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.Game theory is a branch of applied mathematics that is used in the social sciences, most notably in economics, as well as in biology (most notably evolutionary biology and ecology), engineering, political science, international relations, computer science, and philosophy.^ In biology, game theory has been used to understand many different phenomena.
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^ Game theory From Wikipedia, the free encyclopedia Game theory is a branch of applied mathematics that is used in the social sciences, most notably in economics, as well as in biology, engineering, political science, international relations, computer science, and philosophy.
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^ 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.
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.Game theory attempts to mathematically capture behavior in strategic situations, in which an individual's success in making choices depends on the choices of others.^ Game theory and individual motivation .
  • Trust in Game Theory -- Unpublished Talk, May, 1997 8 January 2010 4:14 UTC philosophy.wisc.edu [Source type: FILTERED WITH BAYES]

^ Would this be considered a game theory situation?
  • A Conversation with Len Fisher | Popular Science 8 January 2010 4:14 UTC www.popsci.com [Source type: General]

^ Recall that in games, fitness also depends on the frequency of other behaviors.
  • Introduction to Game Theory -- simple, two-strategy examples 8 January 2010 4:14 UTC www.holycross.edu [Source type: FILTERED WITH BAYES]

.While initially developed to analyze competitions in which one individual does better at another's expense (zero sum games), it has been expanded to treat a wide class of interactions, which are classified according to several criteria.^ While initially developed to analyze competitions in which one individual does better at another's expense (zero sum games), it has been expanded to treat a wide class of interactions, which are classified according to several criteria.
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^ Business is a NON-zero sum game .
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^ Poker exemplifies a zero-sum game (ignoring the possibility of the house's cut), because one wins exactly the amount one's opponents lose.
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.Today, "game theory is a sort of umbrella or 'unified field' theory for the rational side of social science, where 'social' is interpreted broadly, to include human as well as non-human players (computers, animals, plants)" (Aumann 1987).^ Game theory in computer science .
  • Al Roth's game theory, experimental economics, and market design page 8 January 2010 4:14 UTC kuznets.fas.harvard.edu [Source type: FILTERED WITH BAYES]

^ Today, "game theory is a sort of umbrella or 'unified field' theory for the rational side of social science, where 'social' is interpreted broadly, to include human as well as non-human players (computers, animals, plants)" (Aumann 1987).
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^ I call game theory and rational choice theory that includes this assumption "vulgar."
  • Trust in Game Theory -- Unpublished Talk, May, 1997 8 January 2010 4:14 UTC philosophy.wisc.edu [Source type: FILTERED WITH BAYES]

.Traditional applications of game theory attempt to find equilibria in these games.^ Traditional applications of game theory attempt to find equilibria in these games.
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^ The lack of a dynamical theory in the traditional theory of games .
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ Developed by John von Neumann, the theory has applications to real games (cards, chess, etc.
  • In a few words... from Interactive Mathematics Miscellany and Puzzles 8 January 2010 4:14 UTC www.cut-the-knot.org [Source type: Reference]

.In an equilibrium, each player of the game has adopted a strategy that they are unlikely to change.^ Matrices, unlike trees, simply show the outcomes, represented in terms of the players' utility functions, for every possible combination of strategies the players might use.
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^ Nash's idea, based on the idea of equilibrium in a physical system, was that players would adjust their strategies until no player could benefit from changing.
  • Sample Chapter for Camerer, C.F.: Behavioral Game Theory: Experiments in Strategic Interaction. 8 January 2010 4:14 UTC press.princeton.edu [Source type: FILTERED WITH BAYES]

^ Game theory started on this idea that you try to put yourself in the other person's head and think for them, 'What's they're best strategy going to be?'
  • A Conversation with Len Fisher | Popular Science 8 January 2010 4:14 UTC www.popsci.com [Source type: General]

.Many equilibrium concepts have been developed (most famously the Nash equilibrium) in an attempt to capture this idea.^ Nash's idea, based on the idea of equilibrium in a physical system, was that players would adjust their strategies until no player could benefit from changing.
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^ Such remarks ignore the fact that the concept 'Nash equilibrium strategy' is not necessarily synonymous to 'optimal play'.

^ The basic idea behind the concept of a Nash Equilibrium is that for wide classes of games there are ways for the players to play the game, sometimes as pure strategies and sometimes as mixed strategies so that if a player deviates from the Nash Equilibrium strategy, he/she can not improve his/her payoff.
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.These equilibrium concepts are motivated differently depending on the field of application, although they often overlap or coincide.^ In fact, the behaviorist needs the concept of equilibrium in beliefs too, but for different purposes.
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^ Although the game-theoretic equilibrium of zero is a poor guess about initial choices, players are inexorably drawn toward zero as they learn.
  • Sample Chapter for Camerer, C.F.: Behavioral Game Theory: Experiments in Strategic Interaction. 8 January 2010 4:14 UTC press.princeton.edu [Source type: FILTERED WITH BAYES]

^ The movement patterns (or "personalities" as they are often labelled) of the enemy ghosts are different for each ghost and quite complex (Mateas, 2003).
  • Game Studies - The Games Economists Play - Implications of Economic Game Theory for the Study of Computer Games 8 January 2010 4:14 UTC gamestudies.org [Source type: FILTERED WITH BAYES]

.This methodology is not without criticism, and debates continue over the appropriateness of particular equilibrium concepts, the appropriateness of equilibria altogether, and the usefulness of mathematical models more generally.^ So which refinement is more appropriate as a solution concept?
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^ One limitation with this paper is that Shubik worked within the confines of TU games whereas Edgeworth's idea is more appropriately modelled as an NTU game.

^ The introduction of trembling hand perfect equilibria occurred in the paper Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games by Reinhard Selten.

.Although some developments occurred before it, the field of game theory came into being with the 1944 book Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern.^ See economic game theory.
  • Half-Real: A Dictionary of Video Game Theory 8 January 2010 4:14 UTC www.half-real.net [Source type: General]

^ Games and Economic Behavior 1: 60-79.
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^ They published the Theory of Games and Economic Behavior in 1944.
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.This theory was developed extensively in the 1950s by many scholars.^ Many game development methods call for extensive testing of a game on users during all phases of game development.
  • Half-Real: A Dictionary of Video Game Theory 8 January 2010 4:14 UTC www.half-real.net [Source type: General]

.Game theory was later explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s.^ The most well-developed game theory software package is Gambit , a software library developed primarily by Richard McKelvey, Andrew McLennan and Theodore Turocy.
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^ Developed by John von Neumann, the theory has applications to real games (cards, chess, etc.
  • In a few words... from Interactive Mathematics Miscellany and Puzzles 8 January 2010 4:14 UTC www.cut-the-knot.org [Source type: Reference]

^ In the preface to Evolution and the Theory of Games , Maynard Smith notes that “[p]aradoxically, 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.” It is perhaps doubly paradoxical, then, that the subsequent development of evolutionary game theory has produced a theory which holds great promise for social scientists, and is as readily applied to the field of economic behaviour as that for which it was originally designed.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

.Game theory has been widely recognized as an important tool in many fields.^ It is important to distinguish games from game theory .
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^ In the early 1950s, John Nash generalized their results and provided the basis of the modern field of Game Theory.
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^ Interest in data also suffered from the fact that so many interesting mathematical puzzles were open for solution in game theory for such a long time.
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.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 in computer science .
  • Al Roth's game theory, experimental economics, and market design page 8 January 2010 4:14 UTC kuznets.fas.harvard.edu [Source type: FILTERED WITH BAYES]

^ What, then, do these economic game theorists do?
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^ Iyt is interesting that advanced Game Theory was developed by John Nash and he won the Nobel Prize for it.
  • How Israel Wages Game Theory Warfare « INTIFADA 8 January 2010 4:14 UTC intifada-palestine.com [Source type: Original source]

Contents

Representation of games

.The games studied in game theory are well-defined mathematical objects.^ "Game theory" is also a mathematical method of decision-making in which a competitive situation is analyzed to determine the optimal course of action for an interested party.
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^ Interest in data also suffered from the fact that so many interesting mathematical puzzles were open for solution in game theory for such a long time.
  • Sample Chapter for Camerer, C.F.: Behavioral Game Theory: Experiments in Strategic Interaction. 8 January 2010 4:14 UTC press.princeton.edu [Source type: FILTERED WITH BAYES]

^ This broad definition applies to most of the social sciences, but game theory applies mathematical models to this interaction under the assumption that each person's behavior impacts the well-being of all other participants in the game.
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.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.^ Outcome : an assignment of a set of payoffs, one to each player in the game.
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^ A strategy for a player, in this game, consists of an amount of cake that he would like.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ Figure 11 illustrates the state space under the continuous replicator dynamics for the sender-receiver game consisting of two states of the world, two signals, and two responses, where players are restricted to using one of the previous four strategies.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

.Most cooperative games are presented in the characteristic function form, while the extensive and the normal forms are used to define noncooperative games.^ Matrix games are referred to as ‘normal-form’ or ‘strategic-form’ games, and games as trees are referred to as ‘extensive-form’ games.
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^ Keywords: experiments, game theory, extensive form, presentation effects.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ This paper also introduced the extensive form of a game.

Extensive form

An extensive form game
.The extensive form can be used to formalize games with some important order.^ Keywords: experiments, game theory, order of play, extensive form.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ These experiments have worked with the centipede game in extensive form as well as in recent work that gets new insights from using a normal form for the centipede game.
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^ For now, we have described it just in order to use it to introduce one of the two types of mathematical objects used to represent games: game-trees .
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.Games here are often presented as trees (as pictured to the left).^ First, however, here are definitions of some concepts that will be helpful in analyzing game-trees: .
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.Here each vertex (or node) represents a point of choice for a player.^ Node : A point at which a player takes an action.
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^ Strategy : a program instructing a player which action to take at every node in the tree where she could possibly be called on to make a choice.
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^ If the subgame descending from node 3 is played, then Player II will face a choice between a payoff of 4 and a payoff of 3.
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: FILTERED WITH BAYES]
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.The player is specified by a number listed by the vertex.^ The player is specified by a number listed by the vertex.
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^ Each player has two strategies, which are specified by the number of rows and the number of columns.
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.The lines out of the vertex represent a possible action for that player.^ The lines out of the vertex represent a possible action for that player.
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^ Matrices, unlike trees, simply show the outcomes, represented in terms of the players' utility functions, for every possible combination of strategies the players might use.
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^ Of course, not all paths will be possible because the other player has a role in selecting paths too, and won't take actions that lead to less preferred outcomes for him.
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.The payoffs are specified at the bottom of the tree.^ The payoffs are specified at the bottom of the tree.
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^ The numbers in the parentheses at the bottom of the tree are the payoffs at each respective point, in the format (Player 1, Player 2).

.In the game pictured here, there are two players.^ This game involves two players.
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^ Note there are two waveforms here.
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^ In the game pictured here, there are two players.
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.Player 1 moves first and chooses either F or U.^ 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.
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^ Consider what happens if one modifies the one-shot prisoner's dilemma above by having A move first with the knowledge that what she chooses will be common knowledge before B moves.
  • Trust in Game Theory -- Unpublished Talk, May, 1997 8 January 2010 4:14 UTC philosophy.wisc.edu [Source type: FILTERED WITH BAYES]

^ Game Theory predicts that Player 1 will choose D in his first move and thus both players will receive payoff of 1!
  • EconPort - Handbook - Game Theory - Centipede Game 8 January 2010 4:14 UTC www.econport.org [Source type: FILTERED WITH BAYES]

.Player 2 sees Player 1's move and then chooses A or R.^ 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.
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^ Game Theory predicts that Player 1 will choose D in his first move and thus both players will receive payoff of 1!
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^ The top number at the end of each vertical line is a payoff for player 1 and the bottom number is a payoff for player 2.Player 1 has the first move: if she chooses D, both players get 1; if she chooses A, the opportunity to make a decision passes to player 2.
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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.^ When the "moves" and payoffs for a game are displayed in this form, the game is said to be described in "extensive form."
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^ This paper also introduced the extensive form of a game.

^ These experiments have worked with the centipede game in extensive form as well as in recent work that gets new insights from using a normal form for the centipede game.
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.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.^ When players do know each other, they might know what their respective values might be as well as the way they think about situations involved in the game.
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^ An event is common knowledge among a set of agents if all know it and all know that they all know it and so on ad infinitum.

^ In a series of three papers, Games with Incomplete Information Played by 'Bayesian' Players, Parts I, II and III , John Harsanyi constructed the theory of games of incomplete information.

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).^ Matrix games are referred to as ‘normal-form’ or ‘strategic-form’ games, and games as trees are referred to as ‘extensive-form’ games.
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^ Now consider the strategic form of this game: .
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^ Now we can represent your entire situation on a matrix; this is the strategic form of your game: .
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.More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions.^ Matrices, unlike trees, simply show the outcomes, represented in terms of the players' utility functions, for every possible combination of strategies the players might use.
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^ In other words, in zero-sum games the total benefit to all players in the game, for every combination of strategies, always adds to zero.
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^ Each bridge should be thought of as a lottery over the fugitive's possible outcomes, in which each lottery has a different expected payoff in terms of the items in his utility function.
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.In the accompanying example there are two players; one chooses the row and the other chooses the column.^ It will be convenient to refer to the two players involved in a game as Row and Column.
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^ So depending on the perceived probability of those two events it becomes a matter of assessing how the other players view their own sense of "greed".

^ Since each player chooses between two actions at each of two information sets here, each player has four strategies in total.
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.Each player has two strategies, which are specified by the number of rows and the number of columns.^ This number is typically the payoff to the row player.
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^ It will be convenient to refer to the two players involved in a game as Row and Column.
  • Feature Column from the AMS 8 January 2010 4:14 UTC www.ams.org [Source type: FILTERED WITH BAYES]

^ Each player has two strategies, which are specified by the number of rows and the number of columns.
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.The payoffs are provided in the interior.^ The payoffs are provided in the interior.
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.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).^ The first number represents the payoff to A , and the second represents the payoff to B .
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^ For each outcome, Row's payoff is always listed first, followed by Column's.
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^ Player I's payoff appears as the first number of each pair, Player II's as the second.
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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.^ Games in the normal form; 6.
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^ This representation of the game is called the normal form .
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^ 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.
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.If players have some information about the choices of other players, the game is usually presented in extensive form.^ Games in the extensive form.
  • GAME THEORY Outline 8 January 2010 4:14 UTC www.nes.ru [Source type: Reference]

^ In other games, each player's role is different.
  • Game theory - encyclopedia article - Citizendium 8 January 2010 4:14 UTC en.citizendium.org [Source type: FILTERED WITH BAYES]

^ If players have some information about the choices of other players, the game is usually presented in extensive form.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

Characteristic function form

.In cooperative games with transferable utility no individual payoffs are given.^ As well as expounding two-person zero sum theory this book is the seminal work in areas of game theory such as the notion of a cooperative game, with transferable utility (TU), its coalitional form and its von Neumann-Morgenstern stable sets.

^ More specifically, in a social dilemma "each individual always receives a higher payoff for defecting than for cooperating, but all are better off if all cooperate than if all defect" (Dawes and Messick, 2000, p.
  • Game Studies - The Games Economists Play - Implications of Economic Game Theory for the Study of Computer Games 8 January 2010 4:14 UTC gamestudies.org [Source type: FILTERED WITH BAYES]

^ Consequently, no single unit is useful in all circumstances, although the distribution of units in a given game is rarely equal due to production costs, etcetera.
  • Game Studies - The Games Economists Play - Implications of Economic Game Theory for the Study of Computer Games 8 January 2010 4:14 UTC gamestudies.org [Source type: FILTERED WITH BAYES]

Instead, the characteristic function determines the payoff of each coalition. The standard assumption is that the empty coalition obtains a payoff of 0.
.The origin of this form is to be found in the seminal book of von Neumann and Morgenstern who, when studying coalitional normal form games, assumed that when a coalition C forms, it plays against the complementary coalition (N\setminus C) as if they were playing a 2-player game.^ A game played against the computer is considered a single player game.
  • Half-Real: A Dictionary of Video Game Theory 8 January 2010 4:14 UTC www.half-real.net [Source type: General]

^ In the opening pages of their seminal book, von Neumann and Morgenstern (1944, p.
  • Sample Chapter for Camerer, C.F.: Behavioral Game Theory: Experiments in Strategic Interaction. 8 January 2010 4:14 UTC press.princeton.edu [Source type: FILTERED WITH BAYES]

^ Neumann, John von and Oskar Morgenstern.
  • Half-Real: A Dictionary of Video Game Theory 8 January 2010 4:14 UTC www.half-real.net [Source type: General]

The equilibrium payoff of C is characteristic. .Now there are different models to derive coalitional values from normal form games, but not all games in characteristic function form can be derived from normal form games.^ Now consider the strategic form of this game: .
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: FILTERED WITH BAYES]
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^ After all, there is little risk in the game since the worst consequence is merely leaving with $0, which is what you started with.

^ We cannot solve this game, as before, simply on the basis of knowing the players' ordinal utility functions, since the intensities of their respective preferences will now be relevant to their strategies.
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.Formally, a characteristic function form game (also known as a TU-game) is given as a pair (N,v), where N denotes a set of players and v:2^N\longrightarrow\mathbb{R} is a characteristic function.^ Outcome : an assignment of a set of payoffs, one to each player in the game.
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^ The bargaining set includes the core but unlike it, is never empty for TU games.

^ At a given time, a player has a repertoire of skills for playing a given game.
  • Half-Real: A Dictionary of Video Game Theory 8 January 2010 4:14 UTC www.half-real.net [Source type: General]

.The characteristic function form has been generalised to games without the assumption of transferable utility.^ As well as expounding two-person zero sum theory this book is the seminal work in areas of game theory such as the notion of a cooperative game, with transferable utility (TU), its coalitional form and its von Neumann-Morgenstern stable sets.

^ We cannot solve this game, as before, simply on the basis of knowing the players' ordinal utility functions, since the intensities of their respective preferences will now be relevant to their strategies.
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: FILTERED WITH BAYES]
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^ We can solve this new game if we make certain assumptions about the two players' utility functions.
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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).^ Costs may be assigned either positive or negative values depending on the form of the payoff equation.
  • Introduction to Game Theory -- simple, two-strategy examples 8 January 2010 4:14 UTC www.holycross.edu [Source type: FILTERED WITH BAYES]

^ I do not believe that one can explain how people choose if one supposes that their preferences depend only on the material payoffs.
  • Trust in Game Theory -- Unpublished Talk, May, 1997 8 January 2010 4:14 UTC philosophy.wisc.edu [Source type: FILTERED WITH BAYES]

^ (This form does not require that the payoffs for each player be symmetric, only that the proper ordering of the payoffs obtains.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

Application and challenges

.Game theory has been used to study a wide variety of human and animal behaviors.^ Now that's good behavioral game theory!
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^ The results are used to create behavioral game theory.
  • Sample Chapter for Camerer, C.F.: Behavioral Game Theory: Experiments in Strategic Interaction. 8 January 2010 4:14 UTC press.princeton.edu [Source type: FILTERED WITH BAYES]

^ Evolutionary game theory has been used to explain a number of aspects of human behavior.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

.It was initially developed in economics to understand a large collection of economic behaviors, including behaviors of firms, markets, and consumers.^ In economics, game theory is used to analyze behavior of firms that worry about what their competitors will do.
  • Sample Chapter for Camerer, C.F.: Behavioral Game Theory: Experiments in Strategic Interaction. 8 January 2010 4:14 UTC press.princeton.edu [Source type: FILTERED WITH BAYES]

^ Behavioral game theory uses a concept of limited iterated reasoning to understand initial choices and a theory of learning to explain movement across rounds.
  • Sample Chapter for Camerer, C.F.: Behavioral Game Theory: Experiments in Strategic Interaction. 8 January 2010 4:14 UTC press.princeton.edu [Source type: FILTERED WITH BAYES]

^ Some Dynamics of a Strategic Market Game with a Large Number of Agents,” Journal of Economics , 60: 1–28.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

.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.^ Keywords: experiments, game theory, psychological game theory.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ Keywords: experiments, game theory, public, voting, political science.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ This broad definition applies to most of the social sciences, but game theory applies mathematical models to this interaction under the assumption that each person's behavior impacts the well-being of all other participants in the game.
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.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).^ Game theory can be defined as the study of how people interact and make decisions.
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^ If one seeks to use an evolutionary game theoretic model to explain the normativity attached to a social rule, one must explain how such an approach avoids committing the so-called “naturalistic fallacy” of inferring an ought-statement from a conjunction of is-statements.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ Since evolutionary game theory successfully explains the predominance of certain behaviors of insects and animals, where strong rationality assumptions clearly fail, this suggests that rationality is not as central to game theoretic analyses as previously thought.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

.This work predates the name "game theory", but it shares many important features with this field.^ It is important to distinguish games from game theory .
  • Sample Chapter for Camerer, C.F.: Behavioral Game Theory: Experiments in Strategic Interaction. 8 January 2010 4:14 UTC press.princeton.edu [Source type: FILTERED WITH BAYES]

^ In the early 1950s, John Nash generalized their results and provided the basis of the modern field of Game Theory.
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^ Interest in data also suffered from the fact that so many interesting mathematical puzzles were open for solution in game theory for such a long time.
  • Sample Chapter for Camerer, C.F.: Behavioral Game Theory: Experiments in Strategic Interaction. 8 January 2010 4:14 UTC press.princeton.edu [Source type: FILTERED WITH BAYES]

.The developments in economics were later applied to biology largely by John Maynard Smith in his book Evolution and the Theory of Games.^ Keywords: experiments, game theory, evolution.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ Evolution and the Theory of Games” Journal of Theoretical Biology , 1: 382–403.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ See economic game theory.
  • Half-Real: A Dictionary of Video Game Theory 8 January 2010 4:14 UTC www.half-real.net [Source type: General]

.In addition to being used to predict and explain behavior, game theory has also been used to attempt to develop theories of ethical or normative behavior.^ Now that's good behavioral game theory!
  • Sample Chapter for Camerer, C.F.: Behavioral Game Theory: Experiments in Strategic Interaction. 8 January 2010 4:14 UTC press.princeton.edu [Source type: FILTERED WITH BAYES]

^ The results are used to create behavioral game theory.
  • Sample Chapter for Camerer, C.F.: Behavioral Game Theory: Experiments in Strategic Interaction. 8 January 2010 4:14 UTC press.princeton.edu [Source type: FILTERED WITH BAYES]

^ Evolutionary game theory has been used to explain a number of aspects of human behavior.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

.In economics and philosophy, scholars have applied game theory to help in the understanding of good or proper behavior.^ See economic game theory.
  • Half-Real: A Dictionary of Video Game Theory 8 January 2010 4:14 UTC www.half-real.net [Source type: General]

^ They published the Theory of Games and Economic Behavior in 1944.
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^ Now that's good behavioral game theory!
  • Sample Chapter for Camerer, C.F.: Behavioral Game Theory: Experiments in Strategic Interaction. 8 January 2010 4:14 UTC press.princeton.edu [Source type: FILTERED WITH BAYES]

.Game theoretic arguments of this type can be found as far back as Plato.^ Fisher's argument can be understood game theoretically, but he did not state it in those terms.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ Let's see how we formally analyze a game -- how we make theoretical calculations of relative fitness that are based on benefits, costs and frequencies of various types of outcomes.
  • Introduction to Game Theory -- simple, two-strategy examples 8 January 2010 4:14 UTC www.holycross.edu [Source type: FILTERED WITH BAYES]

^ Depending on the type of explanation it seeks to provide, are evolutionary game theoretic explanations of social phenomena irrelevant or mere vehicles for the promulgation of pre-existing values and biases?
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

[1]

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.^ Keywords: experiments, game theory, public, social dilemmas.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ Keywords: experiments, game theory, public.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ Keywords: experiments, game theory, public, voting, political science.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

.In each of these areas, researchers have developed game theoretic models in which the players are often voters, states, special interest groups, and politicians.^ Game-theoretic models of bargaining.
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^ These game types also inspire sudden alliances since one player constructing a wonder will immediately mean a certain convergence of interests of all other players.
  • Game Studies - The Games Economists Play - Implications of Economic Game Theory for the Study of Computer Games 8 January 2010 4:14 UTC gamestudies.org [Source type: FILTERED WITH BAYES]

^ An equilibrium, then, may be thought of as a game state to which the game is likely to gravitate and once reached the players will generally not change their strategies.
  • Game Studies - The Games Economists Play - Implications of Economic Game Theory for the Study of Computer Games 8 January 2010 4:14 UTC gamestudies.org [Source type: FILTERED WITH BAYES]

.For early examples of game theory applied to political science, see the work of Anthony Downs.^ See economic game theory.
  • Half-Real: A Dictionary of Video Game Theory 8 January 2010 4:14 UTC www.half-real.net [Source type: General]

^ Keywords: experiments, game theory, public, voting, political science.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ In the early 1950s, John Nash generalized their results and provided the basis of the modern field of Game Theory.
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.In his book An Economic Theory of Democracy (Downs 1957), he applies the Hotelling firm location model to the political process.^ Few corporations nowadays think about their strategy without adding some game theory models or game elements into their strategy process.
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^ Smith, Vernon L. (1992) Game Theory and Experimental Economics: Beginnings and Early Influences, History of Political Economy , 24:Special Issue 241-282.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ Abstract: This book surveys and develops models of learning and dynamic adjustment in economic games.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

.In the Downsian model, political candidates commit to ideologies on a one-dimensional policy space.^ Figure 3 illustrates one way of representing the replicator dynamical model of the prisoner's dilemma, known as a state-space diagram.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

The theorist shows how the political candidates will converge to the ideology preferred by the median voter. .For more recent examples, see the books by Steven Brams, George Tsebelis, Gene M. Grossman and Elhanan Helpman, or David Austen-Smith and Jeffrey S. Banks.^ Messages are more effective in the treatment where the receiver also sees that sender's message/action decisions from the most recent matching, as compared with a baseline treatment where only the sender's previous action is revealed.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ (Note: a more common type of game is probably "playing the field" (see Maynard Smith, 1982 ) or "n-person game" by Riechert and Hammerstein 1983 ), but we will not consider this model.
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^ (See, for example, the so-called ‘continuous double auction’ experiments discussed in Plott and Smith 1978 and Smith 1962 , 1964 , 1965 , 1976 , 1982 ).
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.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.^ One should however be cautious about extrapolating this explanation to other games where other factors might be involved.
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^ However, many have already indicated far lower values which, once again, illustrates the problem when there is imperfect information regarding the other players.

^ If at least one node shares its information set with another, while others are alone, the game involves both simultaneous and sequential play, and so is still a game of imperfect information.
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.In contrast, it is difficult to know the intentions of nondemocratic leaders, what effect concessions will have, and if promises will be kept.^ In contrast, it is difficult to know the intentions of nondemocratic leaders, what effect concessions will have, and if promises will be kept.
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^ Bobbob321, immediately upon entering, wants to know of rushing is allowed, practically asking for a promise that may or may not be kept once the game starts.
  • Game Studies - The Games Economists Play - Implications of Economic Game Theory for the Study of Computer Games 8 January 2010 4:14 UTC gamestudies.org [Source type: FILTERED WITH BAYES]

Thus there will be mistrust and unwillingness to make concessions if at least one of the parties in a dispute is a nondemocracy (Levy & Razin 2003).

Economics and business

.Economists have long used game theory to analyze a wide array of economic phenomena, including auctions, bargaining, duopolies, fair division, oligopolies, social network formation, and voting systems.^ See economic game theory.
  • Half-Real: A Dictionary of Video Game Theory 8 January 2010 4:14 UTC www.half-real.net [Source type: General]

^ Keywords: experiments, game theory, trust game, fairness, social history.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ Keywords: experiments, game theory, fairness.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

.This research usually focuses on particular sets of strategies known as equilibria in games.^ This research usually focuses on particular sets of strategies known as equilibria in games.
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^ For each player in a two-person zero-sum matrix game, the set of optimal mixed strategies is a closed, convex set (Karlin, 36).
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^ The focus of attention is usually not so much on what is the best way to play such a game, but simply on whether one or the other player has a winning strategy.
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.These "solution concepts" are usually based on what is required by norms of rationality.^ These "solution concepts" are usually based on what is required by norms of rationality.
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^ There is no more ‘refined’ concept of rationality of which this can be argued to be true in general ; and so, according to behaviorists, refinements of NE based on refinements of rationality are likely to be of merely occasional interest.
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^ This is a persuasive solution concept because, again unlike the refinements of Section 2.5, it does not demand ‘more’ rationality of agents, but less .
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.In non-cooperative games, the most famous of these is the Nash equilibrium.^ If all people know that these incentives hold for most others, then cooperation will not only be possible, but will be the expected norm, and the war of all against all becomes a general peace.
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^ We will demonstrate this shortly by reference to the most famous (though not the most typical) game, the so-called Prisoner's Dilemma , and to other, more typical, games.
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^ In 1982, Maynard Smith's seminal text Evolution and the Theory of Games appeared, followed shortly thereafter by Robert Axelrod's famous work The Evolution of Cooperation in 1984.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

.A set of strategies is a Nash equilibrium if each represents a best response to the other strategies.^ A set of strategies is a Nash equilibrium if each represents a best response to the other strategies.
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^ A selection of strategies by a group of agents is said to be in a Nash equilibrium if each agent's strategy is a best-response to the strategies chosen by the other players.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ Since a Nash equilibrium Cooperate -1, -1 -10, 0 of a game constitutes one's best response to the actions of the other players, playing a Defect 0, -10 -5, -5 strategy that is part of a Nash equilibrium seems appropriate.
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.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.^ A selection of strategies by a group of agents is said to be in a Nash equilibrium if each agent's strategy is a best-response to the strategies chosen by the other players.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ In other words, in zero-sum games the total benefit to all players in the game, for every combination of strategies, always adds to zero.
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^ Each power will recognize this incentive on the part of the other, and so will anticipate an attack if they don't preempt it.
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.The payoffs of the game are generally taken to represent the utility of individual players.^ In general, then, a game is partly defined by the payoffs assigned to the players.
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^ Matrices, unlike trees, simply show the outcomes, represented in terms of the players' utility functions, for every possible combination of strategies the players might use.
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^ Outcome : an assignment of a set of payoffs, one to each player in the game.
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.Often in modeling situations the payoffs represent money, which presumably corresponds to an individual's utility.^ Often in modeling situations the payoffs represent money, which presumably corresponds to an individual's utility.
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^ The payoffs of the game are generally taken to represent the utility of individual players.
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^ The number in the upper triangle of each pair indicates the payoff for Player B; the lower triangle, Player A. Higher numbers represent greater payoff for the individual.
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.This assumption, however, can be faulty.^ This assumption, however, can be faulty.
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.A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of some particular economic situation.^ A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of some particular economic situation.
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^ See economic game theory.
  • Half-Real: A Dictionary of Video Game Theory 8 January 2010 4:14 UTC www.half-real.net [Source type: General]

^ Questions about economic or game theory?

.One or more solution concepts are chosen, and the author demonstrates which strategy sets in the presented game are equilibria of the appropriate type.^ Game for more than one player .
  • Half-Real: A Dictionary of Video Game Theory 8 January 2010 4:14 UTC www.half-real.net [Source type: General]

^ One or more solution concepts are chosen, and the author demonstrates which strategy sets in the presented game are equilibria of the appropriate type.
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^ So which refinement is more appropriate as a solution concept?
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: FILTERED WITH BAYES]
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC setis.library.usyd.edu.au [Source type: FILTERED WITH BAYES]

.Naturally one might wonder to what use should this information be put.^ Naturally one might wonder to what use should this information be put.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

^ One natural game to use for investigating the evolution of fairness is divide-the-cake (this is the simplest version of the Nash bargaining game).
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ I tried to spark the use of the term "design economics" in my 2002 manifesto , which might have more naturally included all the things that economists can help design (e.g.
  • Al Roth's game theory, experimental economics, and market design page 8 January 2010 4:14 UTC kuznets.fas.harvard.edu [Source type: FILTERED WITH BAYES]

Economists and business professors suggest two primary uses: descriptive and prescriptive.

Descriptive

A three stage Centipede Game
.The first known use is to describe how human populations behave.^ It all depends, of course, on how accurately the replicator dynamics models the primary evolutionary forces (cultural or biological) acting on human populations.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ First we will need a complete description of the strategy -- how does it behave in regards to other known strategies?
  • Introduction to Game Theory -- simple, two-strategy examples 8 January 2010 4:14 UTC www.holycross.edu [Source type: FILTERED WITH BAYES]

.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.^ 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.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

^ Game theory is a technique for analysing how people, firms and governments should behave in strategic situations (in which they must interact with each other), and in deciding what to do must take into account what others are likely to do and how others might respond to what they do.
  • Economics A-Z | Economist.com 8 January 2010 4:14 UTC www.economist.com [Source type: FILTERED WITH BAYES]

^ Game theorists study the predicted and actual behaviour of individuals in games, as well as optimal strategies.
  • Game theory - Theories Used in IS Research 8 January 2010 4:14 UTC www.fsc.yorku.ca [Source type: Academic]

.This particular view of game theory has come under recent criticism.^ However, this use for game theory has also come under criticism.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

^ This is where game theory comes in.
  • Advanced NFL Stats: Game Theory and Run/Pass Balance 8 January 2010 4:14 UTC www.advancednflstats.com [Source type: General]

^ Recent developments in game theory.
  • Game Theory Books: A Core Collection & More - Business Library - University of Florida 8 January 2010 4:14 UTC www.uflib.ufl.edu [Source type: Academic]

.First, it is criticized because the assumptions made by game theorists are often violated.^ First, it is criticized because the assumptions made by game theorists are often violated.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

^ Shmuel Zamir at the Hebrew University is the last game theorist I know in alphabetical order (but one of the first game theorists I knew).
  • Al Roth's game theory, experimental economics, and market design page 8 January 2010 4:14 UTC kuznets.fas.harvard.edu [Source type: FILTERED WITH BAYES]

^ What we should therefore expect, because it is the only NE of the game, is a race between the two powers to be the first to attack.
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: FILTERED WITH BAYES]
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC setis.library.usyd.edu.au [Source type: FILTERED WITH BAYES]

.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.^ While it is true that every noncooperative game in which players may use mixed strategies has a Nash equilibrium, some have questioned the significance of this for real agents.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ Why has classical game theory helped to predict non-human animal behavior more straightforwardly than it has done most human behavior?
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: FILTERED WITH BAYES]
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC setis.library.usyd.edu.au [Source type: FILTERED WITH BAYES]

^ Also, in darkened arcades many Space Invaders players will have considered their scores highly gratifying even though the game may be technically impossible to win.
  • Game Studies - The Games Economists Play - Implications of Economic Game Theory for the Study of Computer Games 8 January 2010 4:14 UTC gamestudies.org [Source type: FILTERED WITH BAYES]

.Explanations of this phenomenon are many; irrationality, new models of deliberation, or even different motives (like that of altruism).^ There are many models that use altruism to explain apparent cooperation, especially in more complicated games.
  • Trust in Game Theory -- Unpublished Talk, May, 1997 8 January 2010 4:14 UTC philosophy.wisc.edu [Source type: FILTERED WITH BAYES]

.Game theorists respond by comparing their assumptions to those used in physics.^ Game theorists respond by comparing their assumptions to those used in physics.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

^ 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.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

^ Game theorists achieve this mathematically by determining which strategies are numerically dominant and using randomly mixed strategies.
  • A Game Theory Guide to Negotiations | Digital Tonto 8 January 2010 4:14 UTC www.digitaltonto.com [Source type: General]

.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.^ For this reason, economists use ‘overlapping generations’ models when modeling distribution games.
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: FILTERED WITH BAYES]
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC setis.library.usyd.edu.au [Source type: FILTERED WITH BAYES]

^ 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.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

^ The purpose of the course is to introduce basic concepts and results of the modern game theory which are increasingly used in Economics.
  • GAME THEORY Outline 8 January 2010 4:14 UTC www.nes.ru [Source type: Reference]

.However, additional criticism of this use of game theory has been levied because some experiments have demonstrated that individuals do not play equilibrium strategies.^ Keywords: experiments, game theory, mixed strategies.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ Keywords: experiments, game theory, 2x2 games.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ Keywords: experiments, game theory, evolution.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

.For instance, in the centipede game, guess 2/3 of the average game, and the dictator game, people regularly do not play Nash equilibria.^ Other games with these characteristics such as the game Guess 2/3 of the average and Traveler's dilemma invariably lead to experimental results that deviate markedly from the game-theoretical (Nash equilibrium) predictions.

^ It highlights the fact that one hasn't specified what game people are playing until one specifies the players' preferences and perspective.
  • Trust in Game Theory -- Unpublished Talk, May, 1997 8 January 2010 4:14 UTC philosophy.wisc.edu [Source type: FILTERED WITH BAYES]

^ To see this, first notice that there are an infinite number of Nash equilibria for this game.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

.There is an ongoing debate regarding the importance of these experiments.^ There is an ongoing debate regarding the importance of these experiments.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

[2]
.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.^ 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.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

^ One cannot ask why people play as they do without attributing to them subjective states of belief and preference.
  • Trust in Game Theory -- Unpublished Talk, May, 1997 8 January 2010 4:14 UTC philosophy.wisc.edu [Source type: FILTERED WITH BAYES]

^ Human communities evolve cultural norms to select equilibria in these games, and many of these equilibria will be compatible with high levels of apparently altruistic behavior in some (but not all) games.
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: FILTERED WITH BAYES]
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC setis.library.usyd.edu.au [Source type: FILTERED WITH BAYES]

.However, the question of how populations reach those points remains open.^ However, the question of how populations reach those points remains open.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

^ The problem of how the difference [min j max i t ij ] - [max min t ij ] ³ 0 should be subdivided between the players thus remains open.
  • Zero-Sum Games with Applications 8 January 2010 4:14 UTC home.ubalt.edu [Source type: Academic]

^ Notice however, that f rom the point of view of A, interactions with B are extremely rare as compared to those with A. Thus, we will assume that we can ignore the fitness contribution of A vs. B interactions to the overall fitness of strat.
  • Introduction to Game Theory -- simple, two-strategy examples 8 January 2010 4:14 UTC www.holycross.edu [Source type: FILTERED WITH BAYES]

.Some game theorists have turned to evolutionary game theory in order to resolve these worries.^ These are the times that " Game Theory " has been played: .
  • Breakfast of Champions Search Engine 8 January 2010 4:14 UTC wmbr.org [Source type: FILTERED WITH BAYES]

^ VirtualLabs in evolutionary game theory has some tutorials.
  • Al Roth's game theory, experimental economics, and market design page 8 January 2010 4:14 UTC kuznets.fas.harvard.edu [Source type: FILTERED WITH BAYES]

^ What, then, do these economic game theorists do?
  • Game Studies - The Games Economists Play - Implications of Economic Game Theory for the Study of Computer Games 8 January 2010 4:14 UTC gamestudies.org [Source type: FILTERED WITH BAYES]

.These models presume either no rationality or bounded rationality on the part of players.^ Modeling bounded rationality.
  • Game Theory Books: A Core Collection & More - Business Library - University of Florida 8 January 2010 4:14 UTC www.uflib.ufl.edu [Source type: Academic]

^ These models presume either no rationality or bounded rationality on the part of players.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

^ There is no bound on the number of offers that can be made but there is a cost to delay for each player.

.Despite the name, evolutionary game theory does not necessarily presume natural selection in the biological sense.^ Despite the name, evolutionary game theory does not necessarily presume natural selection in the biological sense.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

^ First, the ‘evolution’ treated by evolutionary game theory need not be biological evolution.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ Applications of Evolutionary Game Theory 4.1 A sense of fairness 4.2 The emergence of language.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

.Evolutionary game theory includes both biological as well as cultural evolution and also models of individual learning (for example, fictitious play dynamics).^ Keywords: experiments, game theory, learning, evolution, replicator dynamics.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ Keywords: experiments, game theory, evolution.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ Keywords: experiments, game theory, learning, dynamics, equilibrium selection.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

Prescriptive or normative analysis

Cooperate Defect
Cooperate -1, -1 -10, 0
Defect 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.^ Prescriptive or normative analysis Cooperate Defect 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.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

^ See economic game theory.
  • Half-Real: A Dictionary of Video Game Theory 8 January 2010 4:14 UTC www.half-real.net [Source type: General]

^ Now that's good behavioral game theory!
  • Sample Chapter for Camerer, C.F.: Behavioral Game Theory: Experiments in Strategic Interaction. 8 January 2010 4:14 UTC press.princeton.edu [Source type: FILTERED WITH BAYES]

.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.^ Game for more than one player .
  • Half-Real: A Dictionary of Video Game Theory 8 January 2010 4:14 UTC www.half-real.net [Source type: General]

^ A selection of strategies by a group of agents is said to be in a Nash equilibrium if each agent's strategy is a best-response to the strategies chosen by the other players.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ Game for one player .
  • Half-Real: A Dictionary of Video Game Theory 8 January 2010 4:14 UTC www.half-real.net [Source type: General]

.However, this use for game theory has also come under criticism.^ However, this use for game theory has also come under criticism.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

^ This is where game theory comes in.
  • Advanced NFL Stats: Game Theory and Run/Pass Balance 8 January 2010 4:14 UTC www.advancednflstats.com [Source type: General]

^ This particular view of game theory has come under recent criticism.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

.First, in some cases it is appropriate to play a non-equilibrium strategy if one expects others to play non-equilibrium strategies as well.^ First, in some cases it is appropriate to play a non- The Prisoner's Dilemma equilibrium strategy if one expects others to play non-equilibrium strategies as well.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

^ If not, one strategy will be increasing relative to the other .
  • Introduction to Game Theory -- simple, two-strategy examples 8 January 2010 4:14 UTC www.holycross.edu [Source type: FILTERED WITH BAYES]

^ Since a Nash equilibrium Cooperate -1, -1 -10, 0 of a game constitutes one's best response to the actions of the other players, playing a Defect 0, -10 -5, -5 strategy that is part of a Nash equilibrium seems appropriate.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

For an example, see Guess 2/3 of the average.
.Second, the Prisoner's dilemma presents another potential counterexample.^ Abstract: The prisoner's dilemma is modified by adding an initial stage in which players can precommit to reward the other one for a cooperative decision in the second stage.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ As an example of the second approach, consider the well-known Prisoner's Dilemma.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

.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.^ 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 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

^ Most classical game theoretical analyses predict that rational, self-interested players will make decisions to reach outcomes, known as Nash equilibria ( 4 ), from which no player can increase his or her own payoff unilaterally.
  • Social Decision-Making: Insights from Game Theory and Neuroscience -- Sanfey 318 (5850): 598 -- Science 8 January 2010 4:14 UTC www.sciencemag.org [Source type: Academic]

^ More specifically, in a social dilemma "each individual always receives a higher payoff for defecting than for cooperating, but all are better off if all cooperate than if all defect" (Dawes and Messick, 2000, p.
  • Game Studies - The Games Economists Play - Implications of Economic Game Theory for the Study of Computer Games 8 January 2010 4:14 UTC gamestudies.org [Source type: FILTERED WITH BAYES]

Biology

Hawk Dove
Hawk v−c, v−c 2v, 0
Dove 0, 2v v, v
The hawk-dove game
.Unlike economics, the payoffs for games in biology are often interpreted as corresponding to fitness.^ Zauner, K. G. (1999) A Payoff Uncertainty Explanation of Results in Experimental Centipede Games, Games and Economic Behavior , 26:1 (January), 157-185.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ On Interpreting Payoffs: Morality and Efficiency in Games .
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: FILTERED WITH BAYES]
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC setis.library.usyd.edu.au [Source type: FILTERED WITH BAYES]

.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.^ Again, one may wonder what has been gained by the evolutionary model--would it not have been just as easy to determine the cultural dynamics and initial conditions beforehand, constructing the model afterwards?
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ As we will see after discussing evolutionary game theory in a later section, it is possible to maintain this understanding of equilibria in the case of game theory.
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: FILTERED WITH BAYES]
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC setis.library.usyd.edu.au [Source type: FILTERED WITH BAYES]

^ Institutions and evolutionary processes build many environments, and what counts as rational procedure in one environment may not be favoured in another.
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: FILTERED WITH BAYES]
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC setis.library.usyd.edu.au [Source type: FILTERED WITH BAYES]

.The best known equilibrium in biology is known as the evolutionarily stable strategy (or ESS), and was first introduced in (Smith & Price 1973).^ The first approach derives from the work of Maynard Smith and Price and employs the concept of an evolutionarily stable strategy as the principal tool of analysis.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ In 1972, Maynard Smith defined the concept of an evolutionarily stable strategy (hereafter ESS) in the article “Game Theory and the Evolution of Fighting.” However, it was the publication of “The Logic of Animal Conflict,” by Maynard Smith and Price in 1973 that introduced the concept of an ESS into widespread circulation.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ Maynard Smith, John and George Price (1973).
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

.Although its initial motivation did not involve any of the mental requirements of the Nash equilibrium, every ESS is a Nash equilibrium.^ Although its initial motivation did not involve any of the mental requirements of the Nash equilibrium, every ESS is a Nash equilibrium.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

^ These equilibrium concepts are motivated differently depending on the field of application, although they often overlap or coincide.
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^ Although the game-theoretic equilibrium of zero is a poor guess about initial choices, players are inexorably drawn toward zero as they learn.
  • Sample Chapter for Camerer, C.F.: Behavioral Game Theory: Experiments in Strategic Interaction. 8 January 2010 4:14 UTC press.princeton.edu [Source type: FILTERED WITH BAYES]

.In biology, game theory has been used to understand many different phenomena.^ Evolutionary game theory has been used to explain many seemingly incongruous phenomena in nature.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

^ In biology, game theory has been used to understand many different phenomena.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

^ Some other theorists understand the point of game theory differently.
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: FILTERED WITH BAYES]
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC setis.library.usyd.edu.au [Source type: FILTERED WITH BAYES]

.It was first used to explain the evolution (and stability) of the approximate 1:1 sex ratios.^ It was first used to explain the evolution (and stability) of the approximate 1:1 sex ratios.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

^ This theory of the evolution of the sex ratio is normally attributed to R. A. Fisher (The Genetical Theory of Natural Selection.

^ Using the expression for B vs. A that you just wrote and the matrix above , explain whether or not B is stable against invasion by A. (ANS) .
  • Introduction to Game Theory -- simple, two-strategy examples 8 January 2010 4:14 UTC www.holycross.edu [Source type: FILTERED WITH BAYES]

.(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.^ We can think of each lineage as ‘trying’ to maximize its reproductive fitness (= expected number of grandchildren) through finding strategies that are optimal given the strategies of other lineages.
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: FILTERED WITH BAYES]
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC setis.library.usyd.edu.au [Source type: FILTERED WITH BAYES]

^ It all depends, of course, on how accurately the replicator dynamics models the primary evolutionary forces (cultural or biological) acting on human populations.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ Fisher pointed out that, in such a situation, the evolutionary dynamics lead to the sex ratio becoming fixed at equal numbers of males and females.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

.Additionally, biologists have used evolutionary game theory and the ESS to explain the emergence of animal communication (Harper & Maynard Smith 2003).^ Evolutionary game theory has been used to explain a number of aspects of human behavior.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ Evolutionary game theory owes its explicit genesis to Maynard Smith (1982) (**).
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: FILTERED WITH BAYES]
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC setis.library.usyd.edu.au [Source type: FILTERED WITH BAYES]

^ Applications of Evolutionary Game Theory 4.1 A sense of fairness 4.2 The emergence of language.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

.The analysis of signaling games and other communication games has provided some insight into the evolution of communication among animals.^ Here are some of the main insights from the game: .
  • Game Theory in The Dark Knight: the opening scene (spoilers) - Mind Your Decisions 8 January 2010 4:14 UTC mindyourdecisions.com [Source type: General]

^ The analysis of signaling games and other communication games has provided some insight into the evolution of communication among animals.
  • Game Theory 8 January 2010 4:14 UTC www.slideshare.net [Source type: FILTERED WITH BAYES]

^ The employees in CEE regard the corporate environment as a conflict zone, a zero sum game (in which the gains by some equal the losses to others).
  • Game Theory Applied to Economics 8 January 2010 4:14 UTC samvak.tripod.com [Source type: Original source]

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.^ Games and Economic Behavior has a web page from which you can see tables of contents.
  • Al Roth's game theory, experimental economics, and market design page 8 January 2010 4:14 UTC kuznets.fas.harvard.edu [Source type: FILTERED WITH BAYES]

.Biologists have used the game of chicken to analyze fighting behavior and territoriality.^ The following subsections provide a brief illustration of the use of evolutionary game theoretic models to explain two areas of human behavior.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ Abstract: This paper uses the Luce probabilistic choice rule to analyze behavior in matrix games.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ Seminal texts in neuroeconomics, with extensive use of and implications for behavioral game theory, are Montague and Berns (2002) , Glimcher 2003 (**), and Camerer, Loewenstein and Prelec (2005) .
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: FILTERED WITH BAYES]
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC setis.library.usyd.edu.au [Source type: FILTERED WITH BAYES]

[citation needed]
.Maynard Smith, in the preface to Evolution and the Theory of Games, writes, "[p]aradoxically, 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.^ Keywords: experiments, game theory, evolution.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ Evolutionary game theory has been used to explain a number of aspects of human behavior.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ Evolution and the Theory of Games” Journal of Theoretical Biology , 1: 382–403.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

[3]
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.^ They thus acted so as to create situations in which this was true for other Serbs (Hutus) as well.
  • Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: FILTERED WITH BAYES]
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^ There is no way of applying game theory ‘all by itself’, independently of other modelling technologies.
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^ Briefly, the Prisoner's Dilemma is a situation in which two people are faced with a temptation to act in their personal interest disregarding the interest of the other person.
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.This is distinct from traditional notions of altruism because such actions are not conscious, but appear to be evolutionary adaptations to increase overall fitness.^ The topic that has received most attention from evolutionary game theorists is altruism , defined as any behaviour by an organism that decreases its own expected fitness in a single interaction but increases that of the other interactor.
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^ This is so because in such games (as long as the games are finite, that is, terminate after a known number of actions) players and analysts can use a straightforward procedure for predicting outcomes.
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.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.^ It can be raised with respect to any number of examples, but we will borrow an elegant one from C. Bicchieri ( 1993 ), who also provides the most extensive treatment of the problem found in the literature.
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^ When they ran this game under hyperscanning, the King-Casas and Montague group obtained the following results.
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^ The puzzle Fisher faced was this: why is it that the sex ratio is approximately equal in many species where the majority of males never mate?
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[4] .All of these actions increase the overall fitness of a group, but occur at a cost to the individual.^ Then standing and fighting is each soldier's individually rational course of action after all, because the cost of running is sure to be at least as high as the cost of staying.
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^ The coordination failure occurs because the Sender and Receiver only pair the appropriate action with the state of the world in one instance, as with 〈Sender 1, Receiver 1〉, or not at all, as with 〈Sender 2, Receiver 3〉.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

.Evolutionary game theory explains this altruism with the idea of kin selection.^ Keywords: experiments, game theory, altruism.
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^ The explanatory irrelevance of evolutionary game theory .
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^ The principles of evolutionary game theory are best explained through examples.
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Altruists discriminate between the individuals they help and favor relatives. .Hamilton's rule explains the evolutionary reasoning behind this selection with the equation c<b*r where the cost ( c ) to the altruist must be less than the benefit ( b ) to the recipient multiplied by the coefficient of relatedness ( r ).^ As we saw, the puzzle in that game consists in the fact that if the fugitive's reasoning selects a particular bridge as optimal, his pursuer must be assumed to be able to duplicate that reasoning.
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.The more closely related two organisms are causes the incidences of altruism to increase because they share many of the same alleles.^ Here are two papers on the early history of experimental economics (and its close relation to game theory): .
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^ Because they are more difficult for other players to infer, their use increases the probability of miscommunication.
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^ Often, of course, more than two players compete for the same resource.
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.This means that the altruistic individual, by ensuring that the alleles of its close relative are passed on, (through survival of its offspring) can forgo the option of having offspring itself because the same number of alleles are passed on.^ Distrust towards newcomers is an unfortunate consequence of having any degree of persistent identities, which, as mentioned above, is in itself an essential feature for ensuring trust in many interactions [5] .
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Helping a sibling for example, has a coefficient of ½, because an individual shares ½ of the alleles in its sibling's offspring. Ensuring that enough of a sibling’s offspring survive to adulthood precludes the necessity of the altruistic individual producing offspring.[4] .The coefficient values depend heavily on the scope of the playing field; for example if the choice of whom to favor includes all genetic living things, not just all relatives, we assume the discrepancy between all humans only accounts for approximately 1% of the diversity in the playing field, a co-efficient that was ½ in the smaller field becomes 0.995. Similarly if it is considered that information other than that of a genetic nature (e.g.^ The strategic dynamics may be quite different between games which are only played once (single-shot games) and games that are repeated.
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^ If all people know that these incentives hold for most others, then cooperation will not only be possible, but will be the expected norm, and the war of all against all becomes a general peace.
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^ This is obviously only dangerous in free-for-all games or team games where players can switch between teams.
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epigenetics, religion, science, etc) persisted through time the playing field becomes larger still, and the discrepancies smaller.

Computer science and logic

.Game theory has come to play an increasingly important role in logic and in computer science.^ Game theory in computer science .
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^ Game theory plays a leading role in neuroeconomics at two levels.
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^ Combinatorial Game Theory : Some game theory with connections to operations research and computer science can be found at Combinatorial Game Theory maintained by David Eppstein at UC Irvine.
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.Several logical theories have a basis in game semantics.^ Gintis (2000) (**) has provided a text crammed with terrific problem exercises, which is also unique in that it treats evolutionary game theory as providing the foundational basis for game theory in general.
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^ Prior to the advent of game theory, therefore, economists were severely limited in the class of circumstances to which they could neatly apply their models.
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^ Our last point above opens the way to a philosophical puzzle, one of several that still preoccupy those concerned with the logical foundations of game theory.
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.In addition, computer scientists have used games to model interactive computations.^ Researchers in matching and market design: (game theorists, experimenters and computer scientists interested in matching, auctions and market design) Alphabetically by last name: .
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^ First, game theory has been used to predict the computations that individual neurons and groups of neurons serving the reward system must perform.
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^ For this reason, economists use ‘overlapping generations’ models when modeling distribution games.
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.Also, game theory provides a theoretical basis to the field of multi-agent systems.^ Second, the rationality assumptions underlying evolutionary game theory are, in many cases, more appropriate for the modelling of social systems than those assumptions underlying the traditional theory of games.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ Moreover, most of the evolutionary game theoretic models developed to date have provided the crudest approximations of the real cultural dynamics driving the social phenomenon in question.
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^ Keywords: experiments, game theory, principal agent model, contracts, linear contracts, inequality aversion, fairness.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

.Separately, game theory has played a role in online algorithms.^ Game theory plays a leading role in neuroeconomics at two levels.
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^ With the use of game theory algorithms, those responses become predictable, even foreseeable—within an acceptable range of probabilities.
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^ Keywords: experiments, game theory, coordination, suggested play.
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.In particular, the k-server problem, which has in the past been referred to as games with moving costs and request-answer games (Ben David, Borodin & Karp et al. 1994).^ According to Binmore's ( 1994 , 1998 , 2005a ) model,the basic class of strategic problems facing non-eusocial social animals are coordination games.
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^ Budescu, David V.*, and Amnon Rapoport (1994) Subjective Randomization in One- and Two-Person Games, Journal of Behavioral Decision Making , 7261-278.
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^ Game Theory, Rationality and Evolution,” in M. L. Dalla Chiara et al .
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.Yao's principle is a game-theoretic technique for proving lower bounds on the computational complexity of randomized algorithms, and especially of online algorithms.^ Ross (2005a) studies the game-theoretic foundations of microeconomics in general, but especially behavioral economics and neuroeconomics, from the perspective of cognitive science.
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^ We begin our backward-induction analysis—using a technique called Zermelo's algorithm —with the sub-games that arise last in the sequence of play.
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^ A useful high-level principle for sorting the literature indexes it to the different auxiliary assumptions with which game-theoretic axioms are applied.
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.The field of algorithmic game theory combines computer science concepts of complexity and algorithm design with game theory and economic theory.^ Frdrick Asselin maintains a list of researchers in Game Theory or Economics and Computer Science .
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^ Game theory in computer science .
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^ I n 2005, the Nobel Prize in Economic Science was awarded to Israeli mathematician and game theory specialist Robert J. Aumann, co-founder of the Center for Rationality at Hebrew University.
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.The emergence of the internet has motivated the development of algorithms for finding equilibria in games, markets, computational auctions, peer-to-peer systems, and security and information markets.^ That is, a player can find a set of systems of beliefs for the other players such that any history of the game along an equilibrium path is consistent with that set of systems.
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^ Researchers in matching and market design: (game theorists, experimenters and computer scientists interested in matching, auctions and market design) Alphabetically by last name: .
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^ Keywords: experiments, game theory, mixed strategies, posted offer auctions, markets, risk aversion, capacity constraints, cost asymmetries.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

[5]

Philosophy

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.Game theory has been put to several uses in philosophy.^ With the use of game theory algorithms, those responses become predictable, even foreseeable—within an acceptable range of probabilities.
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^ Evolutionary game theory has been used to explain a number of aspects of human behavior.
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^ Yaw Nyarko is at NYU. Barry O'Neill , at UCLA uses game theory in innovative ways in political science.
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.Responding to two papers by W.V.O. Quine (1960, 1967), Lewis (1969) used game theory to develop a philosophical account of convention.^ However, the philosopher who wants game theory to serve as a descriptive and/or normative theory of strategic rationality cannot rest content with this answer.
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^ At this point, one may see little difference between the two approaches to evolutionary game theory.
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^ Abstract: This paper reports a series of two-person coordination game experiments in which the outcomes are generally well organized by the notion of risk dominance.
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.In so doing, he provided the first analysis of common knowledge and employed it in analyzing play in coordination games.^ Keywords: experiments, game theory, coordination, suggested play.
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^ The first approach can thus be thought of as providing a static conceptual analysis of evolutionary stability.
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^ Abstract: Subjects play several finite prisoner's dilemma games simultaneously, which allows rejection of the common notion that players can be categorized into distinct "types."
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.In addition, he first suggested that one can understand meaning in terms of signaling games.^ Shmuel Zamir at the Hebrew University is the last game theorist I know in alphabetical order (but one of the first game theorists I knew).
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^ First, the game must be repeated, with uncertainty as to which round is the last one.
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^ It is natural, as a first approximation, to think of sequential-move games as being ones in which players choose their strategies one after the other, and of simultaneous-move games as ones in which players choose their strategies at the same time.
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.This later suggestion has been pursued by several philosophers since Lewis (Skyrms (1996), Grim, Kokalis, and Alai-Tafti et al. (2004)).^ Some exciting applications of evolutionary game theory to a range of philosophical issues, on which this article has drawn heavily, is Skyrms (1996) (**).
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^ Lewis (1969) (**) puts game-theoretic equilibrium concepts to wider application in philosophy, a program that is carried a good deal further in Skyrms (1996) (**).
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^ For a selection of some discussion see, in particular, D'Arms (1996, 2000); D'Arms et al ., 1998; Danielson (1998); Bicchieri (1999); Kitcher (1999); Gintis (2000); Harms (2000); Krebs (2000); Alexander and Skyrms (1999); and Alexander (2000, 2007).
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.Following Lewis (1969) game-theoretic account of conventions, Ullmann Margalit (1977) and Bicchieri (2006) have developed theories of social norms that define them as Nash equilibria that result from transforming a mixed-motive game into a coordination game.^ Keywords: experiments, game theory, coordination, convention.
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^ Keywords: experiments, game theory, coordination.
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^ Keywords: experiments, game theory, entry games, coordination, mixed strategies.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

[6]
.Game theory has also challenged philosophers to think in terms of interactive epistemology: what it means for a collective to have common beliefs or knowledge, and what are the consequences of this knowledge for the social outcomes resulting from agents' interactions.^ However, the philosopher who wants game theory to serve as a descriptive and/or normative theory of strategic rationality cannot rest content with this answer.
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^ Keywords: experiments, game theory, beliefs, scoring rule, elicitation.
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^ Second, the rationality assumptions underlying evolutionary game theory are, in many cases, more appropriate for the modelling of social systems than those assumptions underlying the traditional theory of games.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

.Philosophers who have worked in this area include Bicchieri (1989, 1993),[7] Skyrms (1990),[8] and Stalnaker (1999).^ It can be raised with respect to any number of examples, but we will borrow an elegant one from C. Bicchieri ( 1993 ), who also provides the most extensive treatment of the problem found in the literature.
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^ Philosophical puzzles at this foundational level are critically discussed in Bicchieri (1993) (**).
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[9]
.In ethics, some authors have attempted to pursue the project, begun by Thomas Hobbes, of deriving morality from self-interest.^ 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.
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^ Thus, people's choices in the UG do not conform to a model in which decisions are driven by financial self-interest, and neuroscience has begun to offer clues as to the mechanisms underlying these decisions.
  • Social Decision-Making: Insights from Game Theory and Neuroscience -- Sanfey 318 (5850): 598 -- Science 8 January 2010 4:14 UTC www.sciencemag.org [Source type: Academic]

.Since games like the Prisoner's dilemma present an apparent conflict between morality and self-interest, explaining why cooperation is required by self-interest is an important component of this project.^ We will demonstrate this shortly by reference to the most famous (though not the most typical) game, the so-called Prisoner's Dilemma , and to other, more typical, games.
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^ Keywords: experiments, game theory, prisoner's dilemma, communication, gender effects.
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^ Game theorists objected that MAD was mad, because it set up a Prisoner's Dilemma as a result of the fact that the reciprocal threats were incredible.
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.This general strategy is a component of the general social contract view in political philosophy (for examples, see Gauthier (1986) and Kavka (1986).^ For example, two firms might commit to their marketing strategies independently and in secrecy from one another, but thereafter engage in pricing competition in full view of one another.
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^ Précis of Evolution of the Social Contract ,” Philosophy and Phenomenological Research , 59 (1): 217–220.
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^ Critical Notice: Evolution of the Social Contract ,” Canadian Journal of Philosophy , 28 (4): 627–652.
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[10]
.Other authors have attempted to use evolutionary game theory in order to explain the emergence of human attitudes about morality and corresponding animal behaviors.^ Evolutionary game theory has been used to explain a number of aspects of human behavior.
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^ The model is used to explain behavior in games with unique mixed strategies.
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^ Other game theory servers: .
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.These authors look at several games including the Prisoner's dilemma, Stag hunt, and the Nash bargaining game as providing an explanation for the emergence of attitudes about morality (see, e.g., Skyrms (1996, 2004) and Sober and Wilson (1999)).^ We will demonstrate this shortly by reference to the most famous (though not the most typical) game, the so-called Prisoner's Dilemma , and to other, more typical, games.
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^ It's useful to start the discussion here from the case of the Prisoner's Dilemma because it's unusually simple from the perspective of these puzzles.
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^ To see this, first notice that there are an infinite number of Nash equilibria for this game.
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.Some assumptions used in some parts of game theory have been challenged in philosophy; psychological egoism states that rationality reduces to self-interest—a claim debated among philosophers.^ Keywords: experiments, game theory, psychological game theory.
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^ However, the philosopher who wants game theory to serve as a descriptive and/or normative theory of strategic rationality cannot rest content with this answer.
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^ Second, the rationality assumptions underlying evolutionary game theory are, in many cases, more appropriate for the modelling of social systems than those assumptions underlying the traditional theory of games.
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(see Psychological egoism#Criticism)

Types of games

Cooperative or non-cooperative

.A game is cooperative if the players are able to form binding commitments.^ In this game, individuals choose one of two strategies, typically called “Cooperate” and “Defect.” Here is the general form of the payoff matrix for the prisoner's dilemma: .
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^ The two sorts of games are not equivalent, because extensive-form games contain information—about sequences of play and players' levels of information about the game structure—that strategic-form games do not.
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^ The former provides a setwise generalization of the concept of an evolutionarily stable strategy, and the latter extends the concept of an evolutionarily stable strategy to the context of two-player extensive form games.
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For instance the legal system requires them to adhere to their promises. In noncooperative games this is not possible.
.Often it is assumed that communication among players is allowed in cooperative games, but not in noncooperative ones.^ It is also one of the landmarks of "cooperative" game theory.
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^ Outcome : an assignment of a set of payoffs, one to each player in the game.
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^ Abstract: The experiment introduces costless preplay communication into a multi-player coordination game with multiple Pareto-ranked equilibria.
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This classification on two binary criteria has been rejected (Harsanyi 1974).
.Of the two types of games, noncooperative games are able to model situations to the finest details, producing accurate results.^ Moreover, even if an evolutionary game theoretic model indicated that a single historical sequence was capable of producing a given social phenomenon, there remains the important question of why we ought to take this result seriously.
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^ For now, we have described it just in order to use it to introduce one of the two types of mathematical objects used to represent games: game-trees .
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^ Abstract: The paper uses Rabin's fairness model as a basis for explaining results of a standard trust game.
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.Cooperative games focus on the game at large.^ In this game, the basin of attraction for defection is large unless there is a high proportion of cooperators in round one.
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Considerable efforts have been made to link the two approaches. .The so-called Nash-programme has already established many of the cooperative solutions as noncooperative equilibria.^ As a result, when set into what is intended to be a one-shot PD in the experimental laboratory, people tend to initially play as if the game were a single round of a repeated PD. The repeated PD has many Nash equilibria that involve cooperation rather than defection.
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.Hybrid games contain cooperative and non-cooperative elements.^ The Prisoner's Dilemma and Dynamical Systems Associated to Non-cooperative Games,” Econometrica , 48: 1617–1634.
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^ Non-cooperative Games.’ Annals of Mathematics Journal 54:286-295.
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.For instance, coalitions of players are formed in a cooperative game, but these play in a non-cooperative fashion.^ As the values of these two variables increase, game behavior shifts (weakly) in the direction of Nash equilibrium play.
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^ The Prisoner's Dilemma and Dynamical Systems Associated to Non-cooperative Games,” Econometrica , 48: 1617–1634.
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^ Figure 1: The Hawk-Dove Game (The payoffs listed in the matrix are for that of a player using the strategy in the appropriate row, playing against someone using the strategy in the appropriate column.
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Symmetric and asymmetric

E F
E 1, 2 0, 0
F 0, 0 1, 2
An asymmetric game
.A symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them.^ If player I plays strategy 2, his payoff depends on which strategy player II has chosen to play.
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^ The fate (or the payoff) of a player in a game depends not only on the actions of that player but also on the other players!
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^ E F Symmetric and asymmetric E 1, 2 0, 0 Main article: Symmetric game F 0, 0 1, 2 A symmetric game is a game where the payoffs for playing a particular strategy depend only An asymmetric game on the other strategies employed, not on who is playing them.
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.If the identities of the players can be changed without changing the payoff to the strategies, then a game is symmetric.^ In an equilibrium, each player of the game has adopted a strategy that they are unlikely to change.
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^ If the identities of the players can be changed without changing the payoff to the strategies, then a game is symmetric.
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^ Two-Person Zero Sum Game In a two-person zero sum game , each of the two players is given a choice between several prescribed strategies at each turn, and each player's loss is equal to the other player's gain.
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.Many of the commonly studied 2×2 games are symmetric.^ Many of the commonly studied 2×2 games are symmetric.
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^ Most commonly studied asymmetric games are games where there are not identical strategy sets for both players.
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^ But many aspects of strategy can be studied and systematized into a science -- game theory.
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.The standard representations of chicken, the prisoner's dilemma, and the stag hunt are all symmetric games.^ We will demonstrate this shortly by reference to the most famous (though not the most typical) game, the so-called Prisoner's Dilemma , and to other, more typical, games.
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^ Keywords: experiments, game theory, prisoner's dilemma, communication, gender effects.
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^ Game theorists objected that MAD was mad, because it set up a Prisoner's Dilemma as a result of the fact that the reciprocal threats were incredible.
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.Some scholars would consider certain asymmetric games as examples of these games as well.^ Human communities evolve cultural norms to select equilibria in these games, and many of these equilibria will be compatible with high levels of apparently altruistic behavior in some (but not all) games.
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^ For some games, these adjustment paths lead to equilibria that are ruled out by the intuitive criterion and other refinements.
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^ These “failures of the transitivity of preference” would not occur if people had a well-defined consistent set of preferences.
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.However, the most common payoffs for each of these games are symmetric.^ However, most games do not have this property.
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^ However, for most games of reasonable complexity (and hence interest), the extensive form of the game quickly becomes unmanageable.
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^ Clearly, however, if this is applied then a theory of games that incorporated it would not be descriptively true of most people.
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.Most commonly studied asymmetric games are games where there are not identical strategy sets for both players.^ For example, the game pictured to the right is asymmetric despite having identical strategy sets for both players.
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^ Most commonly studied asymmetric games are games where there are not identical strategy sets for both players.
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^ It is possible, however, for a game to have identical strategies for both players, yet be asymmetric.
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.For instance, the ultimatum game and similarly the dictator game have different strategies for each player.^ For instance, the ultimatum game and similarly the dictator game have different strategies for each player.
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^ Game theory is a branch of applied mathematics that studies strategic situations where players choose different actions in an attempt to maximize their returns.
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^ Two-Person Zero Sum Game In a two-person zero sum game , each of the two players is given a choice between several prescribed strategies at each turn, and each player's loss is equal to the other player's gain.
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.It is possible, however, for a game to have identical strategies for both players, yet be asymmetric.^ For example, the game pictured to the right is asymmetric despite having identical strategy sets for both players.
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^ Most commonly studied asymmetric games are games where there are not identical strategy sets for both players.
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^ It is possible, however, for a game to have identical strategies for both players, yet be asymmetric.
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.For example, the game pictured to the right is asymmetric despite having identical strategy sets for both players.^ For example, the game pictured to the right is asymmetric despite having identical strategy sets for both players.
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^ Most commonly studied asymmetric games are games where there are not identical strategy sets for both players.
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^ It is possible, however, for a game to have identical strategies for both players, yet be asymmetric.
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Zero-sum and non-zero-sum

A B
A –1, 1 3, –3
B 0, 0 –2, 2
A zero-sum game
.Zero-sum games are a special case of constant-sum games, in which choices by players can neither increase nor decrease the available resources.^ We were able to solve this game straightforwardly because we set the utility functions in such a way as to make it zero-sum , or strictly competitive .
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^ First, there is the problem that in most non-zero-sum games, there is more than one NE, but not all NE look equally plausible as the solutions upon which strategically rational players would hit.
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^ In an inspection game, one player faces a series of choices either to work for a reward, in which case he is sure to receive it, or to perform another, easier action ("shirking"), in which case he will receive the reward only if the other player (the "inspector") is not monitoring him.
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.In zero-sum games the total benefit to all players in the game, for every combination of strategies, always adds to zero (more informally, a player benefits only at the equal expense of others).^ All other possible combinations of strategies result in the players failing to coordinate.
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^ Each player in a game faces a choice among two or more possible strategies .
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^ Matrices, unlike trees, simply show the outcomes, represented in terms of the players' utility functions, for every possible combination of strategies the players might use.
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.Poker exemplifies a zero-sum game (ignoring the possibility of the house's cut), because one wins exactly the amount one's opponents lose.^ Because nodes 13 and 14 fall inside a common information set, Selten's Horse has only one subgame (namely, the whole game).
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^ We were able to solve this game straightforwardly because we set the utility functions in such a way as to make it zero-sum , or strictly competitive .
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^ First, there is the problem that in most non-zero-sum games, there is more than one NE, but not all NE look equally plausible as the solutions upon which strategically rational players would hit.
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.Other zero-sum games include matching pennies and most classical board games including Go and chess.^ The game “Matching Pennies” illustrates this problem.
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^ Figure 7: Payoff matrix for the game of Matching Pennies (Row wins if the two coins do not match, whereas Column wins if the two coins match).
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^ Keywords: experiments, game theory, bounded rationality, procedural rationality, coordination games, matching pennies, mixed strategies.
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.Many games studied by game theorists (including the famous prisoner's dilemma) are non-zero-sum games, because some outcomes have net results greater or less than zero.^ For this purpose, we'll use the most famous game: the Prisoner's Dilemma.
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^ Game theorists objected that MAD was mad, because it set up a Prisoner's Dilemma as a result of the fact that the reciprocal threats were incredible.
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^ We will demonstrate this shortly by reference to the most famous (though not the most typical) game, the so-called Prisoner's Dilemma , and to other, more typical, games.
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.Informally, in non-zero-sum games, a gain by one player does not necessarily correspond with a loss by another.^ Another famous non-zero sum strategy game is The Prisoner's Dilemma: .
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^ In this case the sum is not fixed; the game is non-zero sum (or variable sum), as the total sum of gains and losses experienced by the players is not necessarily zero.
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^ Such games are termed zero-sum , as the total sum of the players' gains and losses equals zero.
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.Constant-sum games correspond to activities like theft and gambling, but not to the fundamental economic situation in which there are potential gains from trade.^ Keywords: experiments, game theory, centipede games, constant-sum games.
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^ Keywords: experiments, game theory, learning, constant sum games.
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^ Mookherjee, Dilip, and Barry Sopher (1997) Learning and Decision Costs in Experimental Constant Sum Games, Games and Economic Behavior , 19:1 (April), 97-132.
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.It is possible to transform any game into a (possibly asymmetric) zero-sum game by adding an additional dummy player (often called "the board"), whose losses compensate the players' net winnings.^ Abstract: The experiment introduces costless preplay communication into a multi-player coordination game with multiple Pareto-ranked equilibria.
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^ We were able to solve this game straightforwardly because we set the utility functions in such a way as to make it zero-sum , or strictly competitive .
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^ First, there is the problem that in most non-zero-sum games, there is more than one NE, but not all NE look equally plausible as the solutions upon which strategically rational players would hit.
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Simultaneous and sequential

.Simultaneous games are games where both players move simultaneously, or if they do not move simultaneously, the later players are unaware of the earlier players' actions (making them effectively simultaneous).^ For example, if two competing businesses are both planning marketing campaigns, one might commit to its strategy months before the other does; but if neither knows what the other has committed to or will commit to when they make their decisions, this is a simultaneous-move game.
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^ But you will recall from earlier in this section that this is just what defines two moves as simultaneous.
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^ We can solve this new game if we make certain assumptions about the two players' utility functions.
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.Sequential game (or dynamic games) are games where later players have some knowledge about earlier actions.^ We can solve this new game if we make certain assumptions about the two players' utility functions.
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^ Chess, by contrast, is normally played as a sequential-move game: you see what your opponent has done before choosing your own next action.
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^ That is, a player might intend to take an action but then slip up in the execution and send the game down some other path instead.
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.This need not be perfect information about every action of earlier players; it might be very little knowledge.^ Matrices, unlike trees, simply show the outcomes, represented in terms of the players' utility functions, for every possible combination of strategies the players might use.
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^ The simplest games (from the perspective of logical structure) are those in which agents have perfect information , meaning that at every point where each agent's strategy tells her to take an action, she knows everything that has happened in the game up to that point.
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^ Strategy : a program instructing a player which action to take at every node in the tree where she could possibly be called on to make a choice.
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.For instance, a player may know that an earlier player did not perform one particular action, while he does not know which of the other available actions the first player actually performed.^ In the first case, nodes at the top of the page are interpreted as coming earlier in the sequence of actions.
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^ If what the Receiver does is the correct response, given the state of the world, then both players receive a payoff of 1; if the Receiver performed an incorrect response, then both players receive a payoff of 0.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ For example, if two competing businesses are both planning marketing campaigns, one might commit to its strategy months before the other does; but if neither knows what the other has committed to or will commit to when they make their decisions, this is a simultaneous-move game.
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.The difference between simultaneous and sequential games is captured in the different representations discussed above.^ The difference between simultaneous and sequential games is captured in the different representations discussed above.
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^ More interestingly, one could contrast the sequential game described above, in which A knows that her move will be common knowledge before B moves, with a different sequential game in which A is not told that B will know how she moved before he gets to move.
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^ If at least one node shares its information set with another, while others are alone, the game involves both simultaneous and sequential play, and so is still a game of imperfect information.
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.Often, normal form is used to represent simultaneous games, and extensive form is used to represent sequential ones; although this isn't a strict rule in a technical sense.^ Matrix games are referred to as ‘normal-form’ or ‘strategic-form’ games, and games as trees are referred to as ‘extensive-form’ games.
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^ Email Contact: stahl@eco.utexas.edu Stahl, Dale O. (1999) Rule-Learning in Symmetric Normal-Form Games: Theory and Evidence, Games and Economic Behavior , forthcoming.
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^ If one seeks to use an evolutionary game theoretic model to explain the normativity attached to a social rule, one must explain how such an approach avoids committing the so-called “naturalistic fallacy” of inferring an ought-statement from a conjunction of is-statements.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

Perfect information and imperfect information

A game of imperfect information (the dotted line represents ignorance on the part of player 2)
.An important subset of sequential games consists of games of perfect information.^ We will assume that the games involved are games of perfect information .
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^ Perfect information and imperfect information A game of imperfect information (the dotted line represents ignorance on the part of player 2) Main article: Perfect information An important subset of sequential games consists of games of perfect information.
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^ Most games studied in game theory are imperfect-information games, although there are some interesting examples of perfect-information games, including the ultimatum game and centipede game.
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.A game is one of perfect information if all players know the moves previously made by all other players.^ Only if all information sets are inhabited by just one node do we have a game of perfect information.
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^ Game for more than one player .
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^ A game is one of perfect information if all players know the moves previously made by all other players.
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.Thus, only sequential games can be games of perfect information, since in simultaneous games not every player knows the actions of the others.^ Thus, only sequential games can be games of perfect information, since in simultaneous games not every player knows the actions of the others.
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^ The fate (or the payoff) of a player in a game depends not only on the actions of that player but also on the other players!
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^ A game is one of perfect information if all players know the moves previously made by all other players.
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.Most games studied in game theory are imperfect-information games, although there are some interesting examples of perfect-information games, including the ultimatum game and centipede game.^ Other servers devoted to game theory (including some blogs) .
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^ By contrast, the example of the bridge-crossing game from Section 1 above illustrates a game of imperfect information , since the fugitive must choose a bridge to cross without knowing the bridge at which the pursuer has chosen to wait, and the pursuer similarly makes her decision in ignorance of the actions of her quarry.
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^ Only if all information sets are inhabited by just one node do we have a game of perfect information.
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.Perfect-information games include also chess, go, mancala, and arimaa.^ The setting is a two-stage game with perfect information.
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^ Rosenthal, Robert W. (1981) Games of Perfect Information, Predatory Pricing and the Chain Store Paradox, Journal of Economic Theory , 2592-100.
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Perfect information is often confused with complete information, which is a similar concept. .Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions.^ Own strategies/decisions - Other players' strategies/decisions .
  • Game theory - Theories Used in IS Research 8 January 2010 4:14 UTC www.fsc.yorku.ca [Source type: Academic]

^ The fate (or the payoff) of a player in a game depends not only on the actions of that player but also on the other players!
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^ Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions.
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Infinitely long games

.Games, as studied by economists and real-world game players, are generally finished in a finite number of moves.^ For this reason, economists use ‘overlapping generations’ models when modeling distribution games.
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^ A number of game theory papers and abstracts are available from the webpages of the Center for mathematical studies in economics and management science at Northwestern, and from the CentER for Economic Research at Tilburg.
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^ There's even a band named Game Theory , and a Game Theory Guild for World of Warcraft players.
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.Pure mathematicians are not so constrained, and set theorists in particular study games that last for an infinite number of moves, with the winner (or other payoff) not known until after all those moves are completed.^ To see this, first notice that there are an infinite number of Nash equilibria for this game.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ We see that eventually the population, for this particular set of initial conditions, converges to one of the pure Lewisian signalling systems identified above.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ Game theorists objected that MAD was mad, because it set up a Prisoner's Dilemma as a result of the fact that the reciprocal threats were incredible.
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.The focus of attention is usually not so much on what is the best way to play such a game, but simply on whether one or the other player has a winning strategy.^ Figure 11 illustrates the state space under the continuous replicator dynamics for the sender-receiver game consisting of two states of the world, two signals, and two responses, where players are restricted to using one of the previous four strategies.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

^ Matrices, unlike trees, simply show the outcomes, represented in terms of the players' utility functions, for every possible combination of strategies the players might use.
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^ Outcome : an assignment of a set of payoffs, one to each player in the game.
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.(It can be proven, using the axiom of choice, that there are games—even with perfect information, and where the only outcomes are "win" or "lose"—for which neither player has a winning strategy.^ Each player in a game faces a choice among two or more possible strategies .
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^ Figure 11 illustrates the state space under the continuous replicator dynamics for the sender-receiver game consisting of two states of the world, two signals, and two responses, where players are restricted to using one of the previous four strategies.
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^ While it is true that every noncooperative game in which players may use mixed strategies has a Nash equilibrium, some have questioned the significance of this for real agents.
  • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

) .The existence of such strategies, for cleverly designed games, has important consequences in descriptive set theory.^ Keywords: experiments, game theory, mixed strategies.
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^ However, the philosopher who wants game theory to serve as a descriptive and/or normative theory of strategic rationality cannot rest content with this answer.
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^ Keywords: experiments, game theory, mixed strategies, posted offer auctions, markets, risk aversion, capacity constraints, cost asymmetries.
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Discrete and continuous games

.Much of game theory is concerned with finite, discrete games, that have a finite number of players, moves, events, outcomes, etc.^ Outcome : an assignment of a set of payoffs, one to each player in the game.
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^ This brings us right up to the moving frontier of experimental / behavioral applications of classical game theory.
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^ Evolutionary game theory has been used to explain a number of aspects of human behavior.
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Many concepts can be extended, however. .Continuous games allow players to choose a strategy from a continuous strategy set.^ Figure 11 illustrates the state space under the continuous replicator dynamics for the sender-receiver game consisting of two states of the world, two signals, and two responses, where players are restricted to using one of the previous four strategies.
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^ Outcome : an assignment of a set of payoffs, one to each player in the game.
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^ That is, a player can find a set of systems of beliefs for the other players such that any history of the game along an equilibrium path is consistent with that set of systems.
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.For instance, Cournot competition is typically modeled with players' strategies being any non-negative quantities, including fractional quantities.^ How should we interpret the processes being modeled by computations of NE strategy mixes in games like the river-crossing one?
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^ In this figure, the cake is of size C =10 but all strategies between 0 and 10 inclusive are permitted for either player (including fractional demands).
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^ It is natural, as a first approximation, to think of sequential-move games as being ones in which players choose their strategies one after the other, and of simultaneous-move games as ones in which players choose their strategies at the same time.
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Differential games such as the continuous pursuit and evasion game are continuous games.

One-player and many-player games

.Individual decision problems are sometimes considered "one-player games". While these situations are not game theoretical, they are modeled using many of the same tools within the discipline of decision theory.^ Keywords: experiments, game theory, entry game, individual decisions.
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^ Keywords: experiments, game theory, decisions.
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^ It is also one of the landmarks of "cooperative" game theory.
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.It is only with two or more players that a problem becomes game theoretical.^ This game involves two players.
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^ Each player in a game faces a choice among two or more possible strategies .
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^ Now the two leaders can do nothing but watch in dismay as the world is blown up—due to a game-theoretic mistake.
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.A randomly acting player who makes "chance moves", also known as "moves by nature", is often added (Osborne & Rubinstein 1994).^ In some games, players can improve their outcomes by taking actions that makes it impossible for them to take what would be her best actions in the corresponding simultaneous-move games.
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^ It is natural, as a first approximation, to think of sequential-move games as being ones in which players choose their strategies one after the other, and of simultaneous-move games as ones in which players choose their strategies at the same time.
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^ They are cases where the signaling system would not work in a population of players who are pairwise randomly assigned to play the sender-receiver game.
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.This player is not typically considered a third player in what is otherwise a two-player game, but merely serves to provide a roll of the dice where required by the game.^ This game involves two players.
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^ We can solve this new game if we make certain assumptions about the two players' utility functions.
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^ In this game, individuals choose one of two strategies, typically called “Cooperate” and “Defect.” Here is the general form of the payoff matrix for the prisoner's dilemma: .
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.Games with an arbitrary, but finite number of players are often called n-person games (Luce & Raiffa 1957).^ Abstract: Subjects play several finite prisoner's dilemma games simultaneously, which allows rejection of the common notion that players can be categorized into distinct "types."
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^ Abstract: Each subject in the game chooses a number, with a prize going to the person whose choice is closest to two-thirds of the average of all chosen numbers.
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^ (Chess can be turned into a simultaneous-move game if the players each call moves while isolated from the common board; but this is a very different game from conventional chess.
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Metagames

.These are games the play of which is the development of the rules for another game, the target or subject game.^ If at least one node shares its information set with another, while others are alone, the game involves both simultaneous and sequential play, and so is still a game of imperfect information.
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^ For some games, these adjustment paths lead to equilibria that are ruled out by the intuitive criterion and other refinements.
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^ Abstract: Subjects play several finite prisoner's dilemma games simultaneously, which allows rejection of the common notion that players can be categorized into distinct "types."
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

Metagames seek to maximize the utility value of the rule set developed. .The theory of metagames is related to mechanism design theory.^ The 2007 Nobel Prize for "foundations of mechanism design theory " was awarded to Leonid Hurwicz , Eric S. Maskin , and Roger B. Myerson .
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^ Other market design links (researchers, 2007 Nobel Prize for "foundations of mechanism design theory", other designed markets, class materials, presentations, software, conferences...
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^ This is natural: just as chemical engineering is related to but different from chemistry, we can expect "microeconomic engineering" and economic design to be different from but related to game theory.
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.The term metagame analysis is also used to refer to a practical approach developed by Nigel Howard (Howard 1971) whereby a situation is framed as a strategic game in which stakeholders try to realise their objectives by means of the options available to them.^ We mean merely that the action was taken when an alternative action was available, in some sense of ‘available’ normally established by the context of the particular analysis.
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^ I tried to spark the use of the term "design economics" in my 2002 manifesto , which might have more naturally included all the things that economists can help design (e.g.
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^ Matrix games are referred to as ‘normal-form’ or ‘strategic-form’ games, and games as trees are referred to as ‘extensive-form’ games.
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Subsequent developments have led to the formulation of drama theory.

History

.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.^ Keywords: experiments, game theory, mixed strategies.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ Keywords: experiments, game theory, minimax theory.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ Keywords: experiments, game theory, entry games, coordination, mixed strategies.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

.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.^ Now the two leaders can do nothing but watch in dismay as the world is blown up—due to a game-theoretic mistake.
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^ However, two agents in one game, or one agent under different sorts of circumstances, may display very different attitudes to risk.
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^ A useful high-level principle for sorting the literature indexes it to the different auxiliary assumptions with which game-theoretic axioms are applied.
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[11][12]
.It was not until the publication of Antoine Augustin Cournot's Recherches sur les principes mathématiques de la théorie des richesses (Researches into the Mathematical Principles of the Theory of Wealth) in 1838 that a general game theoretic analysis was pursued.^ Keywords: experiments, game theory, altruism, inter-generational transfers.
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^ On the Applicability of Game Theory to Evolution,” Journal of Theoretical Biology , 75 (1): 245–247.
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^ Evolution and the Theory of Games” Journal of Theoretical Biology , 1: 382–403.
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.In this work Cournot considers a duopoly and presents a solution that is a restricted version of the Nash equilibrium.^ The concept of a Nash equilibrium (see the entry on game theory ) has been the most used solution concept in game theory since its introduction by John Nash in 1950.
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^ Yet a difficulty arises with the use of Nash equilibrium as a solution concept for games: if we restrict players to using pure strategies, not every game has a Nash equilbrium.
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^ Unfortunately, so many refinements of the notion of a Nash equilibrium have been developed that, in many games which have multiple Nash equilibria, each equilibrium could be justified by some refinement present in the literature.
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.Although Cournot's analysis is more general than Waldegrave's, game theory did not really exist as a unique field until John von Neumann published a series of papers in 1928. While the French mathematician Émile Borel did some earlier work on games, Von Neumann can rightfully be credited as the inventor of game theory.^ The following paper grew out of our work on game theory: .
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^ The mathematical theory of games was invented by John von Neumann and Oskar Morgenstern ( 1944 ).
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^ In the preface to Evolution and the Theory of Games , Maynard Smith notes that “[p]aradoxically, 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.” It is perhaps doubly paradoxical, then, that the subsequent development of evolutionary game theory has produced a theory which holds great promise for social scientists, and is as readily applied to the field of economic behaviour as that for which it was originally designed.
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.Von Neumann was a brilliant mathematician whose work was far-reaching from set theory to his calculations that were key to development of both the Atom and Hydrogen bombs and finally to his work developing computers.^ A crucial aspect of von Neumann & Morgenstern's (1947) work was the solution to this problem.
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^ At the end of the first chapter of Theory of Games and Economic Behavior , von Neumann and Morgenstern write: .
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^ Prior to the work of von Neumann & Morgenstern (1947) , situations of this sort were inherently baffling to analysts.
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.Von Neumann's work in game theory culminated in the 1944 book Theory of Games and Economic Behavior by von Neumann and Oskar Morgenstern.^ At the end of the first chapter of Theory of Games and Economic Behavior , von Neumann and Morgenstern write: .
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^ The mathematical theory of games was invented by John von Neumann and Oskar Morgenstern ( 1944 ).
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^ Keywords: experiments, game theory, decisions, psychology, behavioral economics.
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.This profound work contains the method for finding mutually consistent solutions for two-person zero-sum games.^ That is, a player can find a set of systems of beliefs for the other players such that any history of the game along an equilibrium path is consistent with that set of systems.
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^ Abstract: This paper reports a series of two-person coordination game experiments in which the outcomes are generally well organized by the notion of risk dominance.
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^ In two-person matrix games with three strategies, the data patterns do not correspond to either Nash or maximin play.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

.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.^ Keywords: experiments, game theory, cooperation.
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^ Game theory and individual motivation .
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^ Keywords: experiments, game theory, cooperative games.
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.In 1950, the first discussion of the prisoner's dilemma appeared, and an experiment was undertaken on this game at the RAND corporation.^ Keywords: experiments, game theory, prisoner's dilemma, incomplete information.
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^ Keywords: experiments, game theory, prisoner's dilemma, number of strategies.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ Keywords: experiments, game theory, prisoner's dilemma, probability of continuation.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

.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.^ The mathematical theory of games was invented by John von Neumann and Oskar Morgenstern ( 1944 ).
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^ A strategy for a player, in this game, consists of an amount of cake that he would like.
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^ Abstract: An unprofitable game is one in which maximin strategies do not constitute a Nash equilibrium, yet they guarantee the same payoff as players earn in a Nash equilibrium.
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.This equilibrium is sufficiently general to allow for the analysis of non-cooperative games in addition to cooperative ones.^ It is also one of the landmarks of "cooperative" game theory.
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^ One strategy is ‘better’ than another if it is likely to leave more copies of itself in the next generation, when the game will be played again.
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^ We can specify one class of games in which NE is always not only necessary but sufficient as a solution concept.
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.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.^ Keywords: experiments, game theory, prisoner's dilemma, repeated play.
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^ Keywords: experiments, game theory, extensive form, presentation effects.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

^ Keywords: experiments, game theory, reciprocity, trust, extensive form game.
  • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

.In addition, the first applications of Game theory to philosophy and political science occurred during this time.^ Game theory in computer science .
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^ Rebecca Morton at NYU applies game theory to political science.
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^ Keywords: experiments, game theory, public, voting, political science.
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.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).^ So which refinement is more appropriate as a solution concept?
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^ Thus even behaviorists who aren't impressed with the project of refinements might make use of the concept of subgame-perfect equilibrium (SPE), as discussed in Section 2.6 , if they think they're dealing with agents who are very well informed (say, because they're in a familiar institutional setting).
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^ Here, we will restrict our attention to the least refined equilibrium-in-belief concept, that obtained when we require players to reason in accordance with Bayes's rule.
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.In 1967, John Harsanyi developed the concepts of complete information and Bayesian games.^ We need a ‘cousin’ concept to SPE that we can apply in cases of imperfect information, and we need a new solution procedure to replace Zermelo's algorithm for such games.
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^ The concept of a Nash equilibrium (see the entry on game theory ) has been the most used solution concept in game theory since its introduction by John Nash in 1950.
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.Nash, Selten and Harsanyi became Economics Nobel Laureates in 1994 for their contributions to economic game theory.^ Smith, Vernon L. (1992) Game Theory and Experimental Economics: Beginnings and Early Influences, History of Political Economy , 24:Special Issue 241-282.
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^ Frdrick Asselin maintains a list of researchers in Game Theory or Economics and Computer Science .
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^ For Paul Walker's outline History of game theory see history of game theory (updated to reflect the 2005 Nobel award to Bob Aumann and Tom Schelling ).
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.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 the preface to Evolution and the Theory of Games , Maynard Smith notes that “[p]aradoxically, 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.” It is perhaps doubly paradoxical, then, that the subsequent development of evolutionary game theory has produced a theory which holds great promise for social scientists, and is as readily applied to the field of economic behaviour as that for which it was originally designed.
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^ The former provides a setwise generalization of the concept of an evolutionarily stable strategy, and the latter extends the concept of an evolutionarily stable strategy to the context of two-player extensive form games.
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^ The first approach derives from the work of Maynard Smith and Price and employs the concept of an evolutionarily stable strategy as the principal tool of analysis.
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In addition, the concepts of correlated equilibrium, trembling hand perfection, and common knowledge[13] were introduced and analyzed.
.In 2005, game theorists Thomas Schelling and Robert Aumann followed Nash, Selten and Harsanyi as Nobel Laureates.^ Robert Aumann and Thomas Schelling shared the Nobel Memorial Prize in Economic Science in 2005 .
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^ For Paul Walker's outline History of game theory see history of game theory (updated to reflect the 2005 Nobel award to Bob Aumann and Tom Schelling ).
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^ But now consider the three-player imperfect-information game below known as ‘Selten's horse’ (for its inventor, Nobel Laureate Reinhard Selten, and because of the shape of its tree; taken from Kreps (1990) , p.
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.Schelling worked on dynamic models, early examples of evolutionary game theory.^ The lack of a dynamical theory in the traditional theory of games .
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^ Schelling worked on dynamic models, early examples of evolutionary game theory.
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^ Evolution, evolutionary game theory, learning, and dynamics .
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.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.^ Equilibrium selection issues are often more fruitfully addressed in the context of extensive-form games.
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^ The news for the fans of justice gets more cheerful still if we introduce correlated play .
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^ Furthermore, experiments with a class of games known as a “beauty pageant” show, quite dramatically, the failure of common knowledge assumptions typically invoked to solve games.
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In 2007, Roger Myerson, together with Leonid Hurwicz and Eric Maskin, 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).

See also

Notes

  1. ^ Ross, Don. "Game Theory". The Stanford Encyclopedia of Philosophy (Spring 2008 Edition). Edward N. Zalta (ed.). http://plato.stanford.edu/archives/spr2008/entries/game-theory/. Retrieved 2008-08-21. 
  2. ^ Experimental work in game theory goes by many names, experimental economics, behavioral economics, and behavioural game theory are several. For a recent discussion on this field see Camerer (2003).
  3. ^ Evolutionary Game Theory (Stanford Encyclopedia of Philosophy)
  4. ^ a b Biological Altruism (Stanford Encyclopedia of Philosophy)
  5. ^ Algorithmic Game Theory. http://www.cambridge.org/journals/nisan/downloads/Nisan_Non-printable.pdf. 
  6. ^ E. Ullmann Margalit, The Emergence of Norms, Oxford University Press, 1977. C. Bicchieri, The Grammar of Society: the Nature and Dynamics of Social Norms, Cambridge University Press, 2006.
  7. ^ "Self-Refuting Theories of Strategic Interaction: A Paradox of Common Knowledge ", Erkenntnis 30, 1989: 69-85. See also Rationality and Coordination, Cambridge University Press, 1993.
  8. ^ The Dynamics of Rational Deliberation, Harvard University Press, 1990.
  9. ^ "Knowledge, Belief, and Counterfactual Reasoning in Games." In Cristina Bicchieri, Richard Jeffrey, and Brian Skyrms, eds., The Logic of Strategy. New York: Oxford University Press, 1999.
  10. ^ For a more detailed discussion of the use of Game Theory in ethics see the Stanford Encyclopedia of Philosophy's entry game theory and ethics.
  11. ^ James Madison, Vices of the Political System of the United States, April, 1787. Link
  12. ^ Jack Rakove, "James Madison and the Constitution", History Now, Issue 13 September 2007. Link
  13. ^ Although common knowledge was first discussed by the philosopher David Lewis in his dissertation (and later book) Convention in the late 1960s, it was not widely considered by economists until Robert Aumann's work in the 1970s.

References

Textbooks and general references

.
  • Aumann, Robert J. (1987), "game theory,", The New Palgrave: A Dictionary of Economics, 2, pp. 460–82 .
  • (2008).^ Smith, Vernon L. (1992) Game Theory and Experimental Economics: Beginnings and Early Influences, History of Political Economy , 24:Special Issue 241-282.
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    ^ Handbook of Game Theory with Economic Applications (Volume 2), Amsterdam: Elsevier Science, pp.
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    ^ Frdrick Asselin maintains a list of researchers in Game Theory or Economics and Computer Science .
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    The New Palgrave Dictionary of Economics, 2nd Edition:
"game theory" by Robert J. Aumann, Abstract.
"game theory in economics, origins of," by Robert Leonard. Abstract.
"behavioural economics and game theory" by Faruk Gul. Abstract.
.
  • Dutta, Prajit K. (1999), Strategies and games: theory and practice, MIT Press, ISBN 978-0-262-04169-0 .^ Keywords: experiments, game theory, mixed strategies.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    ^ Evolutionary Game Theory , Cambridge, MA: The MIT Press.
    • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

    ^ Keywords: experiments, game theory, mixed strategies, posted offer auctions, markets, risk aversion, capacity constraints, cost asymmetries.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    .Suitable for undergraduate and business students.
  • Fernandez, L F.; Bierman, H S. (1998), Game theory with economic applications, Addison-Wesley, ISBN 978-0-201-84758-1 .^ Smith, Vernon L. (1992) Game Theory and Experimental Economics: Beginnings and Early Influences, History of Political Economy , 24:Special Issue 241-282.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    ^ Frdrick Asselin maintains a list of researchers in Game Theory or Economics and Computer Science .
    • Al Roth's game theory, experimental economics, and market design page 8 January 2010 4:14 UTC kuznets.fas.harvard.edu [Source type: FILTERED WITH BAYES]

    ^ Hewlitt Packard has a Game theory and experimental economics research group and a Social Computing Lab .
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    .Suitable for upper-level undergraduates.
  • Fudenberg, Drew; Tirole, Jean (1991), Game theory, MIT Press, ISBN 978-0-262-06141-4 .^ Evolution and the Theory of Games , Cambridge: Cambridge University Press.
    • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

    ^ Friedman, James W. (1986) Game Theory with Applications to Economics, Oxford, U. K.: Oxford University Press.
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    ^ Email Contact: dan@cats.ucsc.edu Coleman, Andrew (1983) Game Theory and Experimental Work, London: Pergamon Press.
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    .Acclaimed reference text, public description.
  • Gibbons, Robert D. (1992), Game theory for applied economists, Princeton University Press, ISBN 978-0-691-00395-5 .^ Keywords: experiments, game theory, public.
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    ^ Princeton: Princeton University Press, 2nd edition.
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    ^ Email Contact: vcrawfor@ucsd.edu Dale, Donald J., John Morgan, and Robert W. Rosenthal (1999) Coordination Through Reputations: A Laboratory Experiment, Boston University, Keywords: experiments, game theory, coordination, history, information, reputations.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    Suitable for advanced undergraduates.
  • Published in Europe as Robert Gibbons (2001), A Primer in Game Theory, London: Harvester Wheatsheaf, ISBN 978-0-7450-1159-2 .
.
  • Gintis, Herbert (2000), Game theory evolving: a problem-centered introduction to modeling strategic behavior, Princeton University Press, ISBN 978-0-691-00943-8 
  • Green, Jerry R.; Mas-Colell, Andreu; Whinston, Michael D. (1995), Microeconomic theory, Oxford University Press, ISBN 978-0-19-507340-9 .^ Princeton: Princeton University Press, 2nd edition.
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    ^ However, the philosopher who wants game theory to serve as a descriptive and/or normative theory of strategic rationality cannot rest content with this answer.
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    ^ Second, the rationality assumptions underlying evolutionary game theory are, in many cases, more appropriate for the modelling of social systems than those assumptions underlying the traditional theory of games.
    • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

    .Presents game theory in formal way suitable for graduate level.
  • edited by Vincent F. Hendricks, Pelle G. Hansen.^ Yaw Nyarko is at NYU. Barry O'Neill , at UCLA uses game theory in innovative ways in political science.
    • Al Roth's game theory, experimental economics, and market design page 8 January 2010 4:14 UTC kuznets.fas.harvard.edu [Source type: FILTERED WITH BAYES]

    ^ Keywords: experiments, game theory, extensive form, presentation effects.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    ^ We have now seen the first level at which neuroeconomics applies game theory.
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    .(2007), Hansen, Pelle G.; Hendricks, Vincent F., eds., Game Theory: 5 Questions, New York, London: Automatic Press / VIP, ISBN 9788799101344 .^ Evolutionary Game Theory , Cambridge, MA: The MIT Press.
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    ^ This is a highly portentous new source of leverage for the empirical application of game theory.
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    ^ New York: New York University Press.
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    Snippets from interviews.
  • Howard, Nigel (1971). .Paradoxes of Rationality: Games, Metagames, and Political Behavior.^ Both players must use backward induction to solve the game; backward induction requires that I know that II knows that I is rational; but II can solve the game only by using a backward induction argument that takes as a premise the irrationality of I. This is the paradox of backward induction .
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    ^ Furthermore, applications of game theory to behavioral topics extend well beyond the political arena.
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    ^ Keywords: experiments, game theory, behavioral game theory, biases, bounded rationality, psychology.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    .Cambridge, Massachusetts: The MIT Press.^ Evolutionary Game Theory , Cambridge, MA: The MIT Press.
    • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

    ^ (Series: Economic Learning and Social Evolution), Cambridge, Massachusetts: MIT Press.
    • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

    ^ Cambridge, MA: MIT Press.
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    .ISBN 978-0262582377. 
  • Isaacs, Rufus (1999), Differential Games: A Mathematical Theory With Applications to Warfare and Pursuit, Control and Optimization, New York: Dover Publications, ISBN 978-0-486-40682-4 
  • Leyton-Brown, Kevin; Shoham, Yoav (2008), Essentials of Game Theory: A Concise, Multidisciplinary Introduction, San Rafael, CA: Morgan & Claypool Publishers, ISBN 978-1-598-29593-1, http://www.gtessentials.org .^ Keywords: experiments, game theory, public.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    ^ First published Mon Jan 14, 2002; substantive revision Sun Jul 19, 2009 Evolutionary game theory originated as an application of the mathematical theory of games to biological contexts, arising from the realization that frequency dependent fitness introduces a strategic aspect to evolution.
    • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

    ^ Binmore (2005b) ; and any economics or game theory text that lets the mathematics do the talking and doesn't insist on ‘spinning’ it in one idealogical direction or another.
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    .An 88-page mathematical introduction; free online at many universities.
  • Miller, James H. (2003), Game theory at work: how to use game theory to outthink and outmaneuver your competition, New York: McGraw-Hill, ISBN 978-0-07-140020-6 .^ There's an edited volume on Algorithmic Game Theory (with a nice foreword by Papadimitriou) [the preface to the book reveals that the full text is online at http://www.cambridge.org/journals/nisan/default.asp , using username: agt1user and password: camb2agt] Chandra Chekuri at the University of Illinois has a nice set of course notes and links for Algorithmic Game Theory David Dowe's chess and games page includes many links to work on game playing programs.
    • Al Roth's game theory, experimental economics, and market design page 8 January 2010 4:14 UTC kuznets.fas.harvard.edu [Source type: FILTERED WITH BAYES]

    ^ (This seems to have been carried out quite independently of the experimental work on market entry using similar games, but looks worth careful study.
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    ^ Second, the rationality assumptions underlying evolutionary game theory are, in many cases, more appropriate for the modelling of social systems than those assumptions underlying the traditional theory of games.
    • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

    .Suitable for a general audience.
  • Osborne, Martin J. (2004), An introduction to game theory, Oxford University Press, ISBN 978-0-19-512895-6 .^ Keywords: experiments, game theory, altruism, inter-generational transfers.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    ^ Martin Osborne at the University of Toronto, has, among other interesting items, corrections to his two textbooks coauthored with Ariel Rubinstein, and to his undergraduate textbook.
    • Al Roth's game theory, experimental economics, and market design page 8 January 2010 4:14 UTC kuznets.fas.harvard.edu [Source type: FILTERED WITH BAYES]

    ^ Email Contact: dan@cats.ucsc.edu Coleman, Andrew (1983) Game Theory and Experimental Work, London: Pergamon Press.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    .Undergraduate textbook.
  • Osborne, Martin J.; Rubinstein, Ariel (1994), A course in game theory, MIT Press, ISBN 978-0-262-65040-3 .^ Evolution and the Theory of Games , Cambridge: Cambridge University Press.
    • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

    ^ Friedman, James W. (1986) Game Theory with Applications to Economics, Oxford, U. K.: Oxford University Press.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    ^ Martin Osborne at the University of Toronto, has, among other interesting items, corrections to his two textbooks coauthored with Ariel Rubinstein, and to his undergraduate textbook.
    • Al Roth's game theory, experimental economics, and market design page 8 January 2010 4:14 UTC kuznets.fas.harvard.edu [Source type: FILTERED WITH BAYES]

    .A modern introduction at the graduate level.
  • Poundstone, William (1992), Prisoner's Dilemma: John von Neumann, Game Theory and the Puzzle of the Bomb, Anchor, ISBN 978-0-385-41580-4 .^ Keywords: experiments, game theory, prisoner's dilemma, incomplete information.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    ^ Keywords: experiments, game theory, prisoner's dilemma, number of strategies.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    ^ Keywords: experiments, game theory, prisoner's dilemma, repeated play.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    .A general history of game theory and game theoreticians.
  • Rasmusen, Eric (2006), Games and Information: An Introduction to Game Theory (4th ed.^ Keywords: experiments, game theory, information.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    ^ Keywords: experiments, game theory, history.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    ^ Keywords: experiments, game theory, altruism, inter-generational transfers.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    ), .Wiley-Blackwell, ISBN 978-1-4051-3666-2, http://www.rasmusen.org/GI/index.html 

Historically important texts

.
  • Cournot, A. Augustin (1838), "Recherches sur les principles mathematiques de la théorie des richesses", Libraire des sciences politiques et sociales (Paris: M. Rivière & C.ie) 
  • Farquharson, Robin (1969).^ Princeton: Princeton University Press, 2nd edition.
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    ^ Email Contact: dan@cats.ucsc.edu Coleman, Andrew (1983) Game Theory and Experimental Work, London: Pergamon Press.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    ^ This bibliography began life as the bibliography of (my) Chapter 4 of The Handbook of Experimental Economics , John H. Kagel and Alvin E. Roth, editors, Princeton University Press, 1995.
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    Theory of Voting. Blackwell (Yale U.P. in the U.S.). ISBN 0631124608. 
  • reprinted edition: R. Duncan Luce ; Howard Raiffa (1989), Games and decisions: introduction and critical survey, New York: Dover Publications, ISBN 978-0-486-65943-5 
.
  • Shapley, L.S. (1953), A Value for n-person Games, In: Contributions to the Theory of Games volume II, H.W. Kuhn and A.W. Tucker (eds.^ Email Contact: vcrawfor@ucsd.edu Dale, Donald J., John Morgan, and Robert W. Rosenthal (1999) Coordination Through Reputations: A Laboratory Experiment, Boston University, Keywords: experiments, game theory, coordination, history, information, reputations.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    ^ The Economy as an Evolving Complex System II (SFI Studies in the Sciences of Complexity: Volume 27), Reading, MA: Addison-Wesley, , pp.
    • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

    ^ At this point, one may see little difference between the two approaches to evolutionary game theory.
    • Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) 8 January 2010 4:14 UTC plato.stanford.edu [Source type: Academic]

    )
  • .
  • Shapley, L.S. (1953), Stochastic Games, Proceedings of National Academy of Science Vol.^ Email Contact: andreoni@facstaff.wisc.edu Andreoni, James, and Hal Varian (1999) Pre-Play Contracting in the Prisoner's Dilemma, Proceedings of the National Academy of Sciences , 96(September), 10933-10938.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    ^ Email Contact: schotter@fasecon.econ.nyu.edu O'Neil, Barry (1987) **Nonmetric Test of the Minimax Theory of Two-Person Zerosum Games, Proceedings of the National Academy of Sciences , 842106-2109.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    ^ McCabe, Kevin A., Stephen J. Rassenti, and Vernon L. Smith (1996) Game Theory and Reciprocity in Some Extensive Form Experimental Games, Proceedings of the National Academy of Sciences , 9313421-13428.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    39, pp. 1095–1100.
  • Zermelo, Ernst (1913), "Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels", Proceedings of the Fifth International Congress of Mathematicians 2: 501–4 

Other print references

.
  • Ben David, S.; Borodin, Allan; Karp, Richard; Tardos, G.; Wigderson, A. (1994), "On the Power of Randomization in On-line Algorithms" (PDF), Algorithmica 11 (1): 2–14, doi:10.1007/BF01294260, http://www.math.ias.edu/~avi/PUBLICATIONS/MYPAPERS/BORODIN/paper.pdf 
  • Bicchieri, Cristina (1993, 2nd.^ Email Contact: rdm@hss.caltech.edu McKelvey, Richard, and John Patty (1999) Quantal Response Voting, California Institute of Technology, Discussion Paper, presented at the Spring 1999 Public Choice Meetings.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    ^ Here's a radio interview about market design , 11/14/07, from public radio station WOSU in Columbus Ohio (about 1 hour).
    • Al Roth's game theory, experimental economics, and market design page 8 January 2010 4:14 UTC kuznets.fas.harvard.edu [Source type: FILTERED WITH BAYES]

    ^ Budescu, David V.*, and Amnon Rapoport (1994) Subjective Randomization in One- and Two-Person Games, Journal of Behavioral Decision Making , 7261-278.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    edition, .1997), Rationality and Coordination, Cambridge University Press, ISBN 0-521-57444-7 
  • Camerer, Colin (2003), Behavioral game theory: experiments in strategic interaction, Russesll Sage Foundation, ISBN 978-0-691-09039-9 
  • Downs, Anthony (1957), An Economic theory of Democracy, New York: Harper 
  • Gauthier, David (1986), Morals by agreement, Oxford University Press, ISBN 978-0-19-824992-4 
  • Allan Gibbard, "Manipulation of voting schemes: a general result", Econometrica, Vol.^ Keywords: experiments, game theory, altruism, inter-generational transfers.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    ^ Keywords: experiments, game theory, voting, quantal response.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    ^ Keywords: experiments, game theory, coordination, history effects.
    • Y2K Bibliography of Experimental Economics and Social Science: Game Theory Experiments 8 January 2010 4:14 UTC people.virginia.edu [Source type: Academic]

    41, No. 4 (1973), pp. .587–601.
  • Grim, Patrick; Kokalis, Trina; Alai-Tafti, Ali; Kilb, Nicholas; St Denis, Paul (2004), "Making meaning happen", Journal of Experimental & Theoretical Artificial Intelligence 16 (4): 209–243, doi:10.1080/09528130412331294715 
  • Harper, David; Maynard Smith, John (2003), Animal signals, Oxford University Press, ISBN 978-0-19-852685-8 
  • Harsanyi, John C. (1974), "An equilibrium point interpretation of stable sets", Management Science 20: 1472–1495, doi:10.1287/mnsc.20.11.1472 
  • Levy, Gilat; Razin, Ronny (2003), "It Takes Two: An Explanation of the Democratic Peace", Working Paper, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=433844 
  • Lewis, David (1969), Convention: A Philosophical Study , ISBN 978-0-631-23257-5 (2002 edition)
  • McDonald, John (1950 - 1996), Strategy in Poker, Business & War, W. W. Norton, ISBN 0-393-31457-X .^ Tatsuyoshi Saijo at Osaka University makes some of his papers available, in English and Japanese.
    • Al Roth's game theory, experimental economics, and market design page 8 January 2010 4:14 UTC kuznets.fas.harvard.edu [Source type: FILTERED WITH BAYES]

    ^ For example, if two competing businesses are both planning marketing campaigns, one might commit to its strategy months before the other does; but if neither knows what the other has committed to or will commit to when they make their decisions, this is a simultaneous-move game.
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    ^ The simplest games (from the perspective of logical structure) are those in which agents have perfect information , meaning that at every point where each agent's strategy tells her to take an action, she knows everything that has happened in the game up to that point.
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    A layman's introduction.
  • Quine, W.v.O (1967), "Truth by Convention", Philosophica Essays for A.N. Whitehead, Russel and Russel Publishers, ISBN 978-0-8462-0970-6 
  • Quine, W.v.O (1960), "Carnap and Logical Truth", Synthese 12 (4): 350–374, doi:10.1007/BF00485423 
  • Mark A. Satterthwaite, "Strategy-proofness and Arrow's Conditions: Existence and Correspondence Theorems for Voting Procedures and Social Welfare Functions", Journal of Economic Theory 10 (April 1975), 187–217.
  • Siegfried, Tom (2006), A Beautiful Math, Joseph Henry Press, ISBN 0-309-10192-1 
  • Skyrms, Brian (1990), The Dynamics of Rational Deliberation, Harvard University Press, ISBN 0-674-21885-X 
  • Skyrms, Brian (1996), Evolution of the social contract, Cambridge University Press, ISBN 978-0-521-55583-8 
  • Skyrms, Brian (2004), The stag hunt and the evolution of social structure, Cambridge University Press, ISBN 978-0-521-53392-8 
  • Sober, Elliott; Wilson, David Alec (1999), Unto others: the evolution and psychology of unselfish behavior, Harvard University Press, ISBN 978-0-674-93047-6 
  • Thrall, Robert M.; Lucas, William F. (1963), "n-person games in partition function form", Naval Research Logistics Quarterly 10 (4): 281–298, doi:10.1002/nav.3800100126 

Websites

.

Simple English

Game theory uses maths to study strategy. Game theory studies more than just board games, sports, and games of luck. It also studies things like business and military decisions. In game theory, people call all of these situations "games." In other words, you can use game theory to study any situation where more than one person makes choices.

The players in a game are not even always people. Players can be people, companies, armies, or other things. Each player wants something: maybe a company wants to make as much money as it can, or a country wants to win a war. Sometimes the players work together, but often they are competing against each other.

Game theory is part of economics.

Prisoner's Dilemma

One important game is the prisoner's dilemma. It's an imaginary situation that shows why sometimes people do not cooperate (help each other).

Imagine this situation: the police catch two criminals after they committed a crime. The police do not know which person committed the crime and which person just helped. They question the two in separate cells. Each prisoner can either stay silent or betray (hurt) the other by blaming the crime on them. If both stay silent, they only go to jail for 6 months. If one betrays and the other stays silent, the one that stays silent goes to jail for 10 years and the other one does not go to jail at all. If they both betray each other, they each go to jail for 2 years. No matter what happens, the prisoners will never see each other again.

If you are a prisoner in this situation and you only care about yourself, the way to get the smallest sentence is to betray the other prisoner. No matter what, you get a shorter sentence when you betray than when you do not. If the other prisoner stays silent and does not betray, then betraying means you do not go to jail at all instead of going to jail for 6 months. If the other prisoner betrays, then betraying lets you go to jail for 2 years instead of 10 years. In other words, it's always best for you to betray, even though the two of you would be better off if you both stayed silent. It is said that betraying the other prisoner is your "dominant strategy" because it is always the best thing for you to do, no matter what the other prisoner does.

The prisoner's dilemma is like a lot of other situations in the real world. For example, if two countries are trying to decide whether to make new weapons, they are both better off if neither country does. But sometimes the countries are in the same situation as the prisoners: each country only cares about itself, and it's better off if it "betrays" the other country by making weapons.

The prisoner's dilemma does not have same result if some of the details are different. If the prisoners (or countries) can talk with each other and plan for the future, they might both decide to cooperate (not betray) because they hope that will make the other country help them in the future. In game theory, this is called a "repeated game." If the players are altruistic (if they care about each other), they might be OK with going to jail so they can help the other person.


Citable sentences

Up to date as of December 14, 2010

Here are sentences from other pages on Game theory, which are similar to those in the above article.








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