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{{short description|The study of mathematical models of strategic interaction between rational decision-makers}}{{about|the mathematical study of optimizing agents|the mathematical study of sequential games|Combinatorial game theory|the study of playing games for entertainment|Game studies|other uses}}{{Use American English|date=July 2018}}{{Economics sidebar}}Game theory is the study of mathematical models of strategic interaction between rational decision-makers.Myerson, Roger B. (1991). Game Theory: Analysis of Conflict, Harvard University Press, p. 1. Chapter-preview links, pp. viiâ€“xi. It has applications in all fields of social science, as well as in logic and computer science. Originally, it addressed zero-sum games, in which each participant's gains or losses are exactly balanced by those of the other participants. Today, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers.Modern game theory began with the idea regarding the existence of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by the 1944 book Theory of Games and Economic Behavior, co-written with Oskar Morgenstern, which considered cooperative games of several players. The second edition of this book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty.Game theory was developed extensively in the 1950s by many scholars. It was later explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields. {{As of|2014}}, with the Nobel Memorial Prize in Economic Sciences going to game theorist Jean Tirole, eleven game theorists have won the economics Nobel Prize. John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology.

History

File:JohnvonNeumann-LosAlamos.gif|thumb|John von NeumannJohn von NeumannFile:John f nash 20061102 3.jpg|thumb|John Nash ]]Early discussions of examples of two-person games occurred long before the rise of modern, mathematical game theory. The first known discussion of game theory occurred in a letter written by Charles Waldegrave, an active Jacobite, and uncle to James Waldegrave, a British diplomat, in 1713.{{citation |url=http://www.jehps.net/Decembre2007/Bellhouse.pdf |title=The Problem of Waldegrave |author=Bellhouse, David |journal=Journal Ã‰lectronique d'Histoire des ProbabilitÃ©s et de la Statistique |year=2007 |volume=3 |issue=2}} In this letter, Waldegrave provides a minimax mixed strategy solution to a two-person version of the card game le Her, and the problem is now known as Waldegrave problem. In his 1838 Recherches sur les principes mathÃ©matiques de la thÃ©orie des richesses (Researches into the Mathematical Principles of the Theory of Wealth), Antoine Augustin Cournot considered a duopoly and presents a solution that is the Nash equilibrium of the considered game.In 1913, Ernst Zermelo published Ãœber eine Anwendung der Mengenlehre auf die Theorie des Schachspiels (On an Application of Set Theory to the Theory of the Game of Chess). It proved that the optimal chess strategy is strictly determined. This paved the way for more general theorems.{{rp|429}}In 1938, the Danish mathematical economist Frederik Zeuthen proved that the mathematical model had a winning strategy by using Brouwer's fixed point theorem.BOOK, Kim, Sungwook, Game theory applications in network design, 3, IGI Global, 2014,weblink 9781466660519, In his 1938 book Applications aux Jeux de Hasard and earlier notes, Ã‰mile Borel proved a minimax theorem for two-person zero-sum matrix games only when the pay-off matrix was symmetric and provides an example of a non-trivial infinite game (known in English as Blotto game) with a solution. Borel conjectured that non-existence of mixed-strategy equilibria in finite two-person zero-sum games would occur, a conjecture that was proved false by von Neumann.Game theory did not really exist as a unique field until John von Neumann published the paper On the Theory of Games of Strategy in 1928.{{citation |first=J. v. |last=Neumann |year=1928 |title=Zur Theorie der Gesellschaftsspiele |journal=Mathematische Annalen |volume=100 |issue=1 |pages=295â€“320 |doi=10.1007/BF01448847}} English translation: {{citation |chapter=On the Theory of Games of Strategy |editor1-first=A. W. |editor1-last=Tucker |editor2-first=R. D. |editor2-last=Luce |year=1959 |title=Contributions to the Theory of Games |volume=4 |pages=13â€“42 |chapterurl=https://books.google.com/books?id=9lSVFzsTGWsC&pg=PA13}} Von Neumann's original proof used Brouwer's fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by his 1944 book Theory of Games and Economic Behavior co-authored with Oskar Morgenstern.BOOK, Philip, Mirowski, What Were von Neumann and Morgenstern Trying to Accomplish?, E. Roy, Weintraub, Toward a History of Game Theory, Durham, Duke University Press, 1992, 978-0-8223-1253-6, 113â€“147,weblink The second edition of this book provided an axiomatic theory of utility, which reincarnated Daniel Bernoulli's old theory of utility (of the money) as an independent discipline. Von Neumann's work in game theory culminated in this 1944 book. This foundational work contains the method for finding mutually consistent solutions for two-person zero-sum games. During the following 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.{{citation |last=Leonard |first=Robert |title=Von Neumann, Morgenstern, and the Creation of Game Theory |location=New York |publisher=Cambridge University Press |year=2010 |isbn=9780521562669}}In 1950, the first mathematical discussion of the prisoner's dilemma appeared, and an experiment was undertaken by notable mathematicians Merrill M. Flood and Melvin Dresher, as part of the RAND Corporation's investigations into game theory. RAND pursued the studies because of possible applications to global nuclear strategy.Prisoner's Dilemma. Plato.stanford.edu (4 September 1997). Retrieved on 3 January 2013. 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. Nash proved that every finite n-player, non-zero-sum (not just 2-player zero-sum) non-cooperative game has what is now known as a Nash equilibrium in mixed strategies.Game theory experienced a flurry of activity in the 1950s, during which time the concepts of the core, the extensive form game, fictitious play, repeated games, and the Shapley value were developed. In addition, the first applications of game theory to philosophy and political science occurred during this time.In 1979 Robert Axelrod tried setting up computer programs as players and found that in tournaments between them the winner was often a simple "tit-for-tat" program that cooperates on the first step, then on subsequent steps just does whatever its opponent did on the previous step. The same winner was also often obtained by natural selection; a fact widely taken to explain cooperation phenomena in evolutionary biology and the social sciences.BOOK, Wolfram, Stephen, A New Kind of Science, Wolfram Media, Inc., 2002, 1104, 978-1-57955-008-0,

Prize-winning achievements

In 1965, Reinhard Selten introduced his solution concept of subgame perfect equilibria, which further refined the Nash equilibrium (later he would introduce trembling hand perfection as well). In 1994 Nash, Selten and Harsanyi became Economics Nobel Laureates for their contributions to economic game theory.In the 1970s, game theory was extensively applied in biology, largely as a result of the work of John Maynard Smith and his evolutionarily stable strategy. In addition, the concepts of correlated equilibrium, trembling hand perfection, and common knowledgeAlthough 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. were introduced and analyzed.In 2005, game theorists Thomas Schelling and Robert Aumann followed Nash, Selten and Harsanyi as Nobel Laureates. Schelling worked on dynamic models, early examples of evolutionary game theory. Aumann contributed more to the equilibrium school, introducing equilibrium coarsening, correlated equilibrium and developing an extensive formal analysis of the assumption of common knowledge and of its consequences.In 2007, Leonid Hurwicz, together with Eric Maskin and Roger Myerson, was awarded the Nobel Prize in Economics "for having laid the foundations of mechanism design theory". Myerson's contributions include the notion of proper equilibrium, and an important graduate text: Game Theory, Analysis of Conflict. Hurwicz introduced and formalized the concept of incentive compatibility.In 2012, Alvin E. Roth and Lloyd S. Shapley were awarded the Nobel Prize in Economics "for the theory of stable allocations and the practice of market design". In 2014, the Nobel went to game theorist Jean Tirole.

Game types

Cooperative / Non-cooperative

A game is cooperative if the players are able to form binding commitments externally enforced (e.g. through contract law). A game is non-cooperative if players cannot form alliances or if all agreements need to be self-enforcing (e.g. through credible threats).[1]Cooperative games are often analysed through the framework of cooperative game theory, which focuses on predicting which coalitions will form, the joint actions that groups take and the resulting collective payoffs. It is opposed to the traditional non-cooperative game theory which focuses on predicting individual players' actions and payoffs and analyzing Nash equilibria.[2][3]Cooperative game theory provides a high-level approach as it only describes the structure, strategies and payoffs of coalitions, whereas non-cooperative game theory also looks at how bargaining procedures will affect the distribution of payoffs within each coalition. As non-cooperative game theory is more general, cooperative games can be analyzed through the approach of non-cooperative game theory (the converse does not hold) provided that sufficient assumptions are made to encompass all the possible strategies available to players due to the possibility of external enforcement of cooperation. While it would thus be optimal to have all games expressed under a non-cooperative framework, in many instances insufficient information is available to accurately model the formal procedures available to the players during the strategic bargaining process, or the resulting model would be of too high complexity to offer a practical tool in the real world. In such cases, cooperative game theory provides a simplified approach that allows analysis of the game at large without having to make any assumption about bargaining powers.

Symmetric / Asymmetric

{{Payoff matrix |Name=An asymmetric game |2L=E |2R=F |1U=E |UL=1, 2 |UR=0, 0 |1D=F |DL=0, 0 |DR=1, 2}}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 the identities of the players can be changed without changing the payoff to the strategies, then a game is symmetric. Many of the commonly studied 2Ã—2 games are symmetric. The standard representations of chicken, the prisoner's dilemma, and the stag hunt are all symmetric games. Some{{Who|date=July 2012}} scholars would consider certain asymmetric games as examples of these games as well. However, the most common payoffs for each of these games are symmetric.Most commonly studied asymmetric games are games where there are not identical strategy sets for both players. For instance, the ultimatum game and similarly the dictator game have different strategies for each player. 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.

Zero-sum / Non-zero-sum

{{Payoff matrix |Name=A zero-sum game |2L=A |2R=B |1U=A |UL=â€“1, 1 |UR=3, â€“3 |1D=B |DL=0, 0 |DR=â€“2, 2}}Zero-sum games are a special case of constant-sum games, in which choices by players can neither increase nor decrease the available resources. 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).BOOK, Game Theory: Third Edition, Owen, Guillermo, Emerald Group Publishing, 1995, 978-0-12-531151-9, Bingley, 11, Poker exemplifies a zero-sum game (ignoring the possibility of the house's cut), because one wins exactly the amount one's opponents lose. Other zero-sum games include matching pennies and most classical board games including Go and chess.Many games studied by game theorists (including the famed prisoner's dilemma) are non-zero-sum games, because the outcome has net results greater or less than zero. Informally, in non-zero-sum games, a gain by one player does not necessarily correspond with a loss by another.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. It is possible to transform any game into a (possibly asymmetric) zero-sum game by adding a dummy player (often called "the board") whose losses compensate the players' net winnings.

Simultaneous / 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). Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions. This need not be perfect information about every action of earlier players; it might be very little knowledge. For instance, a player may know that an earlier player did not perform one particular action, while s/he does not know which of the other available actions the first player actually performed.The difference between simultaneous and sequential games is captured in the different representations discussed above. Often, normal form is used to represent simultaneous games, while extensive form is used to represent sequential ones. The transformation of extensive to normal form is one way, meaning that multiple extensive form games correspond to the same normal form. Consequently, notions of equilibrium for simultaneous games are insufficient for reasoning about sequential games; see subgame perfection.In short, the differences between sequential and simultaneous games are as follows:{| class="wikitable"! !! Sequential !! Simultaneous
Decision trees >Payoff matrix>Payoff matrices style=line-height:1.3em;padding-right:0.65em| No | No style=line-height:1.3emExtensive-form gameExtensive game}} >style=line-height:1.3em|Strategy gameStrategic game}}

Perfect information and imperfect information

File:PD with outside option.svg|thumb|250px|right|A game of imperfect information (the dotted line represents ignorance on the part of player 2, formally called an information set)]]An important subset of sequential games consists of games of perfect information. A game is one of perfect information if all players know the moves previously made by all other players. Most games studied in game theory are imperfect-information games.{{citation needed|date=October 2017}} Examples of perfect-information games include tic-tac-toe, checkers, infinite chess, and Go.WEB,weblink Game Theory, Thomas S., Ferguson, 56â€“57, UCLA Department of Mathematics, WEB, Complete vs Perfect Infomation in Combinatorial Game Theory, June 24, 2014, Stack Exchange,weblink BOOK, Jan, Mycielski, Jan Mycielski, Games with Perfect Information, Handbook of Game Theory with Economic Applications, Volume 1, 1992, 41â€“70, 10.1016/S1574-0005(05)80006-2, 9780444880987, WEB,weblink Infinite Chess, PBS Infinite Series, March 2, 2017, Perfect information defined at 0:25, with academic sources {{ArXiv|1302.4377}} and {{ArXiv|1510.08155}}.Many card games are games of imperfect information, such as poker and bridge.BOOK, Game Theory: Third Edition, Owen, Guillermo, Emerald Group Publishing, 1995, 978-0-12-531151-9, Bingley, 4, Perfect information is often confused with complete information, which is a similar concept.{{citation needed|date=October 2017}} Complete information requires that every player know the strategies and payoffs available to the other players but not necessarily the actions taken. Games of incomplete information can be reduced, however, to games of imperfect information by introducing "moves by nature".{{sfnp|Leyton-Brown|Shoham|2008|p=60}}

{{vanchor|Combinatorial games}}

Games in which the difficulty of finding an optimal strategy stems from the multiplicity of possible moves are called combinatorial games. Examples include chess and go. Games that involve imperfect information may also have a strong combinatorial character, for instance backgammon. There is no unified theory addressing combinatorial elements in games. There are, however, mathematical tools that can solve particular problems and answer general questions.Games of perfect information have been studied in combinatorial game theory, which has developed novel representations, e.g. surreal numbers, as well as combinatorial and algebraic (and sometimes non-constructive) proof methods to solve games of certain types, including "loopy" games that may result in infinitely long sequences of moves. These methods address games with higher combinatorial complexity than those usually considered in traditional (or "economic") game theory.{{citation |last1=Albert |first1=Michael H. |author1-link=Michael H. Albert |last2=Nowakowski |first2=Richard J. |last3=Wolfe |first3=David |isbn=978-1-56881-277-9 |publisher=A K Peters Ltd |title=Lessons in Play: In Introduction to Combinatorial Game Theory |year=2007 |pages=3â€“4}}{{citation |last=Beck |first=JÃ³zsef |author-link=JÃ³zsef Beck |isbn=978-0-521-46100-9 |publisher=Cambridge University Press |title=Combinatorial games: tic-tac-toe theory |year=2008 |pages=1â€“3}} A typical game that has been solved this way is hex. A related field of study, drawing from computational complexity theory, is game complexity, which is concerned with estimating the computational difficulty of finding optimal strategies.{{citation |author1=Robert A. Hearn |author2=Erik D. Demaine |title=Games, Puzzles, and Computation |year=2009 |publisher=A K Peters, Ltd. |isbn=978-1-56881-322-6}}Research in artificial intelligence has addressed both perfect and imperfect information games that have very complex combinatorial structures (like chess, go, or backgammon) for which no provable optimal strategies have been found. The practical solutions involve computational heuristics, like alphaâ€“beta pruning or use of artificial neural networks trained by reinforcement learning, which make games more tractable in computing practice.{{citation |author=JÃ¶rg Bewersdorff |title=Luck, logic, and white lies: the mathematics of games |year=2005 |publisher=A K Peters, Ltd. |isbn=978-1-56881-210-6 |pages=ixâ€“xii and chapter 31|author-link=JÃ¶rg Bewersdorff }}{{citation |author=M. Tim Jones |title=Artificial Intelligence: A Systems Approach |year=2008 |publisher=Jones & Bartlett Learning |isbn=978-0-7637-7337-3 |pages=106â€“118}}

Infinitely long games

Games, as studied by economists and real-world game players, are generally finished in finitely many moves. Pure mathematicians are not so constrained, and set theorists in particular study games that last for infinitely many moves, with the winner (or other payoff) not known until after all those moves are completed.The focus of attention is usually not so much on the best way to play such a game, but whether one player has a winning strategy. (It can be proven, using the axiom of choice, that there are games{{spaced ndash}}even with perfect information and where the only outcomes are "win" or "lose"{{spaced ndash}}for which neither player has a winning strategy.) The existence of such strategies, for cleverly designed games, has important consequences in descriptive set theory.

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. Many concepts can be extended, however. Continuous games allow players to choose a strategy from a continuous strategy set. For instance, Cournot competition is typically modeled with players' strategies being any non-negative quantities, including fractional quantities.

Differential games

Differential games such as the continuous pursuit and evasion game are continuous games where the evolution of the players' state variables is governed by differential equations. The problem of finding an optimal strategy in a differential game is closely related to the optimal control theory. In particular, there are two types of strategies: the open-loop strategies are found using the Pontryagin maximum principle while the closed-loop strategies are found using Bellman's Dynamic Programming method.A particular case of differential games are the games with a random time horizon.{{ru icon}} Petrosjan, L.A. and Murzov, N.V. (1966). Game-theoretic problems of mechanics. Litovsk. Mat. Sb. 6, 423â€“433. In such games, the terminal time is a random variable with a given probability distribution function. Therefore, the players maximize the mathematical expectation of the cost function. It was shown that the modified optimization problem can be reformulated as a discounted differential game over an infinite time interval.

Evolutionary game theory

Evolutionary game theory studies players who adjust their strategies over time according to rules that are not necessarily rational or farsighted.JOURNAL, Newton, Jonathan, 2018, Evolutionary Game Theory: A Renaissance, Games, 9, 2, 31, 10.3390/g9020031, In general, the evolution of strategies over time according to such rules is modeled as a Markov chain with a state variable such as the current strategy profile or how the game has been played in the recent past. Such rules may feature imitation, optimization or survival of the fittest.In biology, such models can represent (biological) evolution, in which offspring adopt their parents' strategies and parents who play more successful strategies (i.e. corresponding to higher payoffs) have a greater number of offspring. In the social sciences, such models typically represent strategic adjustment by players who play a game many times within their lifetime and, consciously or unconsciously, occasionally adjust their strategies.{{sfnp|Webb|2007}}

Stochastic outcomes (and relation to other fields)

Individual decision problems with stochastic outcomes are sometimes considered "one-player games". These situations are not considered game theoretical by some authors.{{By whom|date=March 2011}} They may be modeled using similar tools within the related disciplines of decision theory, operations research, and areas of artificial intelligence, particularly AI planning (with uncertainty) and multi-agent system. Although these fields may have different motivators, the mathematics involved are substantially the same, e.g. using Markov decision processes (MDP).{{Citation needed|date=March 2011}}Stochastic outcomes can also be modeled in terms of game theory by adding a randomly acting player who makes "chance moves" ("moves by nature").{{sfnp|Osborne|Rubinstein|1994}} 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.For some problems, different approaches to modeling stochastic outcomes may lead to different solutions. For example, the difference in approach between MDPs and the minimax solution is that the latter considers the worst-case over a set of adversarial moves, rather than reasoning in expectation about these moves given a fixed probability distribution. The minimax approach may be advantageous where stochastic models of uncertainty are not available, but may also be overestimating extremely unlikely (but costly) events, dramatically swaying the strategy in such scenarios if it is assumed that an adversary can force such an event to happen.Hugh Brendan McMahan (2006), Robust Planning in Domains with Stochastic Outcomes, Adversaries, and Partial Observability, CMU-CS-06-166, pp. 3â€“4 (See Black swan theory for more discussion on this kind of modeling issue, particularly as it relates to predicting and limiting losses in investment banking.)General models that include all elements of stochastic outcomes, adversaries, and partial or noisy observability (of moves by other players) have also been studied. The "gold standard" is considered to be partially observable stochastic game (POSG), but few realistic problems are computationally feasible in POSG representation.

Metagames

These are games the play of which is the development of the rules for another game, the target or subject game. Metagames seek to maximize the utility value of the rule set developed. The theory of metagames is related to mechanism design theory.The term metagame analysis is also used to refer to a practical approach developed by Nigel Howard.{{sfnp|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. Subsequent developments have led to the formulation of confrontation analysis.

Pooling games

These are games prevailing over all forms of society. Pooling games are repeated plays with changing payoff table in general over an experienced path and their equilibrium strategies usually take a form of evolutionary social convention and economic convention. Pooling game theory emerges to formally recognize the interaction between optimal choice in one play and the emergence of forthcoming payoff table update path, identify the invariance existence and robustness, and predict variance over time. The theory is based upon topological transformation classification of payoff table update over time to predict variance and invariance, and is also within the jurisdiction of the computational law of reachable optimality for ordered system.Wenliang Wang (2015). Pooling Game Theory and Public Pension Plan. {{isbn|978-1507658246}}.

Mean field game theory

Mean field game theory is the study of strategic decision making in very large populations of small interacting agents. This class of problems was considered in the economics literature by Boyan Jovanovic and Robert W. Rosenthal, in the engineering literature by Peter E. Caines and by mathematician Pierre-Louis Lions and Jean-Michel Lasry.

Representation of games

{{see also|List of games in game theory}}The games studied in game theory are well-defined mathematical objects. To be fully defined, a game must specify the following elements: the players of the game, the information and actions available to each player at each decision point, and the payoffs for each outcome. (Eric Rasmusen refers to these four "essential elements" by the acronym "PAPI".) A game theorist typically uses these elements, along with a solution concept of their choosing, to deduce a set of equilibrium strategies for each player such that, when these strategies are employed, no player can profit by unilaterally deviating from their strategy. These equilibrium strategies determine an equilibrium to the gameâ€”a stable state in which either one outcome occurs or a set of outcomes occur with known probability.Most cooperative games are presented in the characteristic function form, while the extensive and the normal forms are used to define noncooperative games.

Extensive form

(File:Ultimatum Game Extensive Form.svg|thumb|left|An extensive form game)The extensive form can be used to formalize games with a time sequencing of moves. Games here are played on trees (as pictured here). Here each vertex (or node) represents a point of choice for a player. The player is specified by a number listed by the vertex. The lines out of the vertex represent a possible action for that player. The payoffs are specified at the bottom of the tree. The extensive form can be viewed as a multi-player generalization of a decision tree.{{sfnp|Fudenberg|Tirole|1991|p=67}} To solve any extensive form game, backward induction must be used. It involves working backward up the game tree to determine what a rational player would do at the last vertex of the tree, what the player with the previous move would do given that the player with the last move is rational, and so on until the first vertex of the tree is reached.BOOK, Security Studies: an Introduction (second edition), Williams, Paul D., Routledge, 2013, 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN, 55â€“56, The game pictured consists of two players. The way this particular game is structured (i.e., with sequential decision making and perfect information), Player 1 "moves" first by choosing either F or U (Fair or Unfair). Next in the sequence, Player 2, who has now seen Player 1{{'}}s move, chooses to play either A or R. Once Player 2 has made his/ her choice, the game is considered finished and each player gets their respective payoff. Suppose that Player 1 chooses U and then Player 2 chooses A: Player 1 then gets a payoff of "eight" (which in real-world terms can be interpreted in many ways, the simplest of which is in terms of money but could mean things such as eight days of vacation or eight countries conquered or even eight more opportunities to play the same game against other players) and Player 2 gets a payoff of "two".The extensive form can also capture simultaneous-move games and games with imperfect information. To represent it, either a dotted line connects different vertices to represent them as being part of the same information set (i.e. the players do not know at which point they are), or a closed line is drawn around them. (See example in the imperfect information section.)

Normal form

{{Payoff matrix |Float=right |Width=330| Name = Normal form or payoff matrix of a 2-player, 2-strategy game#900|Player 2chooses Left}}#900|Player 2chooses Right}}#009|Player 1chooses Up}}#009|Player 1chooses Down}}#0094}}, {{color>#900|3}}#009â€“1}}, {{color>#900|â€“1}}#0090}}, {{color>#900|0}}#0093}}, {{color>#900|4}}}}The normal (or strategic form) game is usually represented by a matrix which shows the players, strategies, and payoffs (see the example to the right). More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions. In the accompanying example there are two players; one chooses the row and the other chooses the column. Each player has two strategies, which are specified by the number of rows and the number of columns. The payoffs are provided in the interior. The first number is the payoff received by the row player (Player 1 in our example); the second is the payoff for the column player (Player 2 in our example). Suppose that Player 1 plays Up and that Player 2 plays Left. Then Player 1 gets a payoff of 4, and Player 2 gets 3.When a game is presented in normal form, it is presumed that each player acts simultaneously or, at least, without knowing the actions of the other. If players have some information about the choices of other players, the game is usually presented in extensive form.Every extensive-form game has an equivalent normal-form game, however the transformation to normal form may result in an exponential blowup in the size of the representation, making it computationally impractical.{{sfnp|Leyton-Brown|Shoham|2008|p=35}}

Characteristic function form

In games that possess removable utility, separate rewards are not given; rather, the characteristic function decides the payoff of each unity. The idea is that the unity that is 'empty', so to speak, does not receive a reward at all.The origin of this form is to be found in John von Neumann and Oskar Morgenstern's book; when looking at these instances, they guessed that when a union mathbf{C} appears, it works against the fractionleft(frac{mathbf{N}}{mathbf{C}}right)as if two individuals were playing a normal game. The balanced payoff of C is a basic function. Although there are differing examples that help determine coalitional amounts from normal games, not all appear that in their function form can be derived from such.Formally, a characteristic function is seen as: (N,v), where N represents the group of people and v:2^N to mathbf{R} is a normal utility.Such characteristic functions have expanded to describe games where there is no removable utility.

General and applied uses

As a method of applied mathematics, game theory has been used to study a wide variety of human and animal behaviors. It was initially developed in economics to understand a large collection of economic behaviors, including behaviors of firms, markets, and consumers. The first use of game-theoretic analysis was by Antoine Augustin Cournot in 1838 with his solution of the Cournot duopoly. 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.Although pre-twentieth century naturalists such as Charles Darwin made game-theoretic kinds of statements, the use of game-theoretic analysis in biology began with Ronald Fisher's studies of animal behavior during the 1930s. This work predates the name "game theory", but it shares many important features with this field. The developments in economics were later applied to biology largely by John Maynard Smith in his book Evolution and the Theory of Games.JOURNAL, On economic applications of evolutionary game theory,weblink J Evol Econ, In addition to being used to describe, predict, and explain behavior, game theory has also been used to develop theories of ethical or normative behavior and to prescribe such behavior.Colin F. Camerer (2003). Behavioral Game Theory: Experiments in Strategic Interaction, pp. 5â€“7 (scroll to at 1.1 What Is Game Theory Good For?). In economics and philosophy, scholars have applied game theory to help in the understanding of good or proper behavior. Game-theoretic arguments of this type can be found as far back as Plato.WEB,weblink Game Theory, 21 August 2008, Ross, Don, The Stanford Encyclopedia of Philosophy (Spring 2008 Edition), Edward N. Zalta (ed.), An alternative version of game theory, called chemical game theory, represents the player's choices as metaphorical chemical reactant molecules called â€œknowleculesâ€.JOURNAL, Velegol, Darrell, Suhey, Paul, Connolly, John, Morrissey, Natalie, Cook, Laura, 2018-09-14, Chemical Game Theory, Industrial & Engineering Chemistry Research, en, 57, 41, 13593â€“13607, 10.1021/acs.iecr.8b03835, 0888-5885, Â Chemical game theory then calculates the outcomes as equilibrium solutions to a system of chemical reactions.

Description and modeling

File:Centipede game.svg|thumb|300px|right|A four-stage centipede gamecentipede gameThe primary use of game theory is to describe and model how human populations behave. Some{{Who|date=July 2012}} scholars believe that by finding the equilibria of games they can predict how actual human populations will behave when confronted with situations analogous to the game being studied. This particular view of game theory has been criticized. It is argued that the assumptions made by game theorists are often violated when applied to real-world situations. Game theorists usually assume players act rationally, but in practice human behavior often deviates from this model. Game theorists respond by comparing their assumptions to those used in physics. Thus while their assumptions do not always hold, they can treat game theory as a reasonable scientific ideal akin to the models used by physicists. However, empirical work has shown that in some classic games, such as the centipede game, guess 2/3 of the average game, and the dictator game, people regularly do not play Nash equilibria. There is an ongoing debate regarding the importance of these experiments and whether the analysis of the experiments fully captures all aspects of the relevant situation.Experimental work in game theory goes by many names, experimental economics, behavioral economics, and behavioural game theory are several. For a recent discussion, see Colin F. Camerer (2003). Behavioral Game Theory: Experiments in Strategic Interaction (description and Introduction, pp. 1â€“25).Some game theorists, following the work of John Maynard Smith and George R. Price, have turned to evolutionary game theory in order to resolve these issues. These models presume either no rationality or bounded rationality on the part of players. Despite the name, evolutionary game theory does not necessarily presume natural selection in the biological sense. Evolutionary game theory includes both biological as well as cultural evolution and also models of individual learning (for example, fictitious play dynamics).

Prescriptive or normative analysis

{{Payoff matrix |Name=The Prisoner's Dilemma | 2L=Cooperate |2R=Defect |1U=Cooperate |UL=-1, -1 |UR=-10, 0 |1D=Defect |DL=0, -10 |DR=-5, -5}}Some scholars see game theory not as a predictive tool for the behavior of human beings, but as a suggestion for how people ought to behave. Since a strategy, corresponding to a Nash equilibrium of a game constitutes one's best response to the actions of the other players â€“ provided they are in (the same) Nash equilibrium â€“ playing a strategy that is part of a Nash equilibrium seems appropriate. This normative use of game theory has also come under criticism.{{Anchor|Economics}}

Economics and business

Game theory is a major method used in mathematical economics and business for modeling competing behaviors of interacting agents. Applications include a wide array of economic phenomena and approaches, such as auctions, bargaining, mergers & acquisitions pricing, fair division, duopolies, oligopolies, social network formation, agent-based computational economics, general equilibrium, mechanism design, and voting systems; and across such broad areas as experimental economics, behavioral economics, information economics, industrial organization, and political economy.This research usually focuses on particular sets of strategies known as "solution concepts" or "equilibria". A common assumption is that players act rationally. In non-cooperative games, the most famous of these is the Nash equilibrium. A set of strategies is a Nash equilibrium if each represents a best response to the other strategies. 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.Game-theoretic model to examine the two tradeoffs in the acquisition of information for a careful balancing act {{Webarchive|url=https://web.archive.org/web/20130524231021weblink |date=24 May 2013 }} Research paper INSEADweblink" title="web.archive.org/web/20140327103727weblink">Options Games: Balancing the trade-off between flexibility and commitment {{webarchive |url=https://web.archive.org/web/20130620053305weblink |date=20 June 2013 }}. Europeanfinancialreview.com (15 February 2012). Retrieved on 2013-01-03.The payoffs of the game are generally taken to represent the utility of individual players.A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of a particular economic situation. One or more solution concepts are chosen, and the author demonstrates which strategy sets in the presented game are equilibria of the appropriate type. Naturally one might wonder to what use this information should be put. Economists and business professors suggest two primary uses (noted above): descriptive and prescriptive.

Political science

The application of game theory to political science is focused in the overlapping areas of fair division, political economy, public choice, war bargaining, positive political theory, and social choice theory. In each of these areas, researchers have developed game-theoretic models in which the players are often voters, states, special interest groups, and politicians.Early examples of game theory applied to political science are provided by Anthony Downs. In his book An Economic Theory of Democracy,{{harvard citations|last1=Downs|year=1957}} he applies the Hotelling firm location model to the political process. In the Downsian model, political candidates commit to ideologies on a one-dimensional policy space. Downs first shows how the political candidates will converge to the ideology preferred by the median voter if voters are fully informed, but then argues that voters choose to remain rationally ignorant which allows for candidate divergence. Game Theory was applied in 1962 to the Cuban missile crisis during the presidency of John F. Kennedy.Steven J. Brams, Game theory and the Cuban missile crisis, Plus Magazine, 1 January 2001, accessed 31 January 2016.It has also been proposed that game theory explains the stability of any form of political government. Taking the simplest case of a monarchy, for example, the king, being only one person, does not and cannot maintain his authority by personally exercising physical control over all or even any significant number of his subjects. Sovereign control is instead explained by the recognition by each citizen that all other citizens expect each other to view the king (or other established government) as the person whose orders will be followed. Coordinating communication among citizens to replace the sovereign is effectively barred, since conspiracy to replace the sovereign is generally punishable as a crime. Thus, in a process that can be modeled by variants of the prisoner's dilemma, during periods of stability no citizen will find it rational to move to replace the sovereign, even if all the citizens know they would be better off if they were all to act collectively.JOURNAL, 10.2139/ssrn.2371076, Yes, Law is the Command of the Sovereign, 2013, Morrison, A. S., A game-theoretic explanation for democratic peace is that public and open debate in democracies sends clear and reliable information regarding their intentions to other states. In contrast, it is difficult to know the intentions of nondemocratic leaders, what effect concessions will have, and if promises will be kept. Thus there will be mistrust and unwillingness to make concessions if at least one of the parties in a dispute is a non-democracy.JOURNAL, 10.1162/154247604323015463,weblink 40004867, It Takes Two: An Explanation for the Democratic Peace, Journal of the European Economic Association, 2, 1, 1â€“29, 2004, Levy, G., Razin, R., On the other hand, game theory predicts that two countries may still go to war even if their leaders are cognizant of the costs of fighting. War may result from asymmetric information; two countries may have incentives to mis-represent the amount of military resources they have on hand, rendering them unable to settle disputes agreeably without resorting to fighting. Moreover, war may arise because of commitment problems: if two countries wish to settle a dispute via peaceful means, but each wishes to go back on the terms of that settlement, they may have no choice but to resort to warfare. Finally, war may result from issue indivisibilities.JOURNAL, Fearon, James D., 1 January 1995, Rationalist Explanations for War, International Organization, 49, 3, 379â€“414, 10.1017/s0020818300033324, 2706903, Game theory could also help predict a nation's responses when there is a new rule or law to be applied to that nation. One example would be Peter John Wood's (2013) research when he looked into what nations could do to help reduce climate change. Wood thought this could be accomplished by making treaties with other nations to reduce greenhouse gas emissions. However, he concluded that this idea could not work because it would create a prisoner's dilemma to the nations.JOURNAL, Wood, Peter John, 2011, Climate change and game theory, Ecological Economics Review, 1219, 1, 153â€“70, 10.1111/j.1749-6632.2010.05891.x, 21332497, 1885/67270, 2011NYASA1219..153W,

Biology

{{Payoff matrix |Name=The hawk-dove game |2L=Hawk |2R=Dove |1U=Hawk |UL=20, 20 |UR=80, 40 |1D=Dove |DL=40, 80 |DR=60, 60}}Unlike those in economics, the payoffs for games in biology are often interpreted as corresponding to fitness. In addition, the focus has been less on equilibria that correspond to a notion of rationality and more on ones that would be maintained by evolutionary forces. The best-known equilibrium in biology is known as the evolutionarily stable strategy (ESS), first introduced in {{harv|Smith|Price|1973}}. Although its initial motivation did not involve any of the mental requirements of the Nash equilibrium, every ESS is a Nash equilibrium.In biology, game theory has been used as a model to understand many different phenomena. It was first used to explain the evolution (and stability) of the approximate 1:1 sex ratios. {{harv|Fisher|1930}} suggested that the 1:1 sex ratios are a result of evolutionary forces acting on individuals who could be seen as trying to maximize their number of grandchildren.Additionally, biologists have used evolutionary game theory and the ESS to explain the emergence of animal communication.{{sfnp|Harper|Maynard Smith|2003}} The analysis of signaling games and other communication games has provided insight into the evolution of communication among animals. For example, the mobbing behavior of many species, in which a large number of prey animals attack a larger predator, seems to be an example of spontaneous emergent organization. Ants have also been shown to exhibit feed-forward behavior akin to fashion (see Paul Ormerod's Butterfly Economics).Biologists have used the game of chicken to analyze fighting behavior and territoriality.JOURNAL, Maynard Smith, J., The theory of games and the evolution of animal conflicts, 10.1016/0022-5193(74)90110-6, Journal of Theoretical Biology, 47, 1, 209â€“221, 1974, 4459582, According to Maynard Smith, in the preface to Evolution and the Theory of Games, "paradoxically, it has turned out that game theory is more readily applied to biology than to the field of economic behaviour for which it was originally designed". Evolutionary game theory has been used to explain many seemingly incongruous phenomena in nature.Evolutionary Game Theory (Stanford Encyclopedia of Philosophy). Plato.stanford.edu. Retrieved on 3 January 2013.One such phenomenon is known as biological altruism. This is a situation in which an organism appears to act in a way that benefits other organisms and is detrimental to itself. This is distinct from traditional notions of altruism because such actions are not conscious, but appear to be evolutionary adaptations to increase overall fitness. Examples can be found in species ranging from vampire bats that regurgitate blood they have obtained from a night's hunting and give it to group members who have failed to feed, to worker bees that care for the queen bee for their entire lives and never mate, to vervet monkeys that warn group members of a predator's approach, even when it endangers that individual's chance of survival.Biological Altruism (Stanford Encyclopedia of Philosophy). Seop.leeds.ac.uk. Retrieved on 3 January 2013. All of these actions increase the overall fitness of a group, but occur at a cost to the individual.Evolutionary game theory explains this altruism with the idea of kin selection. Altruists discriminate between the individuals they help and favor relatives. Hamilton's rule explains the evolutionary rationale behind this selection with the equation {{Math|c < b × r}}, where the cost {{varserif|c}} to the altruist must be less than the benefit {{varserif|b}} to the recipient multiplied by the coefficient of relatedness {{varserif|r}}. The more closely related two organisms are causes the incidences of altruism to increase because they share many of the same alleles. 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. For example, helping a sibling (in diploid animals) has a coefficient of {{frac|1|2}}, because (on average) an individual shares {{frac|1|2}} 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. 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 coefficient that was {{frac|1|2}} in the smaller field becomes 0.995. Similarly if it is considered that information other than that of a genetic nature (e.g. 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. Several logical theories have a basis in game semantics. In addition, computer scientists have used games to model interactive computations.{{citation needed|date=March 2019}} Also, game theory provides a theoretical basis to the field of multi-agent systems.BOOK, Yoav Shoham, Kevin Leyton-Brown, Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations,weblink 15 December 2008, Cambridge University Press, 978-1-139-47524-2, Separately, game theory has played a role in online algorithms; in particular, the k-server problem, which has in the past been referred to as games with moving costs and request-answer games.{{harvard citations |last1=Ben David |last2=Borodin |last3=Karp |last4=Tardos |last5=Wigderson |year=1994}} Yao's principle is a game-theoretic technique for proving lower bounds on the computational complexity of randomized algorithms, especially online algorithms.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. Algorithmic game theory and within it algorithmic mechanism design combine computational algorithm design and analysis of complex systems with economic theory.

Philosophy

{{Payoff matrix |Name=Stag hunt | 2L=Stag |2R=Hare |1U=Stag |UL=3, 3 |UR=0, 2 |1D=Hare |DL=2, 0 |DR=2, 2}}Game theory has been put to several uses in philosophy. Responding to two papers by {{harvard citations|txt=yes|first=W.V.O.|last=Quine|author1-link=Willard Van Orman Quine|year=1960|year2=1967}}, {{Harvtxt|Lewis|1969}} used game theory to develop a philosophical account of convention. In so doing, he provided the first analysis of common knowledge and employed it in analyzing play in coordination games. In addition, he first suggested that one can understand meaning in terms of signaling games. This later suggestion has been pursued by several philosophers since Lewis.({{Harvtxt|Skyrms|1996}}, {{harvard citations |txt=yes |last1=Grim |last2=Kokalis |last3=Alai-Tafti |last4=Kilb |last5=St Denis |year=2004}}). Following {{Harvtxt|Lewis|1969}} game-theoretic account of conventions, Edna 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.{{citation |first=E. |last=Ullmann-Margalit |title=The Emergence of Norms |publisher=Oxford University Press |year=1977 |isbn=978-0198244110}}{{citation |first=C. |last=Bicchieri |title=The Grammar of Society: the Nature and Dynamics of Social Norms |publisher=Cambridge University Press |year=2006 |isbn=978-0521573726}}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 the interactions of agents. Philosophers who have worked in this area include Bicchieri (1989, 1993),{{citation |title=Self-Refuting Theories of Strategic Interaction: A Paradox of Common Knowledge |journal=Erkenntnis |volume=30 |issue=1â€“2 |year=1989 |pages=69â€“85 |doi=10.1007/BF00184816 |last1=Bicchieri |first1=Cristina}}{{Citation |last1=Bicchieri |first1=Cristina |author1-link=Cristina Bicchieri |title=Rationality and Coordination |publisher=Cambridge University Press |isbn=978-0-521-57444-0 |year=1993}} Skyrms (1990),{{citation |title=The Dynamics of Rational Deliberation |publisher=Harvard University Press |year=1990 |isbn=978-0674218857}} and Stalnaker (1999).{{citation |chapter=Knowledge, Belief, and Counterfactual Reasoning in Games |editor-first=Cristina |editor-last=Bicchieri |editor2-first=Richard |editor2-last=Jeffrey |editor3-first=Brian |editor3-last=Skyrms |title=The Logic of Strategy |location=New York |publisher=Oxford University Press |year=1999 |isbn=978-0195117158}}In ethics, some (most notably David Gauthier, Gregory Kavka, and Jean Hampton) {{Who|date=July 2012}} authors have attempted to pursue Thomas Hobbes' project of deriving morality from self-interest. 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. This general strategy is a component of the general social contract view in political philosophy (for examples, see {{Harvtxt|Gauthier|1986}} and {{Harvtxt|Kavka |1986}}).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.Other authors have attempted to use evolutionary game theory in order to explain the emergence of human attitudes about morality and corresponding animal behaviors. 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., {{harvard citations|txt=yes|last=Skyrms|year=1996|year2=2004}} and {{harvard citations|txt=yes|last1=Sober|last2=Wilson|year=1999}}).

In popular culture

• Based on the 1998 book by Sylvia Nasar,Nasar, Sylvia (1998) A Beautiful Mind, Simon & Schuster. {{isbn|0-684-81906-6}}. the life story of game theorist and mathematician John Nash was turned into the 2001 biopic A Beautiful Mind, starring Russell Crowe as Nash.Singh, Simon (14 June 1998) "Between Genius and Madness", New York Times.
• The 1959 military science fiction novel Starship Troopers by Robert A. Heinlein mentioned "games theory" and "theory of games".{{citation |last=Heinlein |first=Robert A. |title=Starship Troopers |year=1959}} In the 1997 film of the same name, the character Carl Jenkins referred to his military intelligence assignment as being assigned to "games and theory".
• The 1964 film Dr. Strangelove satirizes game theoretic ideas about deterrence theory. For example, nuclear deterrence depends on the threat to retaliate catastrophically if a nuclear attack is detected. A game theorist might argue that such threats can fail to be credible, in the sense that they can lead to subgame imperfect equilibria. The movie takes this idea one step further, with the Russians irrevocably committing to a catastrophic nuclear response without making the threat public.{{citation needed|date=March 2016}}
• The 1980s power pop band Game Theory was founded by singer/songwriter Scott Miller, who described the band's name as alluding to "the study of calculating the most appropriate action given an adversary... to give yourself the minimum amount of failure."NEWS, Guzman, Rafer, Star on hold: Faithful following, meager sales, March 6, 1996, Pacific Sun (newspaper), Pacific Sun,weblink 25 July 2018,weblink" title="web.archive.org/web/20131106223809weblink">weblink 6 November 2013, yes, dmy-all, .{{cbignore}}
• Liar Game, a 2005 Japanese manga and 2007 television series, presents the main characters in each episode with a game or problem that is typically drawn from game theory, as demonstrated by the strategies applied by the characters.{{citation needed|date=March 2016}}

See also

{{div col end}}Lists

Notes

{{Reflist|30em
| refs =
â€¢ At (JEL classification codes#Mathematical and quantitative methods JEL: C Subcategories|JEL:C7) of the Journal of Economic Literature classification codes.   â€¢ R.J. Aumann (2008). "game theory," The New Palgrave Dictionary of Economics, 2nd Edition. Abstract. {{webarchive |url=https://web.archive.org/web/20110515034120weblink |date=15 May 2011 }}   â€¢ Martin Shubik (1981). "Game Theory Models and Methods in Political Economy," in Kenneth Arrow and Michael Intriligator, ed., Handbook of Mathematical Economics, v. 1, pp. 285â€“330 {{DOI|10.1016/S1573-4382(81)01011-4}}.   â€¢ Carl Shapiro (1989). "The Theory of Business Strategy," RAND Journal of Economics, 20(1), pp. 125â€“137 {{JSTOR|2555656}}.â€¢ Leigh Tesfatsion (2006). "Agent-Based Computational Economics: A Constructive Approach to Economic Theory," ch. 16, Handbook of Computational Economics, v. 2, pp. 831â€“880 {{DOI|10.1016/S1574-0021(05)02016-2}}.   â€¢ Joseph Y. Halpern (2008). "computer science and game theory," The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.â€¢ From The New Palgrave Dictionary of Economics (2008), 2nd Edition:     Roger B. Myerson. "mechanism design." Abstract {{webarchive|url=https://web.archive.org/web/20111123042038weblink |date=23 November 2011 }}.     _____. "revelation principle." Abstract.   â€¢ Tuomas Sandholm. "computing in mechanism design." Abstract. {{webarchive|url=https://web.archive.org/web/20111123042038weblink |date=23 November 2011 }}   â€¢ Noam Nisan and Amir Ronen (2001). "Algorithmic Mechanism Design," Games and Economic Behavior, 35(1â€“2), pp. 166â€“196.   â€¢ Noam Nisan et al., ed. (2007). Algorithmic Game Theory, Cambridge University Press. Description {{webarchive|url=https://web.archive.org/web/20120505140924weblink |date=5 May 2012 }} .BOOK, 10.1016/S1574-0005(05)80062-1, Chapter 30 Voting procedures, 2, 1055â€“1089, Handbook of Game Theory with Economic Applications, 1994, Brams, Steven J., 9780444894274, and BOOK, 10.1016/S1574-0005(05)80063-3, Chapter 31 Social choice, 2, 1091â€“1125, Handbook of Game Theory with Economic Applications, 1994, Moulin, HervÃ©, 9780444894274,
• Vernon L. Smith, 1992. "Game Theory and Experimental Economics: Beginnings and Early Influences," in E. R. Weintraub, ed., Towards a History of Game Theory, pp. 241â€“282
• BOOK,weblink 10.1016/B0-08-043076-7/02232-4, Experimental Economics, International Encyclopedia of the Social & Behavioral Sciences, 5100â€“5108, 2001, Smith, V.L., 9780080430768,
• JOURNAL,weblink
• Vincent P. Crawford (1997). "Theory and Experiment in the Analysis of Strategic Interaction," in Advances in Economics and Econometrics: Theory and Applications, pp. 206â€“242. Cambridge. Reprinted in Colin F. Camerer et al., ed. (2003). Advances in Behavioral Economics, Princeton. 1986â€“2003 papers. Description, preview, Princeton, ch. 12.
• BOOK, 10.1016/S1574-0005(02)03025-4, Chapter 62 Game theory and experimental gaming, Handbook of Game Theory with Economic Applications Volume 3, 3, 2327â€“2351, Handbook of Game Theory with Economic Applications, 2002, Shubik, Martin, 9780444894281,
From The New Palgrave Dictionary of Economics (2008), 2nd Edition:    â€¢ Faruk Gul. "behavioural economics and game theory." Abstract.   â€¢ Colin F. Camerer. "behavioral game theory." Abstract. {{webarchive |url=https://web.archive.org/web/20111123034346weblink |date=23 November 2011 }}   â€¢ _____ (1997). "Progress in Behavioral Game Theory," Journal of Economic Perspectives, 11(4), p. 172, pp. 167â€“188.   â€¢ _____ (2003). Behavioral Game Theory, Princeton. Description, preview ([ctrl]+), and ch. 1 link.   â€¢ _____, George Loewenstein, and Matthew Rabin, ed. (2003). Advances in Behavioral Economics, Princeton. 1986â€“2003 papers. Description, contents, and preview.   â€¢ Drew Fudenberg (2006). "Advancing Beyond Advances in Behavioral Economics," Journal of Economic Literature, 44(3), pp. 694â€“711 {{JSTOR|30032349}}.
• Eric Rasmusen (2007). Games and Information, 4th ed. Description and chapter-preview.
• David M. Kreps (1990). Game Theory and Economic Modelling. Description.
• JOURNAL,weblink
• BOOK,weblink 10.1016/S1574-0005(02)03006-0, Chapter 43 Incomplete information, Handbook of Game Theory with Economic Applications Volume 3, 3, 1665â€“1686, Handbook of Game Theory with Economic Applications, 2002, Aumann, Robert J., Heifetz, Aviad, 9780444894281,
â€¢ Jean Tirole (1988). The Theory of Industrial Organization, MIT Press. Description and chapter-preview links, pp. viiâ€“ix, "General Organization," pp. 5â€“6, and "Non-Cooperative Game Theory: A User's Guide Manual,' " ch. 11, pp. 423â€“59.   â€¢ Kyle Bagwell and Asher Wolinsky (2002). "Game theory and Industrial Organization," ch. 49, Handbook of Game Theory with Economic Applications,v. 3, pp. 1851â€“1895.   â€¢ Martin Shubik (1959). Strategy and Market Structure: Competition, Oligopoly, and the Theory of Games, Wiley. Description and review extract.   â€¢ _____ with Richard Levitan (1980). Market Structure and Behavior, Harvard University Press. Review extract. {{webarchive |url=https://web.archive.org/web/20100315131945weblink |date=15 March 2010 }}â€¢ Martin Shubik (1981). "Game Theory Models and Methods in Political Economy," in Handbook of Mathematical Economics, v. 1, pp. 285â€“330 {{DOI|10.1016/S1573-4382(81)01011-4}}.   â€¢_____ (1987). A Game-Theoretic Approach to Political Economy. MIT Press. Description. {{webarchive |url=https://web.archive.org/web/20110629151809weblink |date=29 June 2011 }}â€¢ Martin Shubik (1978). "Game Theory: Economic Applications," in W. Kruskal and J.M. Tanur, ed., International Encyclopedia of Statistics, v. 2, pp. 372â€“78.   â€¢ Robert Aumann and Sergiu Hart, ed. Handbook of Game Theory with Economic Applications (scrollable to chapter-outline or abstract links):
1992. v. 1; 1994. v. 2; 2002. v. 3.
N. Agarwal and P. Zeephongsekul. Psychological Pricing in Mergers & Acquisitions using Game Theory, School of Mathematics and Geospatial Sciences, RMIT University, MelbourneNoam Nisan et al., ed. (2007). Algorithmic Game Theory, Cambridge University Press. Description. {{webarchive |url=https://web.archive.org/web/20120505140924weblink |date=5 May 2012 }}{{Citation |first=Noam |last=Nisan |first2=Amir |last2=Ronen |year=2001 |title=Algorithmic Mechanism Design |journal=Games and Economic Behavior |volume=35 |issue=1â€“2 |url=http://www.cs.cmu.edu/~sandholm/cs15-892F09/Algorithmic%20mechanism%20design.pdf |pages=166â€“196 |doi=10.1006/game.1999.0790|citeseerx=10.1.1.21.1731 }}â€¢ Joseph Y. Halpern (2008). "computer science and game theory," The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.   â€¢ {{Citation |first=Yoav |last=Shoham |year=2008 |title=Computer Science and Game Theory |journal=Communications of the ACM |volume=51 |issue=8 |pages=75â€“79 |url=http://www.robotics.stanford.edu/~shoham/www%20papers/CSGT-CACM-Shoham.pdf |doi=10.1145/1378704.1378721|citeseerx=10.1.1.314.2936 }}   â€¢ {{Citation |first=Amy |last=Littman |authorlink2=Michael L. Littman |first2=Michael L. |year=2007 |title=Introduction to the Special Issue on Learning and Computational Game Theory |journal=Machine Learning |volume=67 |issue=1â€“2 |pages=3â€“6 |doi=10.1007/s10994-007-0770-1 |last2=Littman}}BOOK, Screpanti, Ernesto, Zamagni, Stefano, An Outline of the History of Economic Thought', 2005, Oxford University Press, 2nd, }}

References and further reading

{{Commons category}}

Textbooks and general references

• {{citation |first=Robert J |last=Aumann |author-link=Robert Aumann |year=1987 |contribution=game theory |title=The New Palgrave: A Dictionary of Economics |volume=2 |pages=460â€“82|title-link=The New Palgrave: A Dictionary of Economics }}.
• {{Citation |last=Camerer |first=Colin |author-link=Colin Camerer |title=Behavioral Game Theory: Experiments in Strategic Interaction |publisher=Russell Sage Foundation |isbn=978-0-691-09039-9 |year=2003 |chapter-url=http://press.princeton.edu/chapters/i7517.html |chapter=Introduction |pages=1â€“25}}, Description.
• {{Citation |last=Dutta |first=Prajit K. |title=Strategies and games: theory and practice |publisher=MIT Press |isbn=978-0-262-04169-0 |year=1999}}. Suitable for undergraduate and business students.
• {{Citation |last1=Fernandez |first1=L F. |last2=Bierman |first2=H S. |title=Game theory with economic applications |publisher=Addison-Wesley |isbn=978-0-201-84758-1 |year=1998}}. Suitable for upper-level undergraduates.
• {{Citation |last=Gibbons |first=Robert D. |title=Game theory for applied economists |publisher=Princeton University Press |isbn=978-0-691-00395-5 |year=1992}}. Suitable for advanced undergraduates.
• Published in Europe as {{Citation |title=A Primer in Game Theory |publisher=Harvester Wheatsheaf |isbn=978-0-7450-1159-2 |location=London |first=Robert |last=Gibbons |year=2001}}.
• {{Citation |last=Gintis |first=Herbert |title=Game theory evolving: a problem-centered introduction to modeling strategic behavior |publisher=Princeton University Press |isbn=978-0-691-00943-8 |year=2000}}
• {{Citation |last1=Green |first1=Jerry R. |last2=Mas-Colell |first2=Andreu |author2-link=Andreu Mas-Colell |last3=Whinston |first3=Michael D. |title=Microeconomic theory |publisher=Oxford University Press |isbn=978-0-19-507340-9 |year=1995}}. Presents game theory in formal way suitable for graduate level.
• Joseph E. Harrington (2008) Games, strategies, and decision making, Worth, {{isbn|0-7167-6630-2}}. Textbook suitable for undergraduates in applied fields; numerous examples, fewer formalisms in concept presentation.
• {{Citation |last=Howard |first=Nigel |title=Paradoxes of Rationality: Games, Metagames, and Political Behavior |publisher=The MIT Press |year=1971 |location=Cambridge, MA |isbn=978-0-262-58237-7}}
• {{Citation |last=Isaacs |first=Rufus |author-link=Rufus Isaacs (game theorist) |title=Differential Games: A Mathematical Theory With Applications to Warfare and Pursuit, Control and Optimization |publisher=Dover Publications |location=New York |isbn=978-0-486-40682-4 |year=1999}}
• {{Citation |last=Miller |first=James H. |title=Game theory at work: how to use game theory to outthink and outmaneuver your competition |publisher=McGraw-Hill |location=New York |isbn=978-0-07-140020-6 |year=2003}}. Suitable for a general audience.
• {{Citation |last=Osborne |first=Martin J. |title=An introduction to game theory |publisher=Oxford University Press |isbn=978-0-19-512895-6 |year=2004}}. Undergraduate textbook.
• {{Citation |last2=Rubinstein |first2=Ariel |author2-link=Ariel Rubinstein |last1=Osborne |first1=Martin J. |title=A course in game theory |publisher=MIT Press |isbn=978-0-262-65040-3 |year=1994}}. A modern introduction at the graduate level.
• {{Citation |last1=Shoham |first1=Yoav |last2=Leyton-Brown |first2=Kevin |title=Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations |publisher=Cambridge University Press |isbn=978-0-521-89943-7 |url=http://www.masfoundations.org/download.html |year=2009 |location=New York |accessdate=8 March 2016}}
• Roger McCain's Game Theory: A Nontechnical Introduction to the Analysis of Strategy{{dead link|date=September 2016|bot=medic|fix-attempted=yes}}{{cbignore|bot=medic}} (Revised Edition)
• {{Citation |last=Webb |first=James N. |year=2007 |title=Game theory: decisions, interaction and evolution |series=Undergraduate mathematics |publisher=Springer |isbn=978-1-84628-423-6}} Consistent treatment of game types usually claimed by different applied fields, e.g. Markov decision processes.

Historically important texts

• Aumann, R.J. and Shapley, L.S. (1974), Values of Non-Atomic Games, Princeton University Press
• {{Citation |last1=Cournot |first1=A. Augustin |author1-link=Antoine Augustin Cournot |title=Recherches sur les principles mathematiques de la thÃ©orie des richesses |year=1838 |journal=Libraire des Sciences Politiques et Sociales}}
• {{Citation |last1=Edgeworth |first1=Francis Y. |author1-link=Francis Ysidro Edgeworth |title=Mathematical Psychics |publisher=Kegan Paul |location=London |year=1881|url=https://books.google.com/books?id=CElYAAAAcAAJ&printsec=frontcoverv=onepage}}
• {{Citation |title=Theory of Voting |first=Robin |last=Farquharson |authorlink=Robin Farquharson |publisher=Blackwell (Yale U.P. in the U.S.) |year=1969 |isbn=978-0-631-12460-3}}
• {{Citation |last1=Luce |first1=R. Duncan |author1-link=R. Duncan Luce |last2=Raiffa |first2=Howard |author2-link=Howard Raiffa |title=Games and decisions: introduction and critical survey |publisher=Wiley |location=New York |year=1957}}

*reprinted edition: {{Citation |title=Games and decisions: introduction and critical survey |publisher=Dover Publications |location=New York |isbn=978-0-486-65943-5 |year=1989 |author=R. Duncan Luce ; Howard Raiffa}}
• {{Citation |last1=Maynard Smith |first1=John |author1-link=John Maynard Smith |title=Evolution and the theory of games |publisher=Cambridge University Press |isbn=978-0-521-28884-2 |year=1982|title-link=Evolution and the Theory of Games }}
• {{Citation |last1=Maynard Smith |first1=John |author1-link=John Maynard Smith |first2=George R. |last2=Price |author2-link=George R. Price |title=The logic of animal conflict |year=1973 |journal=Nature |volume=246 |issue=5427 |pages=15â€“18 |doi=10.1038/246015a0 |bibcode=1973Natur.246...15S}}
• {{Citation |last1=Nash |first1=John |author1-link=John Forbes Nash |title=Equilibrium points in n-person games |year=1950 |journal=Proceedings of the National Academy of Sciences of the United States of America |volume=36 |issue=1 |pages=48â€“49 |doi=10.1073/pnas.36.1.48 |pmc=1063129 |pmid=16588946 |bibcode=1950PNAS...36...48N}}
• 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.)
• Shapley, L.S. (1953), Stochastic Games, Proceedings of National Academy of Science Vol. 39, pp. 1095â€“1100.
• {{Citation |last1=von Neumann |first1=John |title=Zur Theorie der Gesellschaftsspiele |year=1928 |journal=Mathematische Annalen |volume=100 |issue=1 |pages=295â€“320 |doi=10.1007/bf01448847}} English translation: "On the Theory of Games of Strategy," in A. W. Tucker and R. D. Luce, ed. (1959), Contributions to the Theory of Games, v. 4, p. 42. Princeton University Press.
• {{Citation |last1=von Neumann |first1=John |author1-link=John von Neumann |last2=Morgenstern |first2=Oskar |author2-link=Oskar Morgenstern |title=Theory of games and economic behavior |journal=Nature |volume=157 |issue=3981 |pages=172 |publisher=Princeton University Press |year=1944|title-link=Theory of games and economic behavior |bibcode=1946Natur.157..172R |doi=10.1038/157172a0 }}
• {{Citation |last1=Zermelo |first1=Ernst |author1-link=Ernst Zermelo |title=Ãœber eine Anwendung der Mengenlehre auf die Theorie des Schachspiels |year=1913 |journal=Proceedings of the Fifth International Congress of Mathematicians |volume=2 |pages=501â€“4}}

Other print references

• {{Citation |last1=Ben David |first1=S. |last2=Borodin |first2=Allan |author2-link=Allan Borodin |last3=Karp |first3=Richard |author3-link=Richard Karp |last4=Tardos |first4=G. |last5=Wigderson |first5=A. |author5-link=Avi_Wigderson |title=On the Power of Randomization in On-line Algorithms |url=http://www.math.ias.edu/~avi/PUBLICATIONS/MYPAPERS/BORODIN/paper.pdf |journal=Algorithmica |volume=11 |pages=2â€“14 |year=1994 |doi=10.1007/BF01294260 |issue=1}}
• {{Citation |last1=Downs |first1=Anthony |author1-link=Anthony Downs |title=An Economic theory of Democracy |titlelink=An Economic Theory of Democracy |publisher=Harper |location=New York |year=1957}}
• {{Citation |last1=Gauthier |first1=David |author1-link=David Gauthier |title=Morals by agreement |publisher=Oxford University Press |isbn=978-0-19-824992-4 |year=1986}}
• Allan Gibbard, "Manipulation of voting schemes: a general result", Econometrica, Vol. 41, No. 4 (1973), pp. 587â€“601.
• {{Citation |last1=Grim |first1=Patrick |last2=Kokalis |first2=Trina |last3=Alai-Tafti |first3=Ali |last4=Kilb |first4=Nicholas |last5=St Denis |first5=Paul |title=Making meaning happen |year=2004 |journal=Journal of Experimental & Theoretical Artificial Intelligence |volume=16 |issue=4 |pages=209â€“243 |doi=10.1080/09528130412331294715}}
• {{Citation |last1=Harper |first1=David |author1-link=David Harper (biologist) |last2=Maynard Smith |first2=John |author2-link=John Maynard Smith |title=Animal signals |publisher=Oxford University Press |isbn=978-0-19-852685-8 |year=2003}}
• {{Citation |last1=Lewis |first1=David |author1-link=David Kellogg Lewis |title=Convention: A Philosophical Study |year=1969}}, {{isbn|978-0-631-23257-5}} (2002 edition)
• {{Citation |last1=McDonald |first1=John |title=Strategy in Poker, Business & War |publisher=W. W. Norton |isbn=978-0-393-31457-1 |year=1950â€“1996}}. A layman's introduction.
• {{Citation |last1=Papayoanou |first1=Paul |title=Game Theory for Business: A Primer in Strategic Gaming |publisher=Probabilistic |isbn=978-0964793873 |year=2010}}.
• {{Citation |last1=Quine |first1=W.v.O |author1-link=Willard Van Orman Quine |title=Philosophica Essays for A.N. Whitehead |publisher=Russel and Russel Publishers |isbn=978-0-8462-0970-6 |year=1967 |chapter=Truth by Convention}}
• {{Citation |last1=Quine |first1=W.v.O |author1-link=Willard Van Orman Quine |title=Carnap and Logical Truth |year=1960 |journal=Synthese |volume=12 |issue=4 |pages=350â€“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), 187â€“217.
• {{Citation |last1=Siegfried |first1=Tom |title=A Beautiful Math |publisher=Joseph Henry Press |isbn=978-0-309-10192-9 |year=2006}}
• {{Citation |last1=Skyrms |first1=Brian |author1-link=Brian Skyrms |title=The Dynamics of Rational Deliberation |publisher=Harvard University Press |isbn=978-0-674-21885-7 |year=1990}}
• {{Citation |last1=Skyrms |first1=Brian |title=Evolution of the social contract |publisher=Cambridge University Press |isbn=978-0-521-55583-8 |year=1996}}
• {{Citation |last1=Skyrms |first1=Brian |title=The stag hunt and the evolution of social structure |publisher=Cambridge University Press |isbn=978-0-521-53392-8 |year=2004}}
• {{Citation |last1=Sober |first1=Elliott |last2=Wilson |first2=David Sloan |title=Unto others: the evolution and psychology of unselfish behavior |publisher=Harvard University Press |isbn=978-0-674-93047-6 |year=1998}}
• {{Citation |last=Thrall |first=Robert M. |author-link=Robert M. Thrall |last2=Lucas |first2=William F. |author2-link=William F. Lucas |title=n-person games in partition function form |journal=Naval Research Logistics Quarterly |volume=10 |issue=4 |pages=281â€“298 |year=1963 |doi=10.1002/nav.3800100126}}
• {{Citation |last1=Dolev |first1=Shlomi |first2=Panagiota |last2=Panagopoulou |first3=Mikael |last3=Rabie |first4=Elad Michael |last4=Schiller |first5=Paul |last5=Spirakis |chapter=Rationality authority for provable rational behavior |title=Proceedings of the 30th annual ACM SIGACT-SIGOPS symposium on Principles of distributed computing |year=2011 |pages=289â€“290 |doi=10.1145/1993806.1993858 |isbn=9781450307192|chapter-url=https://research.chalmers.se/en/publication/230780 }}
• {{citation | doi = 10.1073/pnas.1406556111 | pmid=24979793 | volume=111 | issue=29 | title=Algorithms, games, and evolution | year=2014 | journal=Proceedings of the National Academy of Sciences | pages=10620â€“10623 | last1 = Chastain | first1 = E.| pmc=4115542 | bibcode=2014PNAS..11110620C }}

External links

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