other communication games has provided some insight into the evolution of communication among animals. For example, the mobbing behavior of many species, in which a large number of prey animals attack a larger predator, seems to be an example of spontaneous emergent organization. Ants have also been shown to exhibit feed-forward behavior akin to fashion, see Butterfly Economics. Biologists have used the game of chicken to analyze fighting behavior and territoriality. Maynard Smith, in the preface to Evolution and the Theory of Games, writes, "paradoxically, it has turned out that game theory is more readily applied to biology than to the field of economic behaviour for which it was originally designed". Evolutionary game theory has been used to explain many seemingly incongruous phenomena in nature.[21] One such phenomenon is known as biological altruism. This is a situation in which an organism appears to act in a way that benefits other organisms and is detrimental to itself. This is distinct from traditional notions of altruism because such actions are not conscious, but appear to be evolutionary adaptations to increase overall fitness. Examples can be found in species ranging from vampire bats that regurgitate blood they have obtained from a night's hunting and give it to group members who have failed to feed, to worker bees that care for the queen bee for their entire lives and never mate, to Vervet monkeys that warn group members of a predator's approach, even when it endangers that individual's chance of survival.[22] All of these actions increase the overall fitness of a group, but occur at a cost to the individual. Evolutionary game theory explains this altruism with the idea of kin selection. Altruists discriminate between the individuals they help and favor relatives. Hamilton's rule explains the evolutionary reasoning behind this selection with the equation c 12, deterministic strategies fail to be optimal. For S = 13, choosing (3, 5, 5), (3, 3, 7) and (1, 5, 7) with probability 1/3 each can be shown to be the optimal probabilistic strategy.
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Blotto games
Application The 2000 US presidential election, one of the closest races in recent history, has been modeled as a Colonel Blotto game.[4] It is argued that Gore could have utilized a strategy that would have won the election, but that such a strategy was not identifiable ex ante.
External links • Ayala Arad and Ariel Rubinstein's article Colonel Blotto's Top secret Files: Multi-Dimensional Iterative Reasoning in Action [5] • Jonathan Partington's Colonel Blotto page [6]
References [1] [2] [3] [4]
The Theory of Play and Integral Equations with Skew Symmetric Kernels (http:/ / www. jstor. org/ stable/ 1906946) The Colonel Blotto game (http:/ / www. springerlink. com/ index/ f22232k0j13816r8. pdf) A Continuous Colonel Blotto Game (http:/ / www. rand. org/ pubs/ research_memoranda/ 2006/ RM408. pdf) Lotto, Blotto, or Frontrunner: An Analysis of Spending Patterns by the National Party Committees in the 2000 Presidential Election (http:/ / www. socsci. duke. edu/ ssri/ federalism/ papers/ tofiasmunger. pdf) [5] http:/ / arielrubinstein. tau. ac. il/ papers/ generals. pdf [6] http:/ / www. amsta. leeds. ac. uk/ ~pmt6jrp/ personal/ blotto. html
War of attrition In game theory, the war of attrition is a model of aggression in which two contestants compete for a resource of value V by persisting while constantly accumulating costs over the time t that the contest lasts. The model was originally formulated by John Maynard Smith[1] , a mixed evolutionary stable strategy (ESS) was determined by Bishop & Cannings[2] . Strategically, the game is an auction, in which the prize goes to the player with the highest bid, and each player pays the loser's low bid (making it an all-pay sealed-bid second-price auction).
Examining the game The war of attrition cannot be properly solved using the payoff matrix. The players' available resources are the only limit to the maximum value of bids; bids can be any number if available resources are ignored, meaning that for any value of α, there is a value β that is greater. Attempting to put all possible bids onto the matrix, however, will result in an ∞×∞ matrix. One can, however, use a pseudo-matrix form of war of attrition to understand the basic workings of the game, and analyze some of the problems in representing the game in this manner. The game works as follows: Each player makes a bid; the one who bids the highest wins a resource of value V. Each player pays the lowest bid, a. The premise that the players may bid any number is important to analysis of the game. The bid may even exceed the value of the resource that is contested over. This at first appears to be irrational, being seemingly foolish to pay more for a resource than its value; however, remember that each bidder only pays the low bid. Therefore, it would seem to be in each player's best interest to bid the maximum possible amount rather than an amount equal to or less than the value of the resource. There is a catch, however; if both players bid higher than V, the high bidder does not so much win as lose less. This situation is commonly referred to as a Pyrrhic victory. In contrast, if each player bids less than V, the player bidding a will lose, and the other player will benefit by an amount of V-a. If each player bids the same amount for a less than V/2, they split the value of V, each gaining V/2-a. For a tie such that a>V/2, they both lose the difference of V/2 and a. Luce and Raiffa
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War of attrition
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(cite_Luce.2C_R._D._and_Raiffa.2C_H._.281957.29._Games_and_Decisions:_Introduction_and_Critical_Survey._Wiley.2C_New_York._Paperback_reprint.2C_Do referred to the latter situation as a "ruinous situation"; the point at which both players suffer, and there is no winner. The conclusion one can draw from this pseudo-matrix is that there is no value to bid which is beneficial in all cases, so there is no dominant strategy. However, this fact and the above argument do not preclude the existence of Nash Equilibria. Any pair of strategies with the following characteristics is a Nash Equilibrium: • One player bids zero • The other player bids any value equal to V or higher, or mixes among any values V or higher. With these strategies, one player wins and pays zero, and the other player loses and pays zero. It is easy to verify that neither player can strictly gain by unilaterally deviating.
Dynamic formulation and Evolutionary stable strategy Another popular formulation of the war of attrition is as follows: Two players are involved in a dispute. The value of the object to each player is . Time is modeled as a continuous variable which starts at zero and runs indefinitely. Each player chooses when to concede the object to the other player. In the case of a tie, each player receives utility. Time is valuable, each player uses one unit of utility per period of time. This formulation is slightly more complex since it allows each player to assign a different value to the object. Its equilibria are not as obvious as the other formulation.The evolutionary stable strategy is a mixed ESS, in which the probability of persisting for a length of time t is: The evolutionary stable strategy below represents the most probable value of a. The value p(t) for a contest with a resource of value V over time t, is the probability that t = a. This strategy does not guarantee the win; rather it is the optimal balance of risk and reward. The outcome of any particular game cannot be predicted as the random factor of the opponent's bid is too unpredictable.
That no pure persistence time is an ESS can be demonstrated simply by considering a putative ESS bid of x, which will be beaten by a bid of x+ .
The ESS in popular culture The evolutionarily stable strategy when playing this game is a probability density of random persistence times which cannot be predicted by the opponent in any particular contest. This result has led to the prediction that threat displays ought not to evolve, and to the conclusion in The Illuminatus! Trilogy that optimal military strategy is to behave in a completely unpredictable, and therefore insane, manner. Neither of these conclusions appear to be truly quantifiably reasonable applications of the model to realistic conditions.
Conclusions By examining the unusual results of this game, it serves to mathematically prove another piece of old wisdom: "Expect the unexpected". By making the assumption that an opponent will act irrationally, one can paradoxically better predict their actions, as they are limited in this game. They will either act rationally, and take the optimal decision, or they will be irrational, and take the non-optimal solution. If one considers the irrational as a bluff and the rational as backing down from a bluff, it transforms the game into another game theory game, Hawk and Dove.
War of attrition
180
References [1] Maynard Smith, J. (1974) Theory of games and the evolution of animal contests. Journal of Theoretical Biology 47: 209-221. [2] Bishop, D.T. & Cannings, C. (1978) A generalized war of attrition. Journal of Theoretical Biology 70: 85-124.
Sources • Bishop, D.T., Cannings, C. & Maynard Smith, J. (1978) The war of attrition with random rewards. Journal of Theoretical Biology 74:377-389. • Maynard Smith, J. & Parker, G. A. (1976). The logic of asymmetric contests. Animal Behaviour. 24:159-175. • Luce,R.D. & Raiffa, H. (1957) "Games and Decisions: Introduction and Critical Survey"(originally published as "A Study of the Behavioral Models Project, Bureau of Applied Social Research") John Wiley & Sons Inc., New York • Rapaport,Anatol (1966) "Two Person Game Theory" University of Michigan Press, Ann Arbor
External links • Exposition of the derivation of the ESS (http://www.holycross.edu/departments/biology/kprestwi/behavior/ ESS/warAtt_mixESS.html) - From Ken Prestwich's Game Theory website at College of the Holy Cross
El Farol Bar problem The El Farol bar problem is a problem in game theory. Based on a bar in Santa Fe, New Mexico, it was created in 1994 by W. Brian Arthur. The problem is as follows: There is a particular, finite population of people. Every Thursday night, all of these people want to go to the El Farol Bar. However, the El Farol is quite small, and it's no fun to go there if it's too crowded. So much so, in fact, that the preferences of the population can be described as follows: • If less than 60% of the population go to the bar, they'll all have a better time than if they stayed at home.
El Farol in Santa Fe, New MexicoSanta Fe
• If more than 60% of the population go to the bar, they'll all have a worse time than if they stayed at home. Unfortunately, it is necessary for everyone to decide at the same time whether they will go to the bar or not. They cannot wait and see how many others go on a particular Thursday before deciding to go themselves on that Thursday. One aspect of the problem is that, no matter what method each person uses to decide if they will go to the bar or not, if everyone uses the same pure strategy it is guaranteed to fail. If everyone uses the same deterministic method, then if that method suggests that the bar will not be crowded, everyone will go, and thus it will be crowded; likewise, if that method suggests that the bar will be crowded, nobody will go, and thus it will not be crowded. Often the solution to such problems in game theory is to permit each player to use a mixed strategy, where a choice is made with a particular probability. In the case of the single-stage El Farol Bar problem, there exists a unique symmetric Nash equilibrium mixed strategy where all players choose to go to the bar with a certain probability that is a function of the number of players, the threshold for crowdedness, and the relative utility of going to a crowded or an uncrowded bar compared to staying home. There are also multiple Nash equilibria where one or more players use a pure strategy, but these equilibria are not symmetric[1] Several variants are considered in [2] .
El Farol Bar problem In some variants of the problem, the people are allowed to communicate with each other before deciding to go to the bar. However, they are not required to tell the truth.
Minority Game One variant of the El Farol Bar problem is the minority game proposed by Yi-Cheng Zhang and Damien Challet from the University of Fribourg. In the minority game, an odd number of players each must choose one of two choices independently at each turn. The players who end up on the minority side win. While the El Farol Bar problem was originally formulated to analyze a decision-making method other than deductive rationality, the minority game examines the characteristic of the game that no single deterministic strategy may be adopted by all participants in equilibrium. Allowing for mixed strategies in the single-stage minority game produces a unique symmetric Nash equilibrium, which is for each player to choose each action with 50% probability, as well as multiple equilibria that are not symmetric. The minority game was featured in the manga Liar Game. In that multi-stage minority game, the majority was eliminated from the game until only one player was left. Players were shown engaging in cooperative strategies.
External links • • • •
An Introductory Guide to the Minority Game [3] Minority game on arxiv.org [4] El Farol bar in Santa Fe, New Mexico [5] Software for Minority Games modelling [6]
References • W. Brian Arthur, “Inductive Reasoning and Bounded Rationality” [7], American Economic Review (Papers and Proceedings), 84,406–411, 1994. [1] Whitehead, Duncan. " The El Farol Bar Problem Revisited: Reinforcement Learning in a Potential Game (http:/ / www. econ. ed. ac. uk/ papers/ The El Farol Bar Problem Revisited. pdf)", University of Edinburgh, September 17, 2008 [2] Gintis, Herbert. Game Theory Evolving (Princeton: Princeton University Press, 2009), Section 6.24: El Farol, p. 134 [3] http:/ / markov. uc3m. es/ last-papers/ the-minority-game-an-introductory-guide. html [4] http:/ / xstructure. inr. ac. ru/ x-bin/ theme3. py?level=1& index1=117820 [5] http:/ / elfarolsf. com [6] http:/ / agf. statsolutions. eu [7] http:/ / www. santafe. edu/ arthur/ Papers/ El_Farol. html
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Fair division
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Fair division Fair division, also known as the cake-cutting problem, is the problem of dividing a resource in such a way that all recipients believe that they have received a fair amount. The problem is easier when recipients have different measures of value of the parts of the resource: in the "cake cutting" version, one recipient may like marzipan, another prefers cherries, and so on—then, and only then, the n recipients may get even more than what would be one n-th of the value of the "cake" for each of them. On the other hand, the presence of different measures opens a vast potential for many challenging questions and directions of further research. There are a number of variants of the problem. The definition of 'fair' may simply mean that they get at least their fair proportion, or harder requirements like envy-freeness may also need to be satisfied. The theoretical algorithms mainly deal with goods that can be divided without losing value. The division of indivisible goods, as in for instance a divorce, is a major practical problem. Chore division is a variant where the goods are undesirable. Fair division is often used to refer to just the simplest variant. That version is referred to here as proportional division or simple fair division. Most of what is normally called a fair division is not considered so by the theory because of the use of arbitration. This kind of situation happens quite often with mathematical theories named after real life problems. The decisions in the Talmud on entitlement when an estate is bankrupt reflect some quite complex ideas about fairness,[1] and most people would consider them fair. However they are the result of legal debates by rabbis rather than divisions according to the valuations of the claimants.
Assumptions Fair division is a mathematical theory based on an idealization of a real life problem. The real life problem is the one of dividing goods or resources fairly between people, the 'players', who have an entitlement to them. The central tenet of fair division is that such a division should be performed by the players themselves, maybe using a mediator but certainly not an arbiter as only the players really know how they value the goods. The theory of fair division provides explicit criteria for various different types of fairness. Its aim is to provide procedures (algorithms) to achieve a fair division, or prove their impossibility, and study the properties of such divisions both in theory and in real life.
Berlin divided by the Potsdam Conference
The assumptions about the valuation of the goods or resources are: • Each player has their own opinion of the value of each part of the goods or resources • The value to a player of any allocation is the sum of his valuations of each part. Often just requiring the valuations be weakly additive is enough. • In the basic theory the goods can be divided into parts with arbitrarily small value. Indivisible parts make the theory much more complex. An example of this would be where a car and a motorcycle have to be shared. This is also an example of where the values may not add up nicely, as either can be used as transport. The use of money can make such problems much easier. The criteria of a fair division are stated in terms of a players valuations, their level of entitlement, and the results of a fair division procedure. The valuations of the other players are not involved in the criteria. Differing entitlements can normally be represented by having a different number of proxy players for each player but sometimes the criteria specify something different.
Fair division In the real world of course people sometimes have a very accurate idea of how the other players value the goods and they may care very much about it. The case where they have complete knowledge of each other's valuations can be modeled by game theory. Partial knowledge is very hard to model. A major part of the practical side of fair division is the devising and study of procedures that work well despite such partial knowledge or small mistakes. A fair division procedure lists actions to be performed by the players in terms of the visible data and their valuations. A valid procedure is one that guarantees a fair division for every player who acts rationally according to their valuation. Where an action depends on a player's valuation the procedure is describing the strategy a rational player will follow. A player may act as if a piece had a different value but must be consistent. For instance if a procedure says the first player cuts the cake in two equal parts then the second player chooses a piece, then the first player cannot claim that the second player got more. What the players do is: • Agree on their criteria for a fair division • Select a valid procedure and follow its rules It is assumed the aim of each player is to maximize the minimum amount they might get, or in other words, to achieve the maximin. Procedures can be divided into finite and continuous procedures. A finite procedure would for instance only involve one person at a time cutting or marking a cake. Continuous procedures involve things like one player moving a knife and the other saying stop. Another type of continuous procedure involves a person assigning a value to every part of the cake.
Criteria for a fair division There are a number of widely used criteria for a fair division. Some of these conflict with each other but often they can be combined. The criteria described here are only for when each player is entitled to the same amount. • A proportional or simple fair division guarantees each player gets his fair share. For instance if three people divide up a cake each gets at least a third by their own valuation. • An envy-free division guarantees no-one will want somebody else's share more than their own. • An exact division is one where every player thinks everyone received exactly their fair share, no more and no less. • An efficient or Pareto optimal division ensures no other allocation would make someone better off without making someone else worse off. The term efficiency comes from the economics idea of the efficient market. A division where one player gets everything is optimal by this definition so on its own this does not guarantee even a fair share. • An equitable division is one where the proportion of the cake a player receives by their own valuation is the same for every player. This is a difficult aim as players need not be truthful if asked their valuation.
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Two players For two people there is a simple solution which is commonly employed. This is the so-called divide and choose method. One person divides the resource into what they believe are equal halves, and the other person chooses the "half" they prefer. Thus, the person making the division has an incentive to divide as fairly as possible: for if they do not, they will likely receive an undesirable portion. This solution gives a proportional and envy-free division. The article on divide and choose describes why the procedure is not equitable. More complex procedures like the adjusted winner procedure are designed to cope with indivisible goods and to be more equitable in a practical context. Austin's moving-knife procedure[2] gives an exact division for two players. The first player places two knives over the cake such that one knife is at the left side of the cake, and one is further right; half of the cake lies between the knives. He then moves the knives right, always ensuring there is half the cake – by his valuation – between the knives. If he reaches the right side of the cake, the leftmost knife must be where the rightmost knife started off. The second player stops the knives when he thinks there is half the cake between the knives. There will always be a point at which this happens, because of the intermediate value theorem. The surplus procedure (SP) achieves a form of equitability called proportional equitability. This procedure is strategy proof and can be generalized to more than two people.[3]
Many players Fair division with three or more players is considerably more complex than the two player case. Proportional division is the easiest and the article describes some procedures which can be applied with any number of players. Finding the minimum number of cuts needed is an interesting mathematical problem. Envy-free division was first solved for the 3 player case in 1960 independently by John Selfridge of Northern Illinois University and John Horton Conway at Cambridge University. The best algorithm uses at most 5 cuts. The Brams-Taylor procedure was the first cake-cutting procedure for four or more players that produced an envy-free division of cake for any number of persons and was published by Steven Brams and Alan Taylor in 1995.[4] This number of cuts that might be required by this procedure is unbounded. A bounded moving knife procedure for 4 players was found in 1997. There are no discrete algorithms for an exact division even for two players, a moving knife procedure is the best that can be done. There are no exact division algorithms for 3 or more players but there are 'near exact' algorithms which are also envy-free and can achieve any desired degree of accuracy. A generalization of the surplus procedure called the equitable procedure (EP) achieves a form of equitability. Equitability and envy-freeness can be incompatible for 3 or more players.[3]
Variants Some cake-cutting procedures are discrete, whereby players make cuts with a knife (usually in a sequence of steps). Moving-knife procedures, on the other hand, allow continuous movement and can let players call "stop" at any point. A variant of the fair division problem is chore division: this is the "dual" to the cake-cutting problem in which an undesirable object is to be distributed amongst the players. The canonical example is a set of chores that the players between them must do. Note that "I cut, you choose" works for chore division. A basic theorem for many person problems is the Rental Harmony Theorem by Francis Su.[5] An interesting application of the Rental Harmony Theorem can be found in the international trade theory.[6] Sperner's Lemma can be used to get as close an approximation as desired to an envy-free solutions for many players. The algorithm gives a fast and practical way of solving some fair division problems.[7] [8] [9]
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Fair division The division of property, as happens for example in divorce or inheritance, normally contains indivisible items which must be fairly distributed between players, possibly with cash adjustments (such pieces are referred to as atoms). A common requirement for the division of land is that the pieces be connected, i.e. only whole pieces and not fragments are allowed. For example the division of Berlin after World War 2 resulted in four connected parts.[10] A consensus halving is where a number of people agree that a resource has been evenly split in two, this is described in exact division.
History According to Sol Garfunkel, the cake-cutting problem had been one of the most important open problems in 20th century mathematics,[11] when the most important variant of the problem was finally solved with the Brams-Taylor procedure by Steven Brams and Alan Taylor in 1995. Divide and choose has probably been used since prehistory . The related activities of bargaining and barter are also ancient. Negotiations involving more than two people are also quite common, the Potsdam Conference is a notable recent example. The theory of fair division dates back only to the end of the second world war. It was devised by a group of Polish mathematicians, Hugo Steinhaus, Bronisław Knaster and Stefan Banach, who used to meet in the Scottish Café in Lvov (then in Poland). A proportional (fair division) division for any number of players called 'last-diminisher' was devised in 1944. This was attributed to Banach and Knaster by Steinhaus when he made the problem public for the first time at a meeting of the Econometric Society in Washington D.C. on 17 September 1947. At that meeting he also proposed the problem of finding the smallest number of cuts necessary for such divisions. Envy-free division was first solved for the 3 player case in 1960 independently by John Selfridge of Northern Illinois University and John Horton Conway at Cambridge University, the algorithm was first published in the 'Mathematical Games' column by Martin Gardner in Scientific American. Envy-free division for 4 or more players was a difficult open problem of the twentieth century. The first cake-cutting procedure that produced an envy-free division of cake for any number of persons was first published by Steven Brams and Alan Taylor in 1995. A major advance on equitable division was made in 2006 by Steven J. Brams, Michael A. Jones, and Christian Klamler.[3]
In popular culture • In Numb3rs season 3 episode "One Hour", Charlie talks about the cake-cutting problem as applied to the amount of money a kidnapper was demanding. • Hugo Steinhaus wrote about a number of variants of fair division in his book Mathematical Snapshots. In his book he says a special three-person version of fair division was devised by G. Krochmainy in Berdechów in 1944 and another by Mrs L Kott.[12] • Martin Gardner and Ian Stewart have both published books with sections about the problem.[13] [14] Martin Gardner introduced the chore division form of the problem. Ian Stewart has popularized the fair division problem with his articles in Scientific American and New Scientist. • A Dinosaur Comics strip is based on the cake-cutting problem.[15]
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References [1] Game Theoretic Analysis of a bankruptcy Problem from the Talmud (http:/ / www. elsevier. com/ framework_aboutus/ Nobel/ Nobel2005/ nobel2005pdfs/ aum16. pdf) Robert J. Aumann and Michael Maschler. Journal of Economic Theory 36, 195-213 (1985) [2] A.K. Austin. Sharing a Cake. Mathematical Gazette 66 1982 [3] Brams, Steven J.; Michael A. Jones and Christian Klamler (December 2006). "Better Ways to Cut a Cake" (http:/ / www. ams. org/ notices/ 200611/ fea-brams. pdf) (PDF). Notices of the American Mathematical Society 53 (11): pp.1314–1321. . Retrieved 2008-01-16. [4] Steven J. Brams; Alan D. Taylor (January 1995). "An Envy-Free Cake Division Protocol". The American Mathematical Monthly (Mathematical Association of America) 102 (1): 9–18. doi:10.2307/2974850. JSTOR 2974850. [5] Francis Edward Su (1999). "Rental Harmony: Sperner's Lemma in Fair Division" (http:/ / www. math. hmc. edu/ ~su/ papers. dir/ rent. pdf). Amer. Math. Monthly 106 (10): 930–942. doi:10.2307/2589747. . [6] Shiozawa, Y. A (2007). "New Construction ofa Ricardian Trade Theory". Evolutionary and Institutional Economics Review 3 (2): 141–187. [7] Francis Edward Su. Cited above. (based on work by Forest Simmons 1980) [8] "The Fair Division Calculator" (http:/ / www. math. hmc. edu/ ~su/ fairdivision/ calc/ ). . [9] Ivars Peterson (March 13, 2000). "A Fair Deal for Housemates" (http:/ / www. maa. org/ mathland/ mathtrek_3_13_00. html). MathTrek. . [10] Steven J. Brams; Alan D. Taylor (1996). Fair division: from cake-cutting to dispute resolution. Cambridge University Press. p. 38. ISBN 978-0521556446. [11] Sol Garfunkel. More Equal than Others: Weighted Voting. For All Practical Purposes. COMAP. 1988 [12] Mathematical Snapshots. H.Steinhaus. 1950, 1969 ISBN 0-19-503267-5 [13] aha! Insight. Martin. Gardner, 1978. ISBN ISBN 978-0716710172 [14] How to cut a cake and other mathematical conundrums. Ian Stewart. 2006. ISBN 978-0199205905 [15] http:/ / www. qwantz. com/ archive/ 001345. html
Further reading • Steven J. Brams and Alan D. Taylor (1996). Fair Division - From cake-cutting to dispute resolution Cambridge University Press. ISBN 0-521-55390-3 • T.P. Hill (2000). "Mathematical devices for getting a fair share", American Scientist, Vol. 88, 325-331. • Jack Robertson and William Webb (1998). Cake-Cutting Algorithms: Be Fair If You Can, AK Peters Ltd, . ISBN 1-56881-076-8.
External links • Short essay about the cake-cutting problem (http://3quarksdaily.blogs.com/3quarksdaily/2005/04/ 3qd_monday_musi.html) by S. Abbas Raza of 3 Quarks Daily. • Fair Division (http://www.colorado.edu/education/DMP/fair_division.html) from the Discrete Mathematics Project at the University of Colorado at Boulder. • The Fair Division Calculator (http://www.math.hmc.edu/~su/fairdivision/calc/) (Java applet) at Harvey Mudd College • Fair Division: Method of Lone Divider (http://www.cut-the-knot.org/Curriculum/SocialScience/LoneDivider. shtml) • Fair Division: Method of Markers (http://www.cut-the-knot.org/Curriculum/SocialScience/Markers.shtml) • Fair Division: Method of Sealed Bids (http://www.cut-the-knot.org/Curriculum/SocialScience/SealedBids. shtml) • Vincent P. Crawford (1987). "fair division," The New Palgrave: A Dictionary of Economics, v. 2, pp. 274–75. • Hal Varian (1987). "fairness," The New Palgrave: A Dictionary of Economics, v. 2, pp. 275–76. • Bryan Skyrms (1996). The Evolution of the Social Contract Cambridge University Press. ISBN 9780521555838
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Cournot competition
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Cournot competition Cournot competition is an economic model used to describe an industry structure in which companies compete on the amount of output they will produce, which they decide on independently of each other and at the same time. It is named after Antoine Augustin Cournot[1] (1801-1877) who was inspired by observing competition in a spring water duopoly. It has the following features: • • • • • •
There is more than one firm and all firms produce a homogeneous product, i.e. there is no product differentiation; Firms do not cooperate, i.e. there is no collusion; Firms have market power, i.e. each firm's output decision affects the good's price; The number of firms is fixed; Firms compete in quantities, and choose quantities simultaneously; The firms are economically rational and act strategically, usually seeking to maximize profit given their competitors' decisions.
An essential assumption of this model is the "not conjecture" that each firm aims to maximize profits, based on the expectation that its own output decision will not have an effect on the decisions of its rivals. Price is a commonly known decreasing function of total output. All firms know , the total number of firms in the market, and take the output of the others as given. Each firm has a cost function
. Normally the cost functions are treated as
common knowledge. The cost functions may be the same or different among firms. The market price is set at a level such that demand equals the total quantity produced by all firms. Each firm takes the quantity set by its competitors as a given, evaluates its residual demand, and then behaves as a monopoly.
Graphically finding the Cournot duopoly equilibrium This section presents an analysis of the model with 2 firms and constant marginal cost. = firm 1 price, = firm 1 quantity,
= firm 2 price = firm 2 quantity
= marginal cost, identical for both firms Equilibrium prices will be:
This implies that firm 1’s profit is given by • Calculate firm 1’s residual demand: Suppose firm 1 believes firm 2 is producing quantity
. What is firm 1's
optimal quantity? Consider the diagram 1. If firm 1 decides not to produce anything, then price is given by . If firm 1 produces then price is given by . More generally, for each quantity that firm 1 might decide to set, price is given by the curve
. The curve
is called firm 1’s
residual demand; it gives all possible combinations of firm 1’s quantity and price for a given value of
.
Cournot competition
188
• Determine firm 1’s optimum output: To do this we must find where marginal revenue equals marginal cost. Marginal cost (c) is assumed to be constant. Marginal revenue is a curve - with twice the slope of and with the same vertical intercept. The point at which the two curves ( corresponds to quantity
. Firm 1’s optimum
. If
) intersect
, depends on what it believes firm 2 is doing. To find
an equilibrium, we derive firm 1’s optimum for other possible values of values of
and
. Diagram 2 considers two possible
, then the first firm's residual demand is effectively the market demand,
The optimal solution is for firm 1 to choose the monopoly quantity;
(
firm 2 were to choose the quantity corresponding to perfect competition,
such that
firm 1’s optimum would be to produce nil: marginal revenue corresponding to
.
is monopoly quantity). If , then
. This is the point at which marginal cost intercepts the .
Cournot competition
189
• It can be shown that, given the linear demand and constant marginal cost, the function Because we have two points, we can draw the entire function has changed, The function
is also linear.
, see diagram 3. Note the axis of the graphs
is firm 1’s reaction function, it gives firm 1’s optimal choice for each possible
choice by firm 2. In other words, it gives firm 1’s choice given what it believes firm 2 is doing.
Cournot competition
190
• The last stage in finding the Cournot equilibrium is to find firm 2’s reaction function. In this case it is symmetrical to firm 1’s as they have the same cost function. The equilibrium is the intersection point of the reaction curves. See diagram 4.
• The prediction of the model is that the firms will choose Nash equilibrium output levels.
Calculating the equilibrium In very general terms, let the price function for the (duopoly) industry be structure
and firm i have the cost
. To calculate the Nash equilibrium, the best response functions of the firms must first be
calculated. The profit of firm i is revenue minus cost. Revenue is the product of price and quantity and cost is given by the firm's cost function, so profit is (as described above): . The best response is to find the value of
that maximises
given
, with
, i.e. given some output of the opponent firm, the output
that maximises profit is found. Hence, the maximum of derivative of
with respect to
with respect to
is to be found. First take the
:
Setting this to zero for maximization:
The values of
that satisfy this equation are the best responses. The Nash equilibria are where both
best responses given those values of
and
.
and
are
Cournot competition
191
An example Suppose the industry has the following price structure: cost structure
such that
and
The profit of firm i (with for ease of computation) is:
The maximization problem resolves to (from the general case):
Without loss of generality, consider firm 1's problem:
By symmetry:
These are the firms' best response functions. For any value of
, firm 1 responds best with any value of
that
satisfies the above. In Nash equilibria, both firms will be playing best responses so solving the above equations simultaneously. Substituting for in firm 1's best response:
The symmetric Nash equilibrium is at
. (See Holt (2005, Chapter 13) for asymmetric examples.) Making
suitable assumptions for the partial derivatives (for example, assuming each firm's cost is a linear function of quantity and thus using the slope of that function in the calculation), the equilibrium quantities can be substituted in the assumed industry price structure to obtain the equilibrium market price.
Cournot competition
192
Cournot competition with many firms and the Cournot Theorem For an arbitrary number of firms, N>1, the quantities and price can be derived in a manner analogous to that given above. With linear demand and identical, constant marginal cost the equilibrium values are as follows: edit; we should specify the constants. Given the following results are these; Market Demand; Cost Function;
, for all i , which is each individual firm's output , which is total industry output , which is the market clearing price
and , which is each individual firm's profit. The Cournot Theorem then states that, in absence of fixed costs of production, as the number of firms in the market, N, goes to infinity, market output, Nq, goes to the competitive level and the price converges to marginal cost.
Hence with many firms a Cournot market approximates a perfectly competitive market. This result can be generalized to the case of firms with different cost structures (under appropriate restrictions) and non-linear demand. When the market is characterized by fixed costs of production, however, we can endogenize the number of competitors imagining that firms enter in the market until their profits are zero. In our linear example with firms, when fixed costs for each firm are
, we have the endogenous number of firms:
and a production for each firm equal to: This equilibrium is usually known as Cournot equilibrium with endogenous entry, or Marshall equilibrium [2] .
Implications • Output is greater with Cournot duopoly than monopoly, but lower than perfect competition. • Price is lower with Cournot duopoly than monopoly, but not as low as with perfect competition. • According to this model the firms have an incentive to form a cartel, effectively turning the Cournot model into a Monopoly. Cartels are usually illegal, so firms might instead tacitly collude using self-imposing strategies to reduce output which, ceteris paribus will raise the price and thus increase profits for all firms involved.
Cournot competition
193
Bertrand versus Cournot Although both models have similar assumptions, they have very different implications: • Since the Bertrand model assumes that firms compete on price and not output quantity, it predicts that a duopoly is enough to push prices down to marginal cost level, meaning that a duopoly will result in perfect competition. • Neither model is necessarily "better." The accuracy of the predictions of each model will vary from industry to industry, depending on the closeness of each model to the industry situation. • If capacity and output can be easily changed, Bertrand is a better model of duopoly competition. If output and capacity are difficult to adjust, then Cournot is generally a better model. • Under some conditions the Cournot model can be recast as a two stage model, where in the first stage firms choose capacities, and in the second they compete in Bertrand fashion. However, when number of firms goes to infinity, Cournot model gives the same result as in Bertrand model: market price is pushed to marginal cost level.
References • Holt, Charles. "Games and Strategic Behavior (PDF version)", http://people.virginia.edu/~cah2k/expbooknsf. pdf • Tirole, Jean. "The Theory of Industrial Organization", MIT Press, 1988. [1] Varian, Hal R. (2006), Intermediate microeconomics: a modern approach (7 ed.), W. W. Norton & Company, p. 490, ISBN 0393927024 [2] Etro, Federico. Simple models of competition (http:/ / dipeco. economia. unimib. it/ persone/ etro/ economia_e_politica_della_concorrenza/ notes. pdf), page 6, Dept. Political Economics -- Università di Milano-Bicocca, November 2006
Deadlock C
D
c 1, 1
0, 3
d 3, 0
2, 2
In game theory, Deadlock is a game where the action that is mutually most beneficial is also dominant. (An example payoff matrix for Deadlock is pictured to the right.) This provides a contrast to the Prisoner's Dilemma where the mutually most beneficial action is dominated. This makes Deadlock of rather less interest, since there is no conflict between self-interest and mutual benefit. The game provides some interest, however, since one has some motivation to encourage one's opponent to play a dominated strategy.
General definition
Deadlock
194
C
D
c a, b
c, d
d e, f
g, h
Any game that satisfies the following two conditions constitutes a Deadlock game: (1) e>g>a>c and (2) d>h>b>f. These conditions require that d and D be dominant. (d, D) be of mutual benefit, and that one prefer one's opponent play c rather than d. Like the Prisoner's Dilemma, this game has one unique Nash equilibrium: (d, D).
References • GameTheory.net [1]
References [1] http:/ / www. gametheory. net/ dictionary/ Games/ Deadlock. html
Unscrupulous diner's dilemma In game theory, the Unscrupulous diner's dilemma (or just Diner's dilemma) is an n-player prisoner's dilemma. The situation imagined is that several individuals go out to eat, and prior to ordering they agree to split the check equally between all of them. Each individual must now choose whether to order the expensive or inexpensive dish. It is presupposed that the expensive dish is better than the cheaper, but not by enough to warrant paying the difference compared to eating alone. Each individual reasons that the expense they add to their bill by ordering the more expensive item is very small, and thus the improved dining experience is worth the money. However, every individual reasons this way and they all end up paying for the cost of the more expensive meal, which, by assumption, is worse for everyone than ordering and paying for the cheaper meal.
Formal definition and equilibrium analysis Let g represent the joy of eating the expensive meal, b the joy of eating the cheap meal, h is the cost of the expensive meal, l the cost of the cheap meal, and n the number of players. From the description above we have the following ordering . Also, in order to make the game sufficiently similar to the Prisoner's dilemma we presume that one would prefer to order the expensive meal given others will help defray the cost,
Consider an arbitrary set of strategies by a player's opponent. Let the total cost of the other player's meals be x. The cost of ordering the cheap meal is utilities for each meal are
and the cost of ordering the expensive meal is for the expensive meal and
. So the
for the cheaper meal. By
assumption, the utility of ordering the expensive meal is higher. Remember that the choice of opponents' strategies was arbitrary and that the situation is symmetric. This proves that the expensive meal is strictly dominant and thus the unique Nash equilibrium. If everyone orders the expensive meal all of the diners pay h and their total utility is suppose that all the individuals had ordered the cheap meal, their utility would have been
. On the other hand . This
demonstrates the similarity between the Diner's dilemma and the Prisoner's dilemma. Like the Prisoner's dilemma,
Unscrupulous diner's dilemma everyone is worse off by playing the unique equilibrium than they would have been if they collectively pursued another strategy.
Experimental evidence Gneezy, Haruvy, and Yafe (2004) tested these results in a field experiment. Groups of six diners faced different billing arrangements. As predicted, subjects consume more when the bill is split equally than when they have to pay individually. Consumption is highest when the meal is free. Finally, members of some groups had to pay only one sixth of their individual costs. There was no difference between the amount consumed by these groups and those splitting the total cost of the meal equally. As the private cost of increased consumption is the same for both treatments but splitting the cost imposes a burden on other group members, this indicates that participants did not take the welfare of others into account when making their choices. This contrasts to a large number of laboratory experiments where subjects face analytically similar choices but the context is more abstract.
References • Glance, Huberman (1994). "The dynamics of social dilemmas" [1]. Scientific American. • Gneezy, U.; Haruvy, E.; Yafe, H. (2004). "The inefficiency of splitting the bill". The Economic Journal 114 (495): 265–280. doi:10.1111/j.1468-0297.2004.00209.x.
References [1] http:/ / www. sciamdigital. com/ index. cfm?fa=Products. ViewIssuePreview& ARTICLEID_CHAR=F76F506E-1A94-4FC6-A44B-C0F31E0F091
Guess 2/3 of the average In game theory, Guess 2/3 of the average is a game where several people guess what 2/3 of the average of their guesses will be, and where the numbers are restricted to the real numbers between 0 and 100, inclusive. The winner is the one closest to the 2/3 average.
Equilibrium analysis In this game there is no strictly dominant strategy. However, there is a unique pure strategy Nash equilibrium. This equilibrium can be found by iterated elimination of weakly dominated strategies. Guessing any number that lies above 66.67 is dominated for every player since it cannot possibly be 2/3 of the average of any guess. These can be eliminated. Once these strategies are eliminated for every player, any guess above 44.45 is weakly dominated for every player since no player will guess above 66.67 and 2/3 of 66.67 is approximately 44.45. This process will continue until all numbers above 0 have been eliminated. This degeneration does not occur in quite the same way if choices are restricted to, for example, the integers between 0 and 100. In this case, all integers except 0 and 1 vanish; it becomes advantageous to select 0 if one expects that at least 1/4 of all players will do so, and 1 if otherwise. (In this way, it is a lopsided version of the so-called "consensus game", where one wins by being in the majority.)
195
Guess 2/3 of the average
196
Experimental results This game is a common demonstration in game theory classes, where even economics graduate students fail to guess 0.[1] When performed among ordinary people it is usually found that the winner guess is much higher than 0, e.g., 21.6 was the winning value in a large internet-based competition organized by the Danish newspaper Politiken. This included 19,196 people and with a prize of 5000 Danish kroner.[2] Creativity Games has an on-line version of the game [3] where you play against the last 100 visitors. The Museum of Money has an interactive flash applet of the game calculate the current outcome.
[4]
, where each given answer will be used to
Rationality versus common knowledge of rationality This game illustrates the difference between perfect rationality of an actor and the common knowledge of rationality of all players. Even perfectly rational players playing in such a game should not guess 0 unless they know that the other players are rational as well and that all players' rationality is common knowledge. If a rational player reasonably believes that other players will not follow the chain of elimination described above, it would be rational for him/her to guess a number above 0. Interestingly, we can suppose that all the players are rational, but they do not have common knowledge of each other's rationality. Even in this case, it is not required that every player guess 0, since they may expect each other to behave irrationally.
Notes [1] Nagel, Rosemarie (1995). "Unraveling in Guessing Games: An Experimental Study". American Economic Review 85 (5): 1313–1326. JSTOR 2950991. [2] (Danish) Astrid Schou, Gæt-et-tal konkurrence afslører at vi er irrationelle (http:/ / politiken. dk/ erhverv/ article123939. ece), Politiken; includes a histogram (http:/ / konkurrence. econ. ku. dk/ distribution?id=1237& d=6655488e6252d35e705500b68a339c50) of the guesses. Note that some of the players guessed close to 100. A large number of players guessed 33.3 (i.e. 2/3 of 50), indicating an assumption that players would guess randomly. A smaller but significant number of players guessed 22.2 (i.e. 2/3 of 33.3), indicating a second iteration of this theory based on an assumption that players would guess 33.3. The final number of 21.6 was slightly below this peak, implying that on average each player iterated their assumption 1.07 times. [3] http:/ / twothirdsofaverage. creativitygames. net [4] http:/ / museumofmoney. org/ exhibitions/ games/ guessnumber. htm
Kuhn poker
Kuhn poker Kuhn poker is a simplified form of poker developed by Dr. Harold W. Kuhn. It is a zero sum two player game. The deck includes only three playing cards, for example a King, Queen, and Jack. One card is dealt to each player, then the first player must bet or pass, then the second player may bet or pass. If any player chooses to bet the opposing player must bet as well ("call") in order to stay in the round. After both players pass or bet the player with the highest card wins the pot. Kuhn demonstrated that there are many game theoretic optimal strategies for the first player in this game, but only one for the second player, and that, when played optimally, the first player should expect to lose at a rate of −1/18 per hand. In more conventional poker terms: • Each player antes 1 • Each player is dealt one of the three cards, and the third is put aside unseen • Player One can check or raise 1 • If Player One checks then Player Two can check or raise 1 • If Player Two checks there is a showdown for the pot of 2 • If Player Two raises then Player One can fold or call • If Player One folds then Player Two takes the pot of 3 • If Player One calls there is a showdown for the pot of 4 • If Player One raises then Player Two can fold or call • If Player Two folds then Player One takes the pot of 3 • If Player Two calls there is a showdown for the pot of 4
References • Kuhn, H. W. (1950). "Simplified Two-Person Poker". In Kuhn, H. W.; Tucker, A. W.. Contributions to the Theory of Games. 1. Princeton University Press. pp. 97–103.
External links • Effective Short-Term Opponent Exploitation in Simplified Poker [1]
References [1] http:/ / www. cs. ualberta. ca/ ~holte/ Publications/ aaai2005poker. pdf
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Nash bargaining game
198
Nash bargaining game The two person bargaining problem is a problem of understanding how two agents should cooperate when non-cooperation leads to Pareto-inefficient results. It is in essence an equilibrium selection problem; Many games have multiple equilibria with varying payoffs for each player, forcing the players to negotiate on which equilibrium to target. The quintessential example of such a game is the Ultimatum game. The underlying assumption of bargaining theory is that the resulting solution should be the same solution an impartial arbitrator would recommend. Solutions to bargaining come in two flavors: an axiomatic approach where desired properties of a solution are satisfied and a strategic approach where the bargaining procedure is modeled in detail as a sequential game.
The bargaining game The bargaining game or Nash bargaining game is a simple two-player game used to model bargaining interactions. In the Nash Bargaining Game two players demand a portion of some good (usually some amount of money). If the total amount requested by the players is less than that available, both players get their request. If their total request is greater than that available, neither player gets their request. A Nash bargaining solution is a (Pareto efficient) solution to a Nash bargaining game. According to Walker (2005), Nash's bargaining solution was shown by John Harsanyi to be the same as Zeuthen's solution of the bargaining problem (Problems of Monopoly and Economic Warfare, 1930).
An example Opera Football Opera
3,2
0,0
Football
0,0
2,3
Battle of the Sexes 1
The Battle of the Sexes, as shown, is a two player coordination game. Both Opera/Opera and Football/Football are Nash equilibria. Any probability distribution over these two Nash equilibria is a correlated equilibrium. The question then becomes which of the infinitely many possible equilibria should be chosen by the two players. If they disagree and choose different distributions, they are likely receive 0 payoffs. In this symmetric case the natural choice is to play Opera/Opera and Football/Football with equal probability. Indeed all bargaining solutions described below prescribe this solution. However, if the game is asymmetric --- for example, Football/Football instead yields payoffs of 2,5 --- the appropriate distribution is less clear. The problem of finding such a distribution is addressed by the bargaining theory.
Formal description A two person bargain problem consists of a disagreement, or threat, point respective payoffs to players 1 and player 2, and a feasibility set which are interpreted as agreements. Set
, a closed convex subset of
and
are the
, the elements of
is convex because an agreement could take the form of a correlated
combination of other agreements. The problem is nontrivial if agreements in disagreement. The goal of bargaining is to choose the feasible agreement negotiations.
, where
are better for both parties than the in
that could result from
Nash bargaining game
Feasibility set Which agreements are feasible depends on whether bargaining is mediated by an additional party. When binding contracts are allowed, any joint action is playable, and the feasibility set consists of all attainable payoffs better than the disagreement point. When binding contracts are unavailable, the players can defect (moral hazard), and the feasibility set is composed of correlated equilibria, since these outcomes require no exogenous enforcement.
Disagreement point The disagreement point is the value the players can expect to receive if negotiations break down. This could be some focal equilibrium that both players could expect to play. This point directly affects the bargaining solution, however, so it stands to reason that each player should attempt to choose his disagreement point in order to maximize his bargaining position. Towards this objective, it is often advantageous to increase one's own disagreement payoff while harming the opponent's disagreement payoff (hence the intrepretation of the disagreement as a threat). If threats are viewed as actions, then one can construct a separate game wherein each player chooses a threat and receives a payoff according to the outcome of bargaining. It is known as Nash's variable threat game. Alternatively, each player could play a minimax strategy in case of disagreement, choosing to disregard personal reward in order to hurt the opponent as much as possible shoud the opponent leave the bargaining table.
Equilibrium analysis Strategies are represented in the Nash bargaining game by a pair (x, y). x and y are selected from the interval [d, z], where z is the total good. If x + y is equal to or less than z, the first player receives x and the second y. Otherwise both get d. d here represents the disagreement point or the threat of the game; often . There are many Nash equilibria in the Nash bargaining game. Any x and y such that x + y = z is a Nash equilibrium. If either player increases their demand, both players receive nothing. If either reduces their demand they will receive less than if they had demanded x or y. There is also a Nash equilibrium where both players demand the entire good. Here both players receive nothing, but neither player can increase their return by unilaterally changing their strategy.
Bargaining solutions Various solutions have been proposed based on slightly different assumptions about what properties are desired for the final agreement point.
Nash bargaining solution John Nash proposed that a solution should satisfy certain axioms: 1. 2. 3. 4.
Invariant to affine transformations or Invariant to equivalent utility representations Pareto optimality Independence of irrelevant alternatives Symmetry
Let u and v be the utility functions of Player 1 and Player 2, respectively. In the Nash bargaining solution, the players will seek to maximize , where and , are the status quo utilities (i.e. the utility obtained if one decides not to bargain with the other player). The product of the two excess utilities is generally referred to as the Nash product.
199
Nash bargaining game
200
Kalai-Smorodinsky bargaining solution Independence of Irrelevant Alternatives can be substituted with a monotonicity condition, as demonstrated by Ehud Kalai and Meir Smorodinsky. It is the point which maintains the ratios of maximal gains. In other words, if player 1 could receive a maximum of with player 2’s help (and vice-versa for ), then the Kalai-Smorodinsky bargaining solution would yield the point
on the Pareto frontier such that
.
Egalitarian bargaining solution The egalitarian bargaining solution, introduced by Ehud Kalai, is a third solution which drops the condition of scale invariance while including both the axiom of Independence of irrelevant alternatives, and the axiom of monotonicity. It is the solution which attempts to grant equal gain to both parties. In other words, it is the point which maximizes the minimum payoff among players.
Applications Some philosophers and economists have recently used the Nash bargaining game to explain the emergence of human attitudes toward distributive justice (Alexander 2000; Alexander and Skyrms 1999; Binmore 1998, 2005). These authors primarily use evolutionary game theory to explain how individuals come to believe that proposing a 50-50 split is the only just solution to the Nash Bargaining Game.
References • Alexander, Jason McKenzie (2000). "Evolutionary Explanations of Distributive Justice". Philosophy of Science 67 (3): 490–516. JSTOR 188629. • Alexander, Jason; Skyrms, Brian (1999). "Bargaining with Neighbors: Is Justice Contagious". Journal of Philosophy 96 (11): 588–598. JSTOR 2564625. • Binmore, K.; Rubinstein, A.; Wolinsky, A. (1986). "The Nash Bargaining Solution in Economic Modelling". RAND Journal of Economics 17: 176–188. JSTOR 2555382. • Binmore, Kenneth (1998). Game Theory and The Social Contract Volume 2: Just Playing. Cambridge: MIT Press. ISBN 0262024446. • Binmore, Kenneth (2005). Natural Justice. New York: Oxford University Press. ISBN 0195178114. • Kalai, Ehud (1977). "Proportional solutions to bargaining situations: Intertemporal utility comparisons". Econometrica 45 (7): 1623–1630. JSTOR 1913954. • Kalai, Ehud & Smorodinsky, Meir (1975). "Other solutions to Nash’s bargaining problem". Econometrica 43 (3): 513–518. JSTOR 1914280. • Nash, John (1950). "The Bargaining Problem". Econometrica 18 (2): 155–162. JSTOR 1907266. • Walker, Paul (2005). "History of Game Theory" [1].
External links • Nash Bargaining Solutions [2]
Screening game
Screening game A screening game is a two-player principal–agent type game used in economic and game theoretical modeling. Principal–agent problems are situations where there are two players whose interests are not necessarily at ends, but where complete honesty is not optimal for one player. This will lead to strategies where the players exchange information based in their actions which is to some degree noisy. This ambiguity prevents the other player from taking advantage of the first. The game is closely related to signaling games, but there is a difference in how information is exchanged. In the principal-agent model, for instance, there is an employer (the principal) and a worker (the agent). The worker has a given skill level, and chooses the amount of effort he will exert. If the worker knows his ability (which is given at the outset, perhaps by nature), and can acquire credentials or somehow signal that ability to the employer before being offered a wage, then the problem is signaling. What sets apart a screening game is that the employer offers a wage level first, at which point the worker chooses the amount of credentials he will acquire (perhaps in the form of education or skills) and accepts or rejects a contract for a wage level. It is called screening, because the worker is screened by the employer in that the offers may be contingent on the skill level of the worker. Some economists use the terms signaling and screening interchangeably, and the distinction can be attributed to Stiglitz and Weiss (1989).
References • Stiglitz, Joseph & Andrew Weiss (1989) “Sorting out the Differences Between Screening and Signalling Models,” in Papers in Commemoration of the Economic Theory Seminar at Oxford University, edited by Michael Dempster, Oxford: Oxford University Press.
Princess and monster game In game theory, the princess and monster game is a pursuit-evasion game played by two players in a region. The game was devised by Rufus Isaacs and published in his book Differential Games (1965) as follows. "The monster searches for the princess, the time required being the payoff. They are both in a totally dark room (of any shape), but they are each cognizant of its boundary. Capture means that the distance between the princess and the monster is within the capture radius, which is assumed to be small in comparison with the dimension of the room. The monster, supposed highly intelligent, moves at a known speed. We permit the princess full freedom of locomotion."[1] This game remained a well known open problem until it was solved by Shmuel Gal in the late 1970s.[2] [3] His optimal strategy for the princess is especially interesting. Go to a random location in the room. Stay still for a time interval which is not too short but not too long, go to another (independent) random location and repeat the procedure.[3] [4] [5] His proposed optimal search strategy is based on subdividing the room into many narrow rectangles, picking a rectangle at random and searching it in some specific way. After some time picking another rectangle randomly and independently, etc. The exact details of the search and evasion strategies are given in the references. Princess and monster games can be played on a pre-selected graph. (A possible simple graph is the circle, suggested by Isaacs as a stepping stone for the game in the region.) It can be demonstrated that for any finite graph an optimal mixed search strategy exists that results in a finite payoff. This game has been solved only for the very simple graph consisting of a single loop (a circle).[6] The value of the game on the unit interval (a graph with two nodes with a link in-between) has been estimated approximatively. This game looks simple but is quite complicated. Surprisingly, the 'obvious' search strategy of starting at one end (chosen at random) and 'sweeping' as fast as possible the whole interval is not optimal. This strategy guarantees 0.75 expected capture time. However, by utilising a more
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Princess and monster game sophisticated mixed searcher and hider strategy, one can reduce the expected capture time by about 8.6%. Actually, this number would be quite close to the value of the game if someone was able to proof the optimality of the related strategy of the Princess. [7] [8]
References [1] R. Isaacs, Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization, John Wiley & Sons, New York (1965), PP 349–350. [2] S. Gal, SEARCH GAMES, Academic Press, New York (1980). [3] Gal Shmuel (1979). "Search games with mobile and immobile hider". SIAM J. Control Optim. 17 (1): 99–122. doi:10.1137/0317009. MR0516859. [4] A. Garnaev (1992). "A Remark on the Princess and Monster Search Game" (http:/ / www. apmath. spbu. ru/ ~kmms/ garnaev/ html/ Downloads/ 1992bGT. pdf). Int. J. Game Theory 20 (3): 269–276. doi:10.1007/BF01253781. . [5] M. Chrobak (2004). "A princess swimming in the fog looking for a monster cow". ACM SIGACT News 35 (2): 74–78. doi:10.1145/992287.992304. [6] S. Alpern (1973). "The search game with mobile hiders on the circle". Proceedings of the Conference on Differential Games and Control Theory. [7] S. Alpern, R. Fokkink, R. Lindelauf, and G. J. Olsder. Numerical Approaches to the 'Princess and Monster' Game on the Interval. (http:/ / www. cdam. lse. ac. uk/ Reports/ Files/ cdam-2006-18. pdf) SIAM J. control and optimization 2008. [8] L. Geupel. The 'Princess and Monster' Game on an Interval. (http:/ / hempelz. de/ lenny/ Leonhard Geupel - Bachelor's Thesis - The 'Princess and Monster' Game on an Interval. pdf)
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Theorems Minimax Minimax (sometimes minmax) is a decision rule used in decision theory, game theory, statistics and philosophy for minimizing the possible loss for a worst case (maximum loss) scenario. Alternatively, it can be thought of as maximizing the minimum gain (maximin). Originally formulated for two-player zero-sum game theory, covering both the cases where players take alternate moves and those where they make simultaneous moves, it has also been extended to more complex games and to general decision making in the presence of uncertainty.
Game theory In the theory of simultaneous games, a minimax strategy is a mixed strategy which is part of the solution to a zero-sum game. In zero-sum games, the minimax solution is the same as the Nash equilibrium.
Minimax theorem The minimax theorem states[1] :22: For every two-person, zero-sum game with finitely many strategies, there exists a value V and a mixed strategy for each player, such that (a) Given player 2's strategy, the best payoff possible for player 1 is V, and (b) Given player 1's strategy, the best payoff possible for player 2 is −V. Equivalently, Player 1's strategy guarantees him a payoff of V regardless of Player 2's strategy, and similarly Player 2 can guarantee himself a payoff of −V. The name minimax arises because each player minimizes the maximum payoff possible for the other—since the game is zero-sum, he also maximizes his own minimum payoff. This theorem was established by John von Neumann,[2] who is quoted as saying "As far as I can see, there could be no theory of games … without that theorem … I thought there was nothing worth publishing until the Minimax Theorem was proved".[3] See Sion's minimax theorem and Parthasarathy's theorem for generalizations; see also example of a game without a value.
Example B chooses B1
B chooses B2
B chooses B3
A chooses A1
+3
−2
+2
A chooses A2
−1
0
+4
A chooses A3
−4
−3
+1
The following example of a zero-sum game, where A and B make simultaneous moves, illustrates minimax solutions. Suppose each player has three choices and consider the payoff matrix for A displayed at right. Assume the payoff matrix for B is the same matrix with the signs reversed (i.e. if the choices are A1 and B1 then B pays 3 to A). Then, the minimax choice for A is A2 since the worst possible result is then having to pay 1, while the simple minimax choice for B is B2 since the worst possible result is then no payment. However, this solution is not stable, since if B believes A will choose A2 then B will choose B1 to gain 1; then if A believes B will choose B1 then A will choose A1 to gain 3; and then B will choose B2; and eventually both players will realize the difficulty of making
Minimax a choice. So a more stable strategy is needed. Some choices are dominated by others and can be eliminated: A will not choose A3 since either A1 or A2 will produce a better result, no matter what B chooses; B will not choose B3 since some mixtures of B1 and B2 will produce a better result, no matter what A chooses. A can avoid having to make an expected payment of more than 1/3 by choosing A1 with probability 1/6 and A2 with probability 5/6, no matter what B chooses. B can ensure an expected gain of at least 1/3 by using a randomized strategy of choosing B1 with probability 1/3 and B2 with probability 2/3, no matter what A chooses. These mixed minimax strategies are now stable and cannot be improved.
Maximin Frequently, in game theory, maximin is distinct from minimax. Minimax is used in zero-sum games to denote minimizing the opponent's maximum payoff. In a zero-sum game, this is identical to minimizing one's own maximum loss, and to maximizing one's own minimum gain. "Maximin" is a term commonly used for non-zero-sum games to describe the strategy which maximizes one's own minimum payoff. In non-zero-sum games, this is not generally the same as minimizing the opponent's maximum gain, nor the same as the Nash equilibrium strategy.
Combinatorial game theory In combinatorial game theory, there is a minimax algorithm for game solutions. A simple version of the minimax algorithm, stated below, deals with games such as tic-tac-toe, where each player can win, lose, or draw. If player A can win in one move, his best move is that winning move. If player B knows that one move will lead to the situation where player A can win in one move, while another move will lead to the situation where player A can, at best, draw, then player B's best move is the one leading to a draw. Late in the game, it's easy to see what the "best" move is. The Minimax algorithm helps find the best move, by working backwards from the end of the game. At each step it assumes that player A is trying to maximize the chances of A winning, while on the next turn player B is trying to minimize the chances of A winning (i.e., to maximize B's own chances of winning).
Minimax algorithm with alternate moves A minimax algorithm[4] is a recursive algorithm for choosing the next move in an n-player game, usually a two-player game. A value is associated with each position or state of the game. This value is computed by means of a position evaluation function and it indicates how good it would be for a player to reach that position. The player then makes the move that maximizes the minimum value of the position resulting from the opponent's possible following moves. If it is A's turn to move, A gives a value to each of his legal moves. A possible allocation method consists in assigning a certain win for A as +1 and for B as −1. This leads to combinatorial game theory as developed by John Horton Conway. An alternative is using a rule that if the result of a move is an immediate win for A it is assigned positive infinity and, if it is an immediate win for B, negative infinity. The value to A of any other move is the minimum of the values resulting from each of B's possible replies. For this reason, A is called the maximizing player and B is called the minimizing player, hence the name minimax algorithm. The above algorithm will assign a value of positive or negative infinity to any position since the value of every position will be the value of some final winning or losing position. Often this is generally only possible at the very end of complicated games such as chess or go, since it is not computationally feasible to look ahead as far as the completion of the game, except towards the end, and instead positions are given finite values as estimates of the degree of belief that they will lead to a win for one player or another.
204
Minimax
205
This can be extended if we can supply a heuristic evaluation function which gives values to non-final game states without considering all possible following complete sequences. We can then limit the minimax algorithm to look only at a certain number of moves ahead. This number is called the "look-ahead", measured in "plies". For example, the chess computer Deep Blue (that beat Garry Kasparov) looked ahead at least 12 plies, then applied a heuristic evaluation function. The algorithm can be thought of as exploring the nodes of a game tree. The effective branching factor of the tree is the average number of children of each node (i.e., the average number of legal moves in a position). The number of nodes to be explored usually increases exponentially with the number of plies (it is less than exponential if evaluating forced moves or repeated positions). The number of nodes to be explored for the analysis of a game is therefore approximately the branching factor raised to the power of the number of plies. It is therefore impractical to completely analyze games such as chess using the minimax algorithm. The performance of the naïve minimax algorithm may be improved dramatically, without affecting the result, by the use of alpha-beta pruning. Other heuristic pruning methods can also be used, but not all of them are guaranteed to give the same result as the un-pruned search. A naïve minimax algorithm may be trivially modified to additionally return an entire Principal Variation along with a minimax score.
Lua example function minimax(node,depth) if depth 2
No
No
No
Cournot game
2
infinite
1
No
No
No
Deadlock
2
2
1
No
No
No
Dictator game
2
infinite
1
N/A
N/A
Yes
Diner's dilemma
N
2
1
No
No
No
Dollar auction
2
2
0
Yes
Yes
No
El Farol bar
N
2
variable
No
No
No
Example of a game without a value
2
infinite
0
No
No
Yes
[2]
[2]
[3]
[3]
List of games in game theory
238 [4]
Guess 2/3 of the average
N
infinite
1
No
No
Maybe
Kuhn poker
2
27 & 64
0
Yes
No
Yes
Matching pennies
2
2
0
No
No
Yes
Minority Game
N
2
variable
No
No
No
Nash bargaining game
2
infinite
infinite
No
No
No
Peace war game
N
variable
>2
Yes
No
No
Pirate game
N
infinite
infinite
Yes
Yes
No
Prisoner's dilemma
2
2
1
No
No
No
Rock, Paper, Scissors
2
3
0
No
No
Yes
Screening game
N
variable
variable
Yes
No
No
Signaling game
N
variable
variable
Yes
No
No
Stag hunt
2
2
2
No
No
No
Traveler's dilemma
2
N >> 1
1
No
No
No
Trust game
2
infinite
1
Yes
Yes
No
Volunteer's dilemma
N
2
2
No
No
No
War of attrition
2
2
0
No
No
No
Ultimatum game
2
infinite
infinite
Yes
Yes
No
Princess and monster game
2
infinite
0
No
No
Yes
[2]
[2]
[2]
[2]
[2]
[2]
External Links • List of games from gametheory.net [5] • A visual index to common 2x2 games [6]
Notes [1] For the cake cutting problem, there is a simple solution if the object to be divided is homogenous; one person cuts, the other choses who gets which piece (continued for each player). With a non-homogenous object, such as a half chocolate/half vanilla cake or a patch of land with a single source of water, the solutions are far more complex. [2] There may be finite strategies depending on how goods are divisible. [3] Since the dictator game only involves one player actually choosing a strategy (the other does nothing), it cannot really be classified as sequential or perfect information. [4] Potentially zero-sum, provided that the prize is split among all players who make an optimal guess. Otherwise non-zero sum. [5] http:/ / www. gametheory. net/ Dictionary/ games/ [6] http:/ / www. lri. fr/ ~dragice/ gameicons/
List of games in game theory
References • Arthur, W. Brian “Inductive Reasoning and Bounded Rationality”, American Economic Review (Papers and Proceedings), 84,406-411, 1994. • Bolton, Katok, Zwick 1998, "Dictator game giving: Rules of fairness versus acts of kindness" International Journal of Game Theory, Volume 27, Number 2 • Gibbons, Robert (1992) A Primer in Game Theory, Harvester Wheatsheaf • Glance, Huberman. (1994) "The dynamics of social dilemmas." Scientific American. • H. W. Kuhn, Simplified Two-Person Poker; in H. W. Kuhn and A. W. Tucker (editors), Contributions to the Theory of Games, volume 1, pages 97–103, Princeton University Press, 1950. • Martin J. Osborne & Ariel Rubinstein: A Course in Game Theory (1994). • McKelvey, R. and T. Palfrey (1992) "An experimental study of the centipede game," Econometrica 60(4), 803-836. • Nash, John (1950) "The Bargaining Problem" Econometrica 18: 155-162. • Ochs, J. and A.E. Roth (1989) "An Experimental Study of Sequential Bargaining" American Economic Review 79: 355-384. • Rapoport, A. (1966) The game of chicken, American Behavioral Scientist 10: 10-14. • Rasmussen, Eric: Games and Information, 2004 • Shor, Mikhael. "Battle of the sexes" (http://www.gametheory.net/dictionary/BattleoftheSexes.html). GameTheory.net. Retrieved September 30, 2006. • Shor, Mikhael. "Deadlock" (http://www.gametheory.net/dictionary/Games/Deadlock.html). GameTheory.net. Retrieved September 30, 2006. • Shor, Mikhael. "Matching Pennies" (http://www.gametheory.net/dictionary/Games/Matchingpennies.html). GameTheory.net. Retrieved September 30, 2006. • Shor, Mikhael. "Prisoner's Dilemma" (http://www.gametheory.net/dictionary/Prisonersdilemma.html). GameTheory.net. Retrieved September 30, 2006. • Shubik, Martin "The Dollar Auction Game: A Paradox in Noncooperative Behavior and Escalation," The Journal of Conflict Resolution, 15, 1, 1971, 109-111. • Sinervo, B., and Lively, C. (1996). "The Rock-Paper-Scissors Game and the evolution of alternative male strategies". Nature Vol.380, pp. 240–243 • Skyrms, Brian. (2003) The stag hunt and Evolution of Social Structure Cambridge: Cambridge University Press.
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Article Sources and Contributors
Article Sources and Contributors Game theory Source: http://en.wikipedia.org/w/index.php?oldid=465599606 Contributors: 63.208.190.xxx, 7&6=thirteen, A F K When Needed, A bit iffy, AMuseo, APH, Aaker, Aapo Laitinen, Abigail-II, Abiyoyo, Abu badali, Acalamari, Acebrock, Acitrano, Action Jackson IV, AdamSmithee, Adashiel, Adoniscik, Adrian.benko, AdultSwim, Ahoerstemeier, Aim Here, Alakon, Aliahmedraza, Aliekens, Amcfreely, Andonic, Andre Dahlke, Andrew Levine, Angus Lepper, AnneDrew55, Anonymous Dissident, Antandrus, Anthere, Anthon.Eff, AnthonyQBachler, Arbor, Arjayay, Arvindn, Avaya1, AxelBoldt, B7T, Bdesham, Behavioralethics, BenFrantzDale, Billymac00, Bjankuloski06en, Bloodshedder, Blow of Light, Bluemoose, Bobblewik, Bogsat, Bongwarrior, Bracton, Brad7777, Brendan Moody, Bret101x, Bridgeplayer, Brighterorange, BrokenSegue, C mon, CRGreathouse, CSTAR, Calabraxthis, Calton, Camrn86, Can't sleep, clown will eat me, Canderson7, CardinalDan, Catgut, Cats and kittens, Cb6, Cdc, Chairboy, Charledl, Charles Matthews, Cheeseisafruit, Chesleya, Childhoodsend, Clay Juicer, CloudNine, Cometstyles, Common Man, Conversion script, Cphay, Cretog8, Curps, Cybercobra, DVdm, Dameyawn, Damian Yerrick, Danger, Dank, Dave101, David Eppstein, David Haslam, David Levy, David Shay, DavidCary, DavidScotson, Davidbod, Davidcarfi, Davidmayberry, Dcabrilo, Delirium, DerHexer, Dexter inside, Dgscott4, Dicklyon, Digitalme, Dionyziz, Discospinster, Diuturno, Dlgiffen, DouglasGreen, DrDentz, DrJonesNelson, Dramaturgid, Drobinsonatlaur, Droll, Dromedary, Drpickem, Duncharris, Dureo, Dysprosia, EachyJ, Eags, Ebricca, EconoPhysicist, Economicprof, EdJohnston, Edward, Edward321, El C, Electricbassguy, Elwikipedista, EmilJ, Encephalon, Enchanter, Enochlau, EnumaElish, Erianna, Escape Orbit, Espetkov, EvanYares, Everyking, Everything counts, Ewlyahoocom, Faintly, Ferstel, FerzenR, Fioravante Patrone en, Firien, Fish and karate, Fixitrich, Flowerpotman, Fnielsen, Fortunecookie289, Francoisvn, Freegoods, Furrykef, GHe, GabrielAPetrie, Gametheoryguy, Gametheoryme, Gary King, Gaurav1146, Gauss, Geometry guy, GeorgeLouis, Gfoley4, Giftlite, Gimboid13, Goarany, Gombang, Googl, Graham87, Gregalton, Grick, Gronky, GroveGuy, Gtcs-student, Guyasleep, Hairy Dude, Hakperest, HappyCamper, Hardy Littlewood, Harryboyles, Haylon357, Hellkillz, Henrygb, Hoodam, Hooverbag, Hoziron, Hqb, Hubriscantilever, Huliosancez, Huon, Hve, Hwansokcho, I203.129.46.242I, II MusLiM HyBRiD II, Ikanreed, Iluvcapra, Indego Prophecy, Infarom, Inkbacker, Isdhbcfj, Isfisk, Isomorphic, Ivemadeahugemistake, J heisenberg, J.delanoy, JDspeeder1, JForget, Jacob Finn, Jakob.scholbach, JamesGecko, JavOs, Jaymay, Jecar, Jeekc, Jfpierce, Jhanley, Jim1138, JimVC3, Jimmaths, Jitse Niesen, Jlairdpdx, Joe.Dirt, JoergenB, John Quiggin, John254, Jon Roland, Jonel, Jpers36, JuPitEer, JuanOso, Jumbuck, Justin W Smith, Jwdietrich2, Kallimina, Kanmalachoa, Kenneth M Burke, Kesten, Khendon, Kiefer.Wolfowitz, KingOfBurgers, Kinos, Kirtag Hratiba, Kizor, Kku, Knowsetfree, Kntg, Kntrabssi, Koczy, Kukuliik, Kungfuadam, Kymacpherson, Kzollman, La Pianista, Lakinekaki, Landroni, Larknoll, Larry Sanger, LeaveSleaves, LeiserJ, Leon math, Lesslame, Levineps, Lexor, Lgallindo, Liftarn, Lights, Lihaas, LizardJr8, Lotje, LukeNukem, Lupin, Luqui, MER-C, MIT Trekkie, MLCommons, Machine Elf 1735, Magmi, Mandarax, MarcoLittel, Marenio, Mark Renier, Marqueed, Martin Kozák, Marykateolson, Matgerke, Mathprog777, Mattisse, Matveims, Maury Markowitz, Mav, Mayumashu, McMorph23, Mcculley, Mdd4696, Meelar, Mentisock, MercyBreeze, Mfandersen, Michael Hardy, Michel Schinz, Mike2vil, Mild Bill Hiccup, Mindmatrix, MisfitToys, Mitsuhirato, Mk270, Mk5384, Mkch, Monjur 99, Moonlight Poor, Morphh, Mousomer, Moxon, Mr. G. 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TowerDragon, Toxic Waste, Treborbassett, Treisijs, Trevor Andersen, Trialsanderrors, Triathematician, Triwbe, Trovatore, Trumpet marietta 45750, Trusilver, Tslocum, Twerges, Twri, UAIED, Ultramarine, Velho, VeroTheMaster, Vervin, Vextron, Viridae, Viridian Moon, Viriditas, Volunteer Marek, Warhol13, Wavelength, Wayward, Wehpudicabok, WeniWidiWiki, Wiki alf, WikiLaurent, Wikiant, Wikidudeman, Wikipelli, Wiknickwik, Will2k, Wilsonater23456, Wolfkeeper, Wragge, Www.msn.com, Wxlfsr, Xanzzibar, Xchatter, Xeworlebi, Yacht, Yhkhoo, Yodler, Yoichi123, Youngtimer, Zafiroblue05, Zeno Gantner, Zoicon5, Zscout370, Zsniew, Ztobor, Zundark, Zvika, Zzuuzz, Саша Стефановић, Ъыь, 750 anonymous edits Nash equilibrium Source: http://en.wikipedia.org/w/index.php?oldid=466030540 Contributors: 7&6=thirteen, Ahoerstemeier, Alansohn, Aleksi aaltonen, Algebraist, Aliekens, Andrewpmk, Angelobear, AnnaFrance, Art Carlson, Ashutosh.gupta.88, AugPi, AxelBoldt, Bartonjs, Beastly21, Benjamin1414141414141414, BiT, Bidabadi, Billlin12345, Birkett, Bissinger, Bluemoose, Brogersoc, Bryan Derksen, Charles Matthews, Chiguel, Chinju, Chrisbbehrens, Christian Kreibich, Christopher Parham, Chuanren, Closedmouth, Conrado, Coppertwig, Counterfact, Crazy coyote, Cretog8, DFRussia, DanTilkin, Danielcherian, Daniellewis24, Darwinek, Daveagp, DavidLevinson, Deb, Debresser, Delaszk, Denisarona, Denisutku, Dirac1933, Discospinster, DougsTech, DrKiranKalidindi, Dratman, Drpaule, Dynamo152, EconoPhysicist, Ecthelion83, Ed Poor, El T, Ern malleyscrub, Eshock, Evercat, Everyking, Falcon8765, Fioravante Patrone en, FizzyP, Forich, Frecklefoot, Fredericksgary, FutharkRed, Gallando, Gauss, Geometry guy, GeorgeLouis, Gerhardvalentin, Ghemachandar, Giftlite, Gillis, Gioto, Golgofrinchian, Gschmidt, Gtcs-student, Guyfromthe80s, Hadal, Hairy Dude, Halcyonhazard, Hede2000, Hedywong, Henrygb, Homunq, Hubriscantilever, Hut 8.5, I am not a dog, ImperfectlyInformed, Inadarei, Jadhavdevendra, Jandalhandler, Japiot, Jazzazi, Jeekc, Jeff Muscato, JesseStone, Johnnygeekiwi, Jonatanj, Joriki, Joro Iliev, Jpgordon, Jyotirmoyb, KDS4444, Kamilborkowski3, Karada, Karl-Henner, Kborer, Kcordina, Kerrylily, Kotu Kubin, Kraftlos, KrakatoaKatie, Krohde, Kwantus, KyleP, Kzollman, Landroni, Lee Daniel Crocker, Leejefon, Lekrecteurmasque, Lendu, LiDaobing, Liftarn, Linuxrocks123, Little-man, Luqui, MONGO, Marcello511, Marqueed, MathiasRav, Maurice Carbonaro, Maximus Rex, Michael Hardy, Mike1024, MikeHearn, MinorContributor, MisterSheik, Msavidge, Muinchille1, Musiphil, Naddy, Naingmoeaung, NashMelo92, Neanderthalcouzin, Nimlot, Novacatz, Obradovic Goran, Oleg Alexandrov, Olivier, Oxymoron83, Paul August, Pde, Pete.Hurd, Petrus, PhilipMW, Phorapples, Pippo2001, PoetblogMatters, Popsofctown, Poromenos, Psiphiorg, Pyschobbens, Razorflame, Rbb l181, Rgclegg, Rich Farmbrough, Riley Schumacher, Rinconsoleao, RobertHannah89, Rrburke, RussianCash, SDY, SMcCandlish, Salgueiro, Salix alba, Samedina, Sampo, SanitySolipsism, Sbisolo, Sfarringvt, Sfiller, ShaunMacPherson, Shoujun, Shrinershriner, Silly rabbit, SlamDiego, Slizyboy, Smalljim, Smmurphy, Socrtwo, Spike Wilbury, SpuriousQ, Steve Quinn, Steverapaport, Student396, SunCreator, Sundar, Sureshpurohit, Tarotcards, Thumperward, Thunk, Tideflat, Titoxd, ToastyKen, Treborbassett, Trialsanderrors, Uhjoebilly, Ulflarsen, UnitedStatesian, User A1, Uucp, Violeaf, Vladimer, Volunteer Marek, Weakerthanyou, Whpq, Willworkforicecream, WojPob, Wolf87, Wolfkeeper, Woodforc, Woudloper, Wragge, Yayay, Yogi de, Zhou Yu, Zkiraly, Zosimos101, Ъыь, 420 anonymous edits Cooperative game Source: http://en.wikipedia.org/w/index.php?oldid=464407864 Contributors: Admiller, Amire80, Anonash, Arthur Rubin, Artman40, CRGreathouse, Ciphers, ClanCC, Cretog8, CronopioFlotante, Dan Polansky, DayReader, Diego Queiroz, Downtown dan seattle, Edward, Epbr123, Ezrarez, Fioravante Patrone en, Gadfium, Gaius Cornelius, Gtcs-student, Gurch, Gwernol, Hughdbrown, Jhertel, John Quiggin, Koczy, Krauss, Kzollman, Littleelf, Lph, Maulattu, Maurice Carbonaro, Mfandersen, Michael Hardy, Mousomer, Mu Gamma, Omkardpd, Ortolan88, Pete.Hurd, RCSB, Rjwilmsi, RobertMeusel, TMott, Theorist2, Toby Bartels, Unschool, VictorAnyakin, Vincent.feltkamp, Wikidea, Wikiwert, Wragge, 38 anonymous edits Information set Source: http://en.wikipedia.org/w/index.php?oldid=419128499 Contributors: AJackl, CRGreathouse, Cretog8, GeorgeLouis, Grubber, Infvwl, Kzollman, LachlanA, Loihsin, Mennonot, Nicoulas, PatrickFlaherty, Pete.Hurd, PigFlu Oink, Tijfo098, Treborbassett, 4 anonymous edits Preference Source: http://en.wikipedia.org/w/index.php?oldid=463792552 Contributors: DocOfSoc, Forich, Mild Bill Hiccup, Netknowle, Xezbeth, 7 anonymous edits Normal-form game Source: http://en.wikipedia.org/w/index.php?oldid=463403625 Contributors: Amire80, Charles Matthews, Ciphers, Counterfact, Cretog8, Earth, Edward, Enochlau, Gillis, Jay Gatsby, Joerite, Kzollman, LOL, Materialscientist, MaxSem, Neifion, Obradovic Goran, Oliphaunt, Pete.Hurd, Starrynte, Treborbassett, Tregoweth, 39 anonymous edits Extensive-form game Source: http://en.wikipedia.org/w/index.php?oldid=455429869 Contributors: Burschik, Camrn86, Cherkash, Ciphers, Counterfact, Cretog8, Dnapoli, GeorgeLouis, Henrygb, Kirtag Hratiba, Kzollman, Mandarax, Obradovic Goran, Oliphaunt, Pete.Hurd, Rich Farmbrough, Rinconsoleao, Tgwizard, Tijfo098, Treborbassett, WhiteC, 36 anonymous edits Succinct game Source: http://en.wikipedia.org/w/index.php?oldid=458807917 Contributors: Bender235, Hermel, Itai, LachlanA, Malcolma, Tijfo098 Trembling hand perfect equilibrium Source: http://en.wikipedia.org/w/index.php?oldid=450779053 Contributors: Abdel Hameed Nawar, Amcfreely, Bromille, Cfp, Giftlite, GregorB, Gtcs-student, Hubriscantilever, IKiddo, Janlo, Kzollman, Meijerink, Pete.Hurd, Quirky, Rozza69, Smmurphy, Trialsanderrors, Yaris678, 23 anonymous edits Proper equilibrium Source: http://en.wikipedia.org/w/index.php?oldid=392553830 Contributors: Bromille, Counterfact, Henriqueroscoe, 2 anonymous edits Evolutionarily stable strategy Source: http://en.wikipedia.org/w/index.php?oldid=466190556 Contributors: Ajbrown141, Aliekens, Anxietycello, Ashlaender, Barticus88, Benja, Benlisquare, Brighterorange, BrokenSegue, Bueller 007, Cat's Tuxedo, Conversion script, Coontie, DFRussia, Darker Dreams, Duncharris, EconoPhysicist, Emperorbma, Erianna, Falcor84, Fredrik, Geeteshgadkari, Geometry guy, Gerrit, Graham87, Hairy Dude, Hannes Röst, Hequba, Homunq, I am not a dog, Igodard, Ilmari Karonen, Jackzhp, Karada, Kelly Martin, Khamsin, Krohde, Kzollman, Landroni, Lexor, Limbo socrates, Lotje, Marc Harper, Matttttt, Memills, Michael Hardy, Michael Rogers, Minesweeper, Monado, Mpadowadierf, NCurse, Nanobri, Noisy, Pete.Hurd, Phoenix-forgotten, Quadell, RDBrown, Rettetast, Rich Farmbrough, Richard001, Rjwilmsi, Seijihyouronka, SidP, Steverapaport, Sundar, TedPavlic, Template namespace initialisation script, The Anome, TimVickers, Treefrog007, Trialsanderrors, Uriobolski, User A1, WillWare, 51 anonymous edits Risk dominance Source: http://en.wikipedia.org/w/index.php?oldid=450136632 Contributors: CambridgeBayWeather, LukeNukem, Novidmarana, Rinconsoleao, Smmurphy, Tabletop, Trialsanderrors, 13 anonymous edits
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Article Sources and Contributors Self-confirming equilibrium Source: http://en.wikipedia.org/w/index.php?oldid=379205831 Contributors: Benja, BlueNovember, Ciphers, Cretog8, Fconaway, GeorgeLouis, 1 anonymous edits Dominance Source: http://en.wikipedia.org/w/index.php?oldid=453868490 Contributors: Afelton, Bender235, Btyner, Bushytails, Cambrasa, CambrianXp, Christopher Parham, DavidCary, Drorsh, Dysepsion, Iota, Isnow, Isomorphic, Jeekc, Jokestress, Kzollman, Madmardigan53, MountainGoat8, MrOllie, Pete.Hurd, PiccoloNamek, Rentier, Shiyang, Skoosh, SmartGuy, Smmurphy, That Guy, From That Show!, Trigger hippie77, Ъыь, 40 anonymous edits Strategy Source: http://en.wikipedia.org/w/index.php?oldid=462985948 Contributors: AdamSmithee, Alexsalgado, Bender235, Benja, Bokken, Cesiumfrog, Chewie, Cretog8, Cureden, David Eppstein, Dirnstorfer, Eastlaw, Elf, Frank Guerin, Ftm.ashari, Giftlite, Gioto, Gwern, Jitse Niesen, Jlmark3, Kilva, Kzollman, Liger2233, Littlebear1227, Llanowan, MercyBreeze, Mousomer, Obradovic Goran, Pb30, Pete.Hurd, Richard Lotspard, Rinconsoleao, Schneelocke, SimonP, SmartGuy, TheOracle23, Tijfo098, Treborbassett, WadeSimMiser, Wragge, Xeeron, ﻣﺎﻧﻲ, 29 anonymous edits Tit for tat Source: http://en.wikipedia.org/w/index.php?oldid=461509857 Contributors: AdamAtlas, Adoniscik, Amorymeltzer, Andycjp, Anetode, Arnoutf, Asrabkin, AstroHurricane001, Averykrouse, AxG, Barboakley, Before My Ken, Bender235, Bengtlueers, Bhadani, Bmdavll, Boyd Reimer, Brichard37, BrokenSegue, Broozo, CWY2190, ChangChienFu, Chealer, CheekyMonkey, Chipmunk393, Colton3x3, Coneslayer, Cretog8, Dangherous, David Latapie, DenisDiderot, Dkristoffersson, Domokato, Dreaded Walrus, Dwarf Kirlston, E=MC^2, Eltener, Etaonish, Eurobas, EvanProdromou, Faradayplank, GRuban, Gary King, Geldart, Goethean, Good Olfactory, Grick, Hadal, Halcyonhazard, Hephaestos, Heyhay, Hughdbrown, Humblefool, ImperatorExercitus, J.Gowers, JForget, JPK, Jfischoff, John Fader, Kzollman, Lifefeed, Lotje, Lovelac7, Luis Dantas, MER-C, Maarten Hermans, Machine Elf 1735, Madmardigan53, Marasmusine, Marc Harper, Marcel Kosko, Maurice Carbonaro, McGeddon, Melsaran, Michael Hardy, Michael Snow, Michaledwardmarks, Misiekuk, Moloch09, Momentfarm, Mr.E, MrSomeone, Nightscream, Nobel prize 4 peace, OlEnglish, Oliphaunt, Omphaloscope, Onodevo, Pete.Hurd, Pinethicket, Pnevares, Punctured Bicycle, Quantling, Quebec99, R, R Lowry, Rumiton, Ryan Norton, Sarranduin, Seans Potato Business, Sgeureka, Simishag, SiriusB, Smmurphy, Spencerk, Stephen B Streater, Stepheng3, Stevechilton, Steverapaport, Superbock, Taejo, Tamfang, TechnoFaye, TelopiaUtopia, TheSix, ThreeDee912, Toytoy, Trialsanderrors, Twas Now, UnDeRsCoRe, XxTimberlakexx, Yayay, ZeroOne, 119 anonymous edits Grim trigger Source: http://en.wikipedia.org/w/index.php?oldid=458782730 Contributors: Austriacus, Bender235, Bfinn, Davetron, Elembis, Gabriel.c.drummond.cole, Giftlite, Gnomus, John Quiggin, Kzollman, Louisng114, Michael Hardy, Primergrey, Spincrisis, Suzanne0-0, Tktktk, Wragge, 3 anonymous edits Collusion Source: http://en.wikipedia.org/w/index.php?oldid=465314112 Contributors: 16@r, Avochelm, Betacommand, BigFatBuddha, Bluemoose, Buckyboy314, CapitalSasha, Captaincoop, Carinasl, Chendy, DarkSaber2k, Eastlaw, F15x28, Falcon8765, Finalius, Freechild, Gabi S., GregorB, Icairns, J.DrWatson, Jezmck, Jguzmanb, Jonkerz, Katach, Kozuch, Kzollman, LaidOff, Lycurgus, Mcdennis13, Neelix, Nick Number, Nightenbelle, Ninly, Operknockity, Pete.Hurd, Philippe, Phydend, R Lowry, RattusMaximus, Rccoms, RekishiEJ, ReluctantPhilosopher, Rich Farmbrough, Ronz, Russ Anderson, Sardanaphalus, Scyth, Serverradar9, Smallman12q, SpuriousQ, Stevec240, Stevertigo, Struway, Tassedethe, Tb, Timc, Uniqueuponhim, Violetriga, Vranak, Wachen, WikHead, Winhunter, Wonglong, Woohookitty, 113 anonymous edits Backward induction Source: http://en.wikipedia.org/w/index.php?oldid=425652305 Contributors: Anomalocaris, Bediako, Diligent, Dnjkirk, EagleFan, EconoPhysicist, Fioravante Patrone en, Giftlite, Gregbard, Hongooi, Intimaralem85, Iridescent, Jersey Devil, Kristinw, Kzollman, Lambiam, Luqui, McSly, MiNombreDeGuerra, Michael Hardy, Mineminemine, Omnipaedista, Paul August, Quarl, Rinconsoleao, Slanderson, Sloth monkey, Tijfo098, User A1, Vagary, William M. Connolley, 13 anonymous edits Markov strategy Source: http://en.wikipedia.org/w/index.php?oldid=411897358 Contributors: AndrewHowse, LachlanA, Malcolma, Megaloxantha, Severo, Thekohser, Tim1357, Tktktk, 2 anonymous edits Symmetric game Source: http://en.wikipedia.org/w/index.php?oldid=428208111 Contributors: Antimatroid, Ciphers, Dark.knight ayush, HelicopterCrisps, Humanengr, Kzollman, Lionelkarman, Michael Hardy, Obradovic Goran, PV=nRT, Pete.Hurd, Robinh, Tijfo098, 7 anonymous edits Perfect information Source: http://en.wikipedia.org/w/index.php?oldid=455340706 Contributors: Alexsoddy, Apothecia, Arvindn, Batmanand, Bluemoose, Bryan Derksen, Byelf2007, Ciphers, Commander Keane, Cretog8, DirkOliverTheis, Dreispt, Grick, John Quiggin, King brosby, Kmweber, Kzollman, La goutte de pluie, Levineps, Lowellian, M.nelson, Madmardigan53, Maurreen, Michael Hardy, Mikhail Dvorkin, Morven, Mozzie, Nomen4Omen, Pearle, Pete.Hurd, Petter Strandmark, Rast, RayBirks, Rich Farmbrough, Rjanag, TallulahBelle, Tetron76, Tobias Bergemann, Toobaz, Tornadowhiz, Treborbassett, Vincom2, 19 anonymous edits Simultaneous game Source: http://en.wikipedia.org/w/index.php?oldid=418494059 Contributors: Ciphers, GoingBatty, RomualdoGrillo Sequential game Source: http://en.wikipedia.org/w/index.php?oldid=429186552 Contributors: Calvinballing, Ciphers, Henrygb, Isomorphic, Kzollman, Lord Hidelan, Martpol Repeated game Source: http://en.wikipedia.org/w/index.php?oldid=447282159 Contributors: AMuseo, Aiken drum, Andropod, Bender235, Ccerer, Ciphers, Dreadstar, EconoPhysicist, Gaius Cornelius, Gertasik, HieronymousCrowley, JerroldPease-Atlanta, Kzollman, M3taphysical, Msoos, Pete.Hurd, Phyte, Smmurphy, Sudderth1, User A1, Yatinkr, Ъыь, 30 anonymous edits Signaling games Source: http://en.wikipedia.org/w/index.php?oldid=419258654 Contributors: Antique cuckhoo clock, Bender235, Chuunen Baka, Father Goose, Fieldday-sunday, Gbeeker, Jni, John Quiggin, Kolyma, Kzollman, Linas, Luk, Neelix, Pete.Hurd, Rhbest, Rjwilmsi, Signalsbanduk, Smmurphy, Smoothhenry, Some standardized rigour, Tgetty, Timothyjlayton, Treborbassett, 28 anonymous edits Cheap talk Source: http://en.wikipedia.org/w/index.php?oldid=453887719 Contributors: 6birc, Ashmoo, Bender235, Bluemoose, CloudNine, Cogiati, David Sneek, Dr Greg, Ephraim33, Gontrode, Headbomb, Iyerkri, John Quiggin, Koavf, Kzollman, Lexor, MangoWong, MountainGoat8, Pearle, Pete.Hurd, RJFJR, Reetep, Rjwilmsi, Treborbassett, 9 anonymous edits Zero-sum Source: http://en.wikipedia.org/w/index.php?oldid=419976035 Contributors: 164.58.10.xxx, Alexf, Alfanje, Alon, Andre Engels, Andycjp, AnonMoos, Arthur Rubin, Ash211, August1991, BD2412, Bakanov, Bakerstmd, Bender235, BiT, Bromille, Bryan Derksen, CRGreathouse, Ciphers, Comfortably Paranoid, Complex (de), Conversion script, Cougarbate, Courcelles, Cretog8, DavidScotson, Derek Ross, Dgsaunders, Djozwebo, Donfbreed, DrDentz, E23, Egriffin, ElectricRay, Ellywa, Emperorbma, Emurphy42, Erianna, Fioravante Patrone en, Frankk74, Gappiah, Geir Gundersen, George Richard Leeming, Giftlite, Glengordon01, Goethean, Grafen, Graham87, Greenhope01, Guswen, Handcuffed, Henrygb, Hobo loquens, Hq3473, Hutschi, Ihope127, Ixphin, JHunterJ, JamesLucas, Jan Hidders, Jel, Jmrowland, Karlscherer3, KennethJ, Kim Bruning, Klanda, Kocio, Koveras, Kzollman, La goutte de pluie, Lacrimosus, LokiClock, Lord of the down, Luna Santin, Marc K, Marcus Brute, Moopiefoof, Mousomer, Myasuda, Mydogategodshat, Nate1481, Netoholic, Nowordneeded, Obradovic Goran, OldNick, PCM2, Peregrine981, Personman, Pete.Hurd, Prolinol, Psztorc, Pyb, R Lowry, RTV User 545631548625, Risi, Silverxxx, Smallbones, Smmurphy, Solipsist, Starryboy, Suisui, Svetovid, Tajsis, Talldean, Tennekis, Tha human, This, that and the other, Thomasmeeks, Tijfo098, Timrollpickering, Tmh, Tobias Hoevekamp, VictorAnyakin, Volunteer Marek, Weathereye, Weien, Welsh, Wik, Williamlindgren, Wordyness, Yatinkr, Zakuragi, Zoganes, Zsniew, 120 anonymous edits Mechanism design Source: http://en.wikipedia.org/w/index.php?oldid=454428961 Contributors: Aitch Eye, Aknxy, Akriasas, Alexwch, Cretog8, David Eppstein, Dexter inside, DocWatson42, F.khanmirzaee, Gary King, Grochim, Gtcs-student, Halcyonhazard, Isomorphic, Jamesontai, Jfraffen, John Quiggin, Jon Roland, KimvdLinde, Liberal Saudi, Lycurgus, MaCRoEco, Masterpiece2000, Mattisse, Michael Hardy, Neilbeach, Nonempty, Ogo, Oleg Alexandrov, Oliphaunt, Panda, Pcm.nitdgp, Pcm1.nitdgp, Pde, Ph.eyes, Rjwilmsi, SOFTBOOK, Shoeofdeath, StephenWeber, The Anome, TheTito, Theorist2, Thomasmeeks, Tobacman, Toytoy, Unitanode, UnitedStatesian, Utilitus, Voodoom, Wikispan, WillWare, Xenon54, Zeno Gantner, 33 anonymous edits Bargaining Problem Source: http://en.wikipedia.org/w/index.php?oldid=424188631 Contributors: Akalai, Allliam, Bender235, Cfp, Cretog8, Dondegroovily, E.qrqy, Ever wonder, Floquenbeam, Giraffedata, Jessieliaosha, MGM08314, PigFlu Oink, 22 anonymous edits Stochastic game Source: http://en.wikipedia.org/w/index.php?oldid=461678052 Contributors: Aneyman, Bender235, Cuaxdon, David Eppstein, Dcoetzee, Keynes.john.maynard, Kiefer.Wolfowitz, PhS, Rjwilmsi, Sandius, Sudde001, Tijfo098, 17 anonymous edits Large poisson game Source: http://en.wikipedia.org/w/index.php?oldid=417965209 Contributors: Bdemeshev, Bender235, Dyaa, Mild Bill Hiccup, 3 anonymous edits Nontransitive game Source: http://en.wikipedia.org/w/index.php?oldid=412330152 Contributors: Bender235, Gregbard, Jitse Niesen, JocK, Michael Hardy, Sam Staton Global game Source: http://en.wikipedia.org/w/index.php?oldid=379060891 Contributors: Amalas, Bearcat, CronopioFlotante, Michael Hardy, Rinconsoleao, Tobacman Prisoner's dilemma Source: http://en.wikipedia.org/w/index.php?oldid=465265517 Contributors: 129.186.19.xxx, 12mmclean, 7&6=thirteen, 848219pineapple, AdRock, Adam Conover, Akerans, Alai, Alaiche, AlanM1, Alex3917, Alison, AllanLee, Alsandro, Amead, Americanhero, Andrejj, Anetode, Angelbo, Annielogue, Aranherunar, Artoasis, Arvindn, Asaadi, Ascorbic, Atlant, Audacity, AxelBoldt, Az7997, Babij, Baccyak4H, Badger Drink, Bando26, Barrrower, Behavioralethics, Benjamin H-W, Bhause, BigK HeX, Bigturtle, Blackmetalbaz, BlueNovember, Blueyoshi321, Bona Fides, BookgirlST, Bossrat, Boud, Bouke, Bracton, BracusAnguis, Brainpo, Brian Kendig, Brichard37, Brighterorange, Brokenfixer, Bruguiea, Bryan Derksen, Btwied, BurdetteLamar, CBM, CRGreathouse, Calabe1992, Calculuslover800, Can't sleep, clown will eat me, CapitalR, Carloszgz, Causa sui, Cedrus-Libani, Chinju, Chipuni, Chris 73, ChrisG, Chrishmt0423, Christofurio, Chriswaterguy, Chromaticity, Ciaran H, Cmh, Cntras, Cokaban, Colonies Chris, Conversion script, Cpiral, Cpt, Cretog8, Cryptic C62, Cyrius, Cyrus Grisham, DEDemeza, DHN, DTM, Da Vynci, Damuna, Dan East, Dandrake, Daniel Mahu, David Gerard, DavidScotson, Dcoetzee, Dcsohl, Deadbath, Deltabeignet, Denis Diderot, DenisDiderot,
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Article Sources and Contributors Dimitridobrasil, Dogcow, Dominus, Doradus, Dr.K., Drmies, Duncharris, Dunhere, Dying, Dysprosia, EPM, EchetusXe, Edcolins, Eebster the Great, Emersoni, Emhoo, Eric119, Erin Billy, Erroneous01, Eruantalon, Estudiarme, Etaonish, Eurosong, Evercat, Every name is taken12345, FWBOarticle, FatalError, Fedfiore, Fenwayguy, Fiskbil, Flambelle, Foobar, Fourohfour, Freddy engels, Fredrik, Frostlion, Frymaster, Furrykef, Gaius Cornelius, Gakmo, Geekguy02, Giftlite, Givegains, Glasbak, Glummy, Goarany, Graham87, Gregvs3, Grick, Griffin147, Gronky, Grosscha, Guido del Confuso, Gus Polly, H2g2bob, Hadžija, Haonhien, Hawkinsbiz, Haya shiloh, Helfrich, Henry Flower, Hibernian, Hiroshi-br, Hodg, Hu, Huliosancez, Hurmata, Ilmari Karonen, Inkbacker, InverseHypercube, Iridescent, Iusewiki, IxnayOnTheTimmay, J.delanoy, JRSpriggs, Jason.grossman, Jay32183, JayJasper, JebJoya, Jedibob5, Jedonnelley, JeeAge, Jeff0106, Jmcc150, Jmrowland, Jnestorius, Joao, JocK, JohnAlbertRigali, Johnleemk, Jonathanstray, Jpkotta, Jsevilla, Jtneill, Junes, Jwrosenzweig, KConWiki, Kencf0618, Kenneth M Burke, Khoikhoi, King of Hearts, KnightRider, Kozuch, Kubigula, Kurt Jansson, Kzollman, LC, LMB, Lambertman, Lamentation, Lawrencekhoo, LeonardoRob0t, Ligulem, Likebox, Lindosland, Logophile, Loom91, Lotje, Lpetrazickis, Luk, LukeNukem, LukeSurl, Luna Santin, Lupin, Lyran, Maduskis, Magister Mathematicae, Malleus Fatuorum, MapsMan, MarSch, Mariojalves, Marskell, Martijn faassen, Masterofpsi, Materialscientist, Matthew Stannard, Mattisse, Mav, Maximus Rex, Michael Hardy, Michael Rogers, Michelle eris, Mikhailfranco, Mikolik, Mindmatrix, Mkcmkc, Mlm42, Moon light shadow, Morgrim, Mrand, Mtu, Mullibok, NeonMerlin, Neutrality, NickLinsky, Nikodemos, Nlasbo, Noisyboy1234, Nwe, Ocam, Octopuppy, Ojigiri, Oliphaunt, Olivier, Orange Suede Sofa, Orbst, Osterczyk, OwenBlacker, OwenX, PamD, Pandacomics, Pathless, PaulStephens, Pete.Hurd, Pfortuny, Philippe277, Plumbago, Ponder, Poor Yorick, Psb777, Quantling, Qwerty1234, R Lowry, R3m0t, RJII, RZ heretic, Radeks, Ral315, Ramorum, Randomblue, Raul654, Ravik, Reagle, Redrocket, RexNL, Reywas92, Rgoodermote, Rich Farmbrough, Richard001, Richfife, Rik G., Robinh, Romanm, Rompe, Rosuav, Royan, Royboycrashfan, Rracecarr, Ruud Koot, Ruy Lopez, RyanDesign, Ryanhanson, Ryguasu, Saforrest, Samohyl Jan, SandyGeorgia, Sbloch, Sdorrance, Seabreezes1, SeptimusOrcinus, Shaggorama, Sikelianos, Simishag, Simultaneous movement, Smmurphy, Snied, Snoyes, Socrtwo, Solitude, SpNeo, Space Pirate 3000AD, Spencerk, SpookyMulder, Sslevine, Starpollen, Stazed, SteinbDJ, Stekat, Stephen B Streater, Syzygy, TRBP, TWCarlson, Tabletop, Tacomaster4, Tad Lincoln, Taejo, TakuyaMurata, Tedernst, Template namespace initialisation script, Tempshill, Tfll, ThAtSo, That Guy, From That Show!, The Anome, Thehornet, Themissinglint, Tide rolls, Timwi, Tloc, Tluckie13, Toi, Tom harrison, Treborbassett, Trialsanderrors, Tribaal, Trovatore, TurilCronburg, Tyciol, Uniqueuponhim, Urgos, Vapour, Verloren, Vintermann, ViperSnake151, Volunteer Marek, Vorapsak, Vt-aoe, W anthro, Walkie, Wanani, Webponce, Westsider, Wfeidt, WhatamIdoing, Whisky drinker, Wile E. Heresiarch, Wilfried Elmenreich, Wolfkeeper, Wolfman, Woohookitty, Wragge, Wwengr, XLerate, Xyzzy n, Xyzzyplugh, Yacht, Yvh11a, Zafiroblue05, Zsniew, Ъыь, 591 anonymous edits Traveler's dilemma Source: http://en.wikipedia.org/w/index.php?oldid=456029225 Contributors: Another Believer, C-randles, CRGreathouse, Chipuni, Conical Johnson, Connelly, Giftlite, INVERTED, JocK, Kzollman, Megaloxantha, Michael Hardy, Miss Madeline, R.e.b., Reywas92, Rjwilmsi, RucasHost, SneakyTodd, Troped, Venado, 17 anonymous edits Coordination game Source: http://en.wikipedia.org/w/index.php?oldid=456027848 Contributors: Buckyboy314, Discospinster, EAderhold, JaGa, KrakatoaKatie, Krauss, Kzollman, Maurice Carbonaro, NE2, Pete.Hurd, Rich Farmbrough, Rinconsoleao, Roisterer, ST47, Signalhead, SunCreator, Trialsanderrors, Vipinhari, Wragge, 27 anonymous edits Chicken Source: http://en.wikipedia.org/w/index.php?oldid=465044621 Contributors: Aardark, Aaron Brenneman, Abljkgjkf, AdmiralHood, Alansohn, Aleph Infinity, Alfanje, Aliekens, AnOddName, Antandrus, Apelbaum, Apparition11, Aris Katsaris, Avengerx, Babaroberto, Bkell, BrotherE, Bryan Derksen, Burner0718, Ceetar, CesarB, Complex01, Cookiemobsta, Cretog8, DFRussia, Dcoetzee, Dhp1080, Discospinster, Dismas, Donreed, E1890, East718, Edratzer, Elizabeyth, Emurphy42, Ergotius, Evercat, Ezadarque, Flyguy649, Gakmo, Gamer123456754321, Geniusdude247, Geometry guy, Ginsengbomb, Glane23, GregorB, HiDrNick, HieronymousCrowley, Hut 6.5, Hydrogen Iodide, Invertzoo, Isopropyl, JHMM13, Jak86, Jamie C, JocK, John Link, Jorend, Kateshortforbob, KnightofNEE, Kolobochek, Kuru, Kylu, Kzollman, Ld100, LilHelpa, Luis r izquierdo, Luna Santin, Malcohol, Meelar, Michael Hardy, Mikedelsol, Mindmatrix, Nice poa, Ninakscgirl21, Novamo, Orudge, Ospalh, Pete.Hurd, Pils, Pinkkeith, Pnm, Polynomial4456, Preposterone, Pretzelpaws, RPGmaple, Rasmus Faber, RexNL, Reywas92, Rgoodermote, Ricardolacombe, Rjwilmsi, Savidan, SchmuckyTheCat, Scorpion451, Sir Trollsalot, Spacey, Spencerk, Stevechilton, Stevertigo, Superm401, Tali.g, The ed17, TimVickers, Timwi, Trialsanderrors, Ultrahaggis, Unyoyega, Vanished user 03, Volunteer Marek, Vroo, Walkiped, Zanuga, Zoltan271828, Zsniew, 140 anonymous edits Centipede game Source: http://en.wikipedia.org/w/index.php?oldid=427609988 Contributors: Action Jackson IV, Agamemnus, Arthur Rubin, Bender235, Blahedo, C-randles, CRGreathouse, Charvest, Christopher Parham, Chwech, Crazy Boris with a red beard, D.brodale, DerBorg, Elembis, Enochlau, Gioto, Glane23, GregorB, Ihope127, JimVC3, Jrouquie, Kzollman, LarryJeff, Maimone, Matt314, Mau.zachrisson, Michael Hardy, Michael Rogers, Mushyrulez Alot, NYKevin, Parcoj, Pengo, Perryar, Psztorc, Qe2eqe, Rbrwr, Rsmead, SE16, Vagary, Voidmaw, Xs935, 36 anonymous edits Volunteer's dilemma Source: http://en.wikipedia.org/w/index.php?oldid=461728270 Contributors: Amalas, Bender235, Cybercobra, Edward, EffeX2, JeffreyGomez, JocK, M3taphysical, Michael Hardy, Niceguyedc, Nixeagle, Reinyday, Remember, Sbloch, Tesseran, VascoAmaral, Xanzzibar, 18 anonymous edits Dollar auction Source: http://en.wikipedia.org/w/index.php?oldid=438186721 Contributors: Acroterion, Aliasad, Alta-Snowbird, Bender235, Canned Soul, Capricorn42, Cmdrjameson, Comet Tuttle, Coppertwig, Cretog8, DMCer, Ddstretch, Dejan Jovanović, DropDeadGorgias, Ejwong, ElectricRay, Ettrig, Happyhappyallhappy!, J.delanoy, Kent Wang, Kzollman, Lambiam, Mtjaws, NawlinWiki, Oerjan, OneWeirdDude, Parhamr, Payam prz, Pete.Hurd, Philwelch, R'n'B, Rasmus Faber, Rufasto, SURIV, Scorpion451, Sebastark, Turkeyphant, Vahe Kharazyan, Volunteer Marek, Whosasking, Xaosflux, 34 anonymous edits Battle of the sexes Source: http://en.wikipedia.org/w/index.php?oldid=459878008 Contributors: Akerans, Alexis Brooke M, Arthur Rubin, Bluemoose, Btwied, Buckyboy314, Cheater no1, Cyrus Grisham, Danh, Davidbod, Elexhobby, Ergotius, Eve Teschlemacher, Gcc111, Greg Tyler, Jrouquie, Jswitzer, KingShibby, Kulyuhkldffsdsfesfsdf, Kzollman, Melchoir, Musiphil, Pete.Hurd, Quest for Truth, Reyk, Reywas92, Ringbang, Robbie314, Ruinia, Sam Hocevar, Scorpion451, SemanticMantis, Sewebster, Stpasha, Trialsanderrors, Yausmaam, Zsniew, 37 ,דניאל צבי anonymous edits Stag hunt Source: http://en.wikipedia.org/w/index.php?oldid=456934348 Contributors: AlexP, Applejuicefool, BD2412, Burpen, Claymore, Clearlyhidden19, CommonsDelinker, Cruci, Derek Ross, Enfascination, Everyking, Hmains, Kzollman, Mindmatrix, Psiphiorg, Rinconsoleao, Rsmead, Stephen B Streater, TheoClarke, Trialsanderrors, Xiangjw, 37 anonymous edits Matching pennies Source: http://en.wikipedia.org/w/index.php?oldid=429714692 Contributors: Art LaPella, AySz88, Bender235, Cretog8, Culix, Evercat, Fsufezzik, Kzollman, Lessthanideal, Light current, McGeddon, Melchoir, Mild Bill Hiccup, NYKevin, Pepeeg, Pete.Hurd, Pseudoquark, Tankparksalute, Tiger Khan, Tijfo098, Trialsanderrors, Yausmaam, Zvika, 17 anonymous edits Ultimatum game Source: http://en.wikipedia.org/w/index.php?oldid=464875259 Contributors: AMuseo, Aardwolf, AjitPD, Alai, Antonio Lopez, Astanton, AxelBoldt, B4hand, BanyanTree, Behavioralethics, Binks, Bjgyg, BobHackett, Brossow, Btwied, CRGreathouse, Chris V. W., Comfortably Paranoid, Dcoetzee, DocWatson42, EPM, Emersoni, Ettrig, EventHorizon, Geometry guy, Gizmo II, Gveret Tered, Hamiltondaniel, Headbomb, InverseHypercube, Isomorphic, Jason Recliner, Esq., Jivecat, Jjamison, Karijne, Kzollman, Landroni, Lingust, Matthewcgirling, Michael Hardy, MountainGoat8, MrOllie, Nbearden, Osvaldi, Pascal666, Protonk, Qmwne235, Quasarblue, Rjanag, Rjwilmsi, Rlove, RyanCross, Shalom Yechiel, ShelfSkewed, Shentino, Smmurphy, Solitude, Stolsvik, Tabletop, Technopat, Theodore Kloba, Thomasmeeks, Tokorode, Tommy, Trialsanderrors, VeritasEtLuz, VernoWhitney, Volfy, WikiSlasher, WoodenTaco, Wragge, Yaroslav Blanter, Zachaysan, Zenomax, Zingo75, 61 anonymous edits Rock-paper-scissors Source: http://en.wikipedia.org/w/index.php?oldid=466081714 Contributors: .V., 0dd1, 11pnelson, 1ForTheMoney, 293.xx.xxx.xx, 3frenchhens2turtledoves1cup, 63.12.132.xxx, 65.96.213.xxx, 75th Trombone, 8012ED0177, ALargeElk, AaRH, Aaron carass, Aaronstj, Abigail-II, Acather96, Achoo5000, Admiral Norton, Adriaan, Adrian J. Hunter, Aepryus, Aeroknight, Agvulpine, Ahoerstemeier, Ainlina, Aitias, Ajsh, Alansohn, AlexHOUSE, Alison22, All we did was die..., Amazing backslash, Anaxial, Anchors, AndrewvdBK, Andycjp, Anetode, Angelbo, Angeldeb82, Anomie, Antediluvian67, Anthony Appleyard, Anárion, Apfox, Archangel127, ArielGold, Arjayay, ArqMage, ArthurDenture, Arvindn, Ashesindust, Ashmodai, Ashmoo, Asiaindigo, Atarr, Augiedog2010, Awthur, Azndragon126, Azul, BACbKA, BD2412, Bakheer, Balloonman, Bamber100, Bamber101, Batmanand, Beach drifter, Beland, Benbest, Bender235, Benjaminb, Benjaminhammond, Benlisquare, Beyond My Ken, BigEyedFish, Bilderbikkel, BillFlis, Billymac00, Bladonad, Blank Frank, Bloodshedder, Bo Lindbergh, Bob the ducq, Bobo192, Bongwarrior, Bonus Onus, Borgx, Brian Crawford, BrianKnez, Brw12, Bryan Derksen, Btljs, Btyner, Bucketsofg, Bulbaboy, Burke Libbey, Burningview, Burschik, Bwfdc, Byankno1, Byronknoll, CDThieme, CJC47, CWY2190, CalBears99, Caleby, Can't sleep, clown will eat me, CanadianLinuxUser, Canderson7, Captain Zyrain, CaptainDDL, Carioca, CarmelitaCharm, Carribus, Carroy, Carter, Catdude, Cattus, Caz999, Cdc, Chadloder, Charles Matthews, CharlesHBennett, Charliewynn, Chealar, Check ya noggin, Cheetahstu, Chetvorno, ChildofMidnight, Chinasaur, Chingchongcha, Chogno98, Chris Hanson, Christopherlin, Chuck Smith, Ckatz, Cloud13, Cmsg, Cokoli, Cold Season, Colonies Chris, Commander Zulu, Conti, Conversion script, Cormorant, Coryshrmn, Cosmetor, CosmicJake, Couldntthinkofanusername, Cptmurdok, Crazytales, Cretog8, Crossmr, Cruci, Cyberia23, Cyde, Cyp, DARTH SIDIOUS 2, DMCer, DMacks, Da Joe, DaMeanHippo, Dalereese, Damien Prystay, Dan100, Dandv, Daniel Mahu, Daniel Olsen, Daniel,levine, Danielt998, Danny Fenton, Dante Alighieri, Daqu, Darthalex314, DarylNickerson, Dasondas, Dave Runger, David Gerard, David spector, Davidhbolton, Davidizer13, Dcoetzee, Ddpwns, Decumanus, Dejitaru, Deltabeignet, Dex1337, Dhlstrm, Dicklyon, Dico Veritas, Diego Moya, Discospinster, Dismas, Dissident, Diz, Djmerlin3, Dlohcierekim, Dmmaus, Dockingman, Dogman15, Dominus, DoriSmith, Dpakdel, Dragoonmac, DreamGuy, Dreamyshade, Drilnoth, DropDeadGorgias, Druff, Dtrimm88, Dude902, Dysmorodrepanis, Dysprosia, E-Kartoffel, Eagle9141, EamonnPKeane, Earl CG, Editor510, Eeekster, Einar Myre, Elassint, Electricbassguy, Elendal, Elonka, Eluchil, EoGuy, Eouw0o83hf, Equazcion, Erachima, EronMain, Error, Eskandarany, Esperant, Esquire1386, Euryalus, Everyking, Evil Egg, Excirial, FWBOarticle, Fantusta, Faradayplank, FastLizard4, FatzooRPS, Favonian, Feezo, Felix Dance, Felix Wiemann, Feudonym, Fg2, FiP, Fieari, Filzstift, Flarn2006, Flyguy649, Formeruser0910, FoxInShoes, Franklint, Frazzydee, Frederick Spek, Fredrik, Frencheigh, Fritzpoll, Frizero, FrozenUmbrella, Frungi, Fuhghettaboutit, Funandtrvl, Funk2010, Furrykef, Fæ, GRuban, GT3, Gail, Gaius Cornelius, Gallocher E, Galzigler, Garethfoot, Gawaxay, Geir Arne, Gerwalker, Gesiwuj, Gilliam, Gkerster, Glane23, Glenn, Gmonfils, Gogo Dodo, GoodBooksMelbourne, Gracefool, Gracenotes, Graham87, Greatal386, Gscshoyru, Gtrmp, Guliolopez, Gvf, Gzornenplatz, Habj, Hairy Dude, Halo, Halosix, Hamilton burr, Hangways, Happy-melon, Harryboyles, Haukurth, Hayama Akito, Hcane55, Henning Makholm, Hessamnia, Heyesy, Hgilbert, Higherfrequencies, Histrion, Hogyn Lleol, Hooperbloob, Hotcoffeegirl, Hq3473, Htmnssn, Hydrargyrum, INVERTED, IVinshe, Iago4096, Ian Pitchford, Ice Jedi5, Icundell, Idleguy, Illuvatar,, Imperial Star Destroyer, Imperialles, InShaneee, Insomniacpuppy, Inverarity, Ipsenaut, Irishguy, Isomorphic, Ivanip, Ixfd64, J. 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W., Cleisthenes2, Dysmorodrepanis, Econobbler, Econobuster, Ecthelion83, Elembis, Evercat, FT2, False vacuum, Grutness, Igodard, Jason Recliner, Esq., Joetroll, Joolsa123, Kzollman, Messy Thinking, Michael Hardy, Mikespoff, Nb6, Ngchen, Pesilion, Reywas92, Rjwilmsi, Robin S, Rosiestep, SalmonHelmet, Thomasmeeks, Trialsanderrors, VanHelsing23, Wfisher, Wragge, 32 anonymous edits Public goods game Source: http://en.wikipedia.org/w/index.php?oldid=464749063 Contributors: AleXd, Anthon.Eff, Cmdrjameson, Craw-daddy, Darkildor, FrankTobia, Glen, IslandHopper973, Jantin, Kzollman, Longhair, MrBoo, Optikos, PhysPhD, Quarl, Rinconsoleao, RobyWayne, Sean0608, Silverhelm, Tiger Khan, UnknowableSelf, Vinophil, Wang.Zhijian.Zju, Wang.zhijian, Wragge, 15 anonymous edits Blotto games Source: http://en.wikipedia.org/w/index.php?oldid=463986182 Contributors: CRGreathouse, GregorB, INVERTED, JocK, Kilva, Petter Strandmark, R.e.b., Smacdonell, Synergy, The Anome, 16 anonymous edits War of attrition Source: http://en.wikipedia.org/w/index.php?oldid=434686842 Contributors: Arob33, Canned Soul, Czar Kirk, David Schaich, Exomnium, GregorB, Gt.kls, JoeSmack, John Quiggin, Kzollman, Loopy48, Mcdennis13, Njk92, NorthernThunder, Pete.Hurd, Rspeer, Scorpion451, Smmurphy, TelopiaUtopia, Tregoweth, Trialsanderrors, Volemak, WinterSpw, 12 anonymous edits El Farol Bar problem Source: http://en.wikipedia.org/w/index.php?oldid=456648115 Contributors: Bjp716, Brogersoc, Cobain, Dreadstar, Faolin42, GregorB, Hgintis, Hyphz, Kzollman, Matwood, Myglesias, Quuxplusone, S.MahdiRazavi, Saint141, Trialsanderrors, Txomin, Vanished User 1004, Wireader, 11 anonymous edits Fair division Source: http://en.wikipedia.org/w/index.php?oldid=460536737 Contributors:
[email protected], Anonymous Dissident, Buenasdiaz, Cactusthorn, Cacycle, Calvinballing, D6, David Eppstein, Delaszk, Dfeldmann, Dmcq, Eequor, Enragedeconomist, Ergotius, Euphrosyne, Gdr, Huw Powell, Igodard, Infrogmation, InverseHypercube, Itai, Ixfd64, Jitse Niesen, John Reid, Jonkerz, Kzollman, LeeJacksonKing, Malcohol, Matt Crypto, Melchoir, Miken32, Nbatra, Noca2plus, Norvy, Ntsimp, Oleg Alexandrov, Oxymoron83, PMajer, Pete.Hurd, Piotrus, Psiphiorg, R'n'B, Rdancer, Rjwilmsi, Robinh, Rrh02, Shanel, Spike Wilbury, Stephen Bain, Tali.g, Thomasmeeks, Triathematician, Vicarious, Volunteer Marek, Wavelength, Wlod, Woohookitty, 41 anonymous edits Cournot competition Source: http://en.wikipedia.org/w/index.php?oldid=465688491 Contributors: Antonorsi, Asav, Barcturus, Bluemoose, Brisvegas, Clsrskv, Common Man, Coolian, Coppertwig, Cretog8, Eastlaw, Fluffernutter, Frank MacCrory, Gabriel.c.drummond.cole, GregorB, Huax, Iminto, Indirap, Isnow, Jackzhp, Jagerman, Jebba, Jeff3000, Jonkerz, Karada, Katieishot, Kelly Martin, Kochiuyu, Kylu, Kzollman, LachlanA, Landroni, Luke wainscoat, Maurreen, Melfassy, Nikit16, Protonk, Radell, Rinconsoleao, Rl, Ruarrimactire, Sergiodf, Shreevatsa, SimonP, TheStarter, Trammerman, Treborbassett, Urbansuperstar, Vgnohz, Viridae, Volunteer Marek, Zachrome, 90 anonymous edits Deadlock Source: http://en.wikipedia.org/w/index.php?oldid=436435529 Contributors: Coslenchip, Cretog8, Kelseyxckannibal, Kzollman, Mnh, Richard New Forest, Vermin1302, Ziggurat, 4 anonymous edits Unscrupulous diner's dilemma Source: http://en.wikipedia.org/w/index.php?oldid=465522827 Contributors: Apollo Augustus Koo, Ben Standeven, Bender235, DEDemeza, Diego Queiroz, Hede2000, Heron, Kzollman, Lusanaherandraton, OwenX, Pnm, Rich Farmbrough, Tktktk, User6985, Vroo, 8 anonymous edits Guess 2/3 of the average Source: http://en.wikipedia.org/w/index.php?oldid=461227873 Contributors: AmigoCgn, Bender235, Blueviking, EdC, Fnielsen, Halcyonhazard, Henrygb, Icestorm815, Iohannes Animosus, Isomorphic, JaGa, Kortaggio, Kzollman, Leo leo, Markhurd, Mipmip, Nickybutt, Rinconsoleao, ST47, SmartGuy, Tesseran, Twthmoses, UnMatChedProWess, 22 anonymous edits Kuhn poker Source: http://en.wikipedia.org/w/index.php?oldid=416537759 Contributors: 2005, Alai, Bender235, Enkrates, Evercat, Ezrakilty, GregRobson, Grutness, Henrygb, Kzollman, Lomn, Nintendere, Quest for Truth, Woohookitty, 9 anonymous edits Nash bargaining game Source: http://en.wikipedia.org/w/index.php?oldid=416513323 Contributors: Akalai, Allliam, Bender235, Cfp, Cretog8, Dondegroovily, E.qrqy, Ever wonder, Floquenbeam, Giraffedata, Jessieliaosha, MGM08314, PigFlu Oink, 22 anonymous edits Screening game Source: http://en.wikipedia.org/w/index.php?oldid=462464163 Contributors: Dave Rebecca, Djdoobwah, Dmr2, Lmatt, Mild Bill Hiccup, Smmurphy, 2 anonymous edits Princess and monster game Source: http://en.wikipedia.org/w/index.php?oldid=461741048 Contributors: Amalas, Dmcq, Gnomus, Guoguo12, Headbomb, Imnotoneofyou, Jason Quinn, JocK, Julmonn, Leonhard geupel, Maxal, Michael Hardy, Rich Farmbrough, Shuroo, Thorseth, Xnn, 41 anonymous edits Minimax Source: http://en.wikipedia.org/w/index.php?oldid=465609615 Contributors: A. Pichler, AllUltima, Andre Engels, Andresambrois, Anne Bauval, Artichoker, Arvindn, BKfi, Beta16, Borgx, Bpeps, Brainix, Brews ohare, Cburnett, ChangChienFu, CharlesC, CharlesGillingham, Conversion script, Cretog8, Dante Shamest, David Haslam, DavidCary, Dr. Persi, Długosz, El C, Erasmussen, Foobar, Gametheorist77, Garygagliardi, Geometry guy, Giftlite, Glengordon01, Grick, Henrygb, Honza Záruba, Hunyadym, Icaoberg, Imran, Jitse Niesen, Karenjc, Karl-Henner, Kiefer.Wolfowitz, Kku, LOL, MH, Maschelos, Mat-C, MatrixFrog, MattGiuca, Maximin, Michael Hardy, Moneky, Monty845, Nbarth, NeonMerlin, Nmnogueira, OckRaz, Ohconfucius, OliAtlason, Pegua, Pete.Hurd, PhilKnight, Philip Trueman, PsiXi, Qwertyus, Qwfp, R3m0t, RalfKoch, Remy B, Rich Farmbrough, Riitoken, Robert Dober, RobertHannah89, Robinh, Sam Hocevar, Sampo, Scott sauyet, Sean Kelly, Shuroo, SlamDiego, Smmurphy, Sniedo, Spitfire, Syr0, Telespiza, Terry0201, Tghe-retford, The Anome, Thinboy00P, Tijfo098, Tobias Hoevekamp, Trieper, Trovatore, UkPaolo, Wereon, Will Beback, WriterHound, Xijiahe, ZeroOne, Zundark, Zvika, 166 anonymous edits Purification theorem Source: http://en.wikipedia.org/w/index.php?oldid=422065771 Contributors: Cretog8, J04n, Kzollman, Lionelkarman, Profundity06, Rodii, 7 anonymous edits
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Article Sources and Contributors Folk theorem Source: http://en.wikipedia.org/w/index.php?oldid=462458197 Contributors: Adoniscik, Alberto Chilosi, Bgold, David Eppstein, Gdr, Gt.kls, Jergen, Kjlewis, Kzollman, Michael Hardy, Paine Ellsworth, Pete.Hurd, Philwelch, Reverend T. R. Malthus, Shenanenigans, Smmurphy, 8 anonymous edits Revelation principle Source: http://en.wikipedia.org/w/index.php?oldid=416754409 Contributors: Bmcnamee, Btyner, CRGreathouse, Ccerer, Counterfact, GRBerry, GregorB, Kzollman, Lingust, Mdmcginn, Ph.eyes, Sander87, Smmurphy, Thijswijs, Volunteer Marek, 12 anonymous edits Arrow's impossibility theorem Source: http://en.wikipedia.org/w/index.php?oldid=463843048 Contributors: A poor workman blames, AMH-DS, AaronSw, Abd, Akriasas, Alighodsi2, Arnob1, Arrrgggument, Ashley Y, AxelBoldt, Bongomatic, Bromskloss, Bryan Derksen, CRGreathouse, Cancan101, CanisRufus, Carl.bunderson, Cconnett, Colignatus, Commadot, D6, DaGizza, Dan Wylie-Sears 2, DanKeshet, David Eppstein, Dclo, Delikedi, Derek Ross, Dissident, DocGov, Dr. I .D. A. MacIntyre, Draicone, Eberhard Wesche, Edward, Elmju, Enchanter, Euchrid, Frango com Nata, Gartogg, Geoffrey, Geoffrey.landis, Giftlite, GreatBigCircles, Gregbard, Grick, Guanaco, Gwern, Hairy Dude, HalfDome, Henrygb, Homunq, HorsePunchKid, Icairns, Infovarius, JRR Trollkien, JRSpriggs, Jdlh, John Quiggin, Joriki, Josh Cherry, Jsnx, KSmrq, Karada, Kelson, Kevin, Khazar, Kukkurovaca, LC, LamilLerran, Laurusnobilis, Liftarn, Liko81, Lussmu, MarkusSchulze, MartinHarper, Masterpiece2000, Mateo SA, Matt Gies, Matt me, Matthew Woodcraft, Maurreen, Melchoir, Mezzaluna, Miranche, Mousomer, MrOllie, Mschamis, Natnatonline, Nbarth, NeilTarrant, Neilc, NilEinnoc, Oliphaunt, Osndok, Pace212, Paladinwannabe2, Patrick, Paul Stansifer, PhilipMW, Punctured Bicycle, RDBury, Rangek, Rhobite, Rl, Rmharman, RobLa, Rspeer, Ruakh, Sararkd, Sbyrnes321, Scott Ritchie, Sdalva, Sf talkative, Shreevatsa, Slipperyweasel, Smmurphy, Solarapex, Spacethingy, Spot, Svick, TUF-KAT, Tbouricius, Tenmei, Teply, Tesseran, The Anome, Theorist2, Thomasmeeks, Thumperward, Tiger Khan, Tim Ivorson, TittoAssini, Tom harrison, Unweaseler, Vadim Makarov, Villarinho, VoteFair, Waisbrot, Wclark, Wikiborg, Wikidea, William Avery, Wknight94, X1011, Y2y, Zanaq, Zarvok, Zundark, Zvika, 177 anonymous edits Tragedy of the commons Source: http://en.wikipedia.org/w/index.php?oldid=466188961 Contributors: *jb, 64.34.161.xxx, Abu badali, Adamsan, Addshore, Aetheling, Akadruid, Akhil 0950, Al Lemos, Alan Liefting, Ale jrb, AlecStewart, Alejos, Aliveboy, Amritasenray, Anarcho-capitalism, Angela, AngoraFish, Anlace, Antandrus, Anthony, Ashernm, Ashmoo, AtD, BAxelrod, BIL, Banaticus, Batmanand, Bcasterline, Belg4mit, BenBildstein, Bender235, Bergsten, Bharral, Bhuston, Bibble Bibble, BigK HeX, Bobby H. 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Connolley, Wizofaus, Woz2, Wragge, Writershorse, Ww, YUL89YYZ, Yueni, Zach99998, Zerokitsune, Zodon, Ъыь, 505 anonymous edits Tyranny of small decisions Source: http://en.wikipedia.org/w/index.php?oldid=464097939 Contributors: AlexanderKaras, Byelf2007, Charles Matthews, Cretog8, CrusaderForCommonEra, Denny, Eastlaw, Edward, Epipelagic, Gadget850, Good Olfactory, Guillaume2303, Guslacerda, Hmains, John Darrow, LilHelpa, Mediation4u, Mjpollard, Wareh, Will Beback, Wragge, 4 anonymous edits All-pay auction Source: http://en.wikipedia.org/w/index.php?oldid=455033846 Contributors: Amalas, Brian Gunderson, Cretog8, Cronholm144, Edward Curran, GregorB, Haidongwang, Homunq, Ikiwi, Jmayer, Pete.Hurd, Pingveno, Remember, The Anome, Vlad, Widget90, Wikiofdoom, 10 anonymous edits List of games in game theory Source: http://en.wikipedia.org/w/index.php?oldid=443589503 Contributors: Admiller, Ben Standeven, Common Man, Dak, DropDeadGorgias, Eggman64, George Richard Leeming, Hede2000, JocK, Kumioko, Kzollman, Michael Hardy, Perryar, Phoz, Qblik, Quest for Truth, Robinh, Scorpion451, Smmurphy, Zephyrus67, 25 , דניאל צביanonymous edits
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Image Sources, Licenses and Contributors File:JohnvonNeumann-LosAlamos.gif Source: http://en.wikipedia.org/w/index.php?title=File:JohnvonNeumann-LosAlamos.gif License: Public Domain Contributors: LANL Image:Ultimatum Game Extensive Form.svg Source: http://en.wikipedia.org/w/index.php?title=File:Ultimatum_Game_Extensive_Form.svg License: Creative Commons Attribution-ShareAlike 3.0 Unported Contributors: Kevin Zollman --Kzollman Image:Centipede game.svg Source: http://en.wikipedia.org/w/index.php?title=File:Centipede_game.svg License: Creative Commons Attribution-ShareAlike 3.0 Unported Contributors: MaxDZ8, based on work from Kzollman Image:PD with outside option.svg Source: http://en.wikipedia.org/w/index.php?title=File:PD_with_outside_option.svg License: Creative Commons Attribution-ShareAlike 3.0 Unported Contributors: Kevin Zollman --Kzollman File:Nash graph equilibrium.png Source: http://en.wikipedia.org/w/index.php?title=File:Nash_graph_equilibrium.png License: Public Domain Contributors: Luis von Ahn, Andrew Krieger File:SGPNEandPlainNE explainingexample.svg Source: http://en.wikipedia.org/w/index.php?title=File:SGPNEandPlainNE_explainingexample.svg License: Creative Commons Attribution-Sharealike 3.0 Contributors: Me, Gillis Danielsen Image:Battle of the sexes - 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Image Sources, Licenses and Contributors Image:Pyle pirates burying2.jpg Source: http://en.wikipedia.org/w/index.php?title=File:Pyle_pirates_burying2.jpg License: Public Domain Contributors: Beej71, BrokenSphere, Jappalang, Mattes, Quibik, Wolfmann, 1 anonymous edits Image:El Farol Restaurant and Cantina, Santa Fe NM.jpg Source: http://en.wikipedia.org/w/index.php?title=File:El_Farol_Restaurant_and_Cantina,_Santa_Fe_NM.jpg License: Creative Commons Attribution 3.0 Contributors: John Phelan Image:Berlin Blockade-map.svg Source: http://en.wikipedia.org/w/index.php?title=File:Berlin_Blockade-map.svg License: Creative Commons Attribution-ShareAlike 3.0 Unported Contributors: historicair 23:55, 11 September 2007 (UTC) Image:economics cournot diag1 svg.svg Source: http://en.wikipedia.org/w/index.php?title=File:Economics_cournot_diag1_svg.svg License: Public Domain Contributors: Twisp,Bluemoose Image:economics cournot diag2 svg.svg Source: http://en.wikipedia.org/w/index.php?title=File:Economics_cournot_diag2_svg.svg License: Public Domain Contributors: Twisp,Bluemoose Image:economics cournot diag3 svg.svg Source: http://en.wikipedia.org/w/index.php?title=File:Economics_cournot_diag3_svg.svg License: Public Domain Contributors: Twisp,Bluemoose Image:economics cournot diag4 svg.svg Source: http://en.wikipedia.org/w/index.php?title=File:Economics_cournot_diag4_svg.svg License: Public Domain Contributors: Twisp,Bluemoose Image:Minimax.svg Source: http://en.wikipedia.org/w/index.php?title=File:Minimax.svg License: Creative Commons Attribution-Sharealike 2.5 Contributors: Nuno Nogueira (Nmnogueira) File:Cows on Selsley Common - geograph.org.uk - 192472.jpg Source: http://en.wikipedia.org/w/index.php?title=File:Cows_on_Selsley_Common_-_geograph.org.uk_-_192472.jpg License: Creative Commons Attribution-Share Alike 2.0 Generic Contributors: Lamiot File:Lacanja burn.JPG Source: http://en.wikipedia.org/w/index.php?title=File:Lacanja_burn.JPG License: Public domain Contributors: Jami Dwyer File:Twin Glens abutment.jpg Source: http://en.wikipedia.org/w/index.php?title=File:Twin_Glens_abutment.jpg License: Creative Commons Attribution-Share Alike Contributors: Choess File:Jones River marshland near mouth.JPG Source: http://en.wikipedia.org/w/index.php?title=File:Jones_River_marshland_near_mouth.JPG License: Creative Commons Attribution-Sharealike 3.0,2.5,2.0,1.0 Contributors: OldPine
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