Game Theory

Game theory is the mathematical study of strategic decisions, situations in which the outcome for each participant depends not only on their own choice but also on the choices of others. It models "players", the "strategies" they can choose, and the "payoffs" each receives, and asks what rational players will do when they know others are reasoning in the same way. It is used in economics, finance, political science, biology, computer science and military planning.

Modern game theory began with John von Neumann and Oskar Morgenstern's book Theory of Games and Economic Behavior (1944) and was transformed by John Nash's equilibrium concept in 1950. It provides the analytical core for the study of oligopoly, auctions, bargaining, contracts and voting. Several Nobel Memorial Prizes in Economic Sciences have honored game theorists, including John Nash, John Harsanyi and Reinhard Selten in 1994, Robert Aumann and Thomas Schelling in 2005, Alvin Roth and Lloyd Shapley in 2012, and Paul Milgrom and Robert Wilson in 2020.

Think of two cafés opening on the same street. Each must decide its prices, but the best price for one depends on what the other charges. If one cuts prices, the other may lose customers unless it cuts too. Each owner has to guess the rival's move and plan around it, much as chess players think a few moves ahead.

Game theory is a toolkit for that kind of thinking. It writes down who is involved, what each can do and what each gets from every combination of choices. Then it looks for outcomes where nobody would want to change their decision once they see what everyone else did. That stable point helps explain price wars, negotiations, elections and even why some animals share food.

• Players, strategies and payoffs. Every game specifies who decides, what options they have and what each combination of options yields.
• Simultaneous and sequential games. In simultaneous games, like sealed-bid auctions, players move without seeing others' choices; in sequential games, like chess or a market-entry decision, later players observe earlier moves.
• Information. Games of complete information assume everyone knows the payoffs; Harsanyi's work on "Bayesian" games handles cases where players hold private information, which links to asymmetric information, adverse selection and moral hazard.
• Cooperative and non-cooperative games. Cooperative theory studies how coalitions divide gains (the Shapley value is a famous rule); non-cooperative theory studies individual strategies.
• Zero-sum and non-zero-sum games. In a zero-sum game one player's gain equals another's loss; most economic interactions, including trade, are non-zero-sum.
• Equilibrium. The central solution concept is the Nash equilibrium, where no player can gain by changing strategy alone. Refinements such as Selten's subgame perfection rule out threats that would not be carried out.

The most famous game was devised at the RAND Corporation in 1950 by Merrill Flood and Melvin Dresher and given its name by Albert Tucker. Two suspects, held separately, can each stay silent (cooperate) or testify against the other (defect). The table shows years in prison for (Player A, Player B):

Whatever B does, A serves less time by testifying (0 instead of 1, or 5 instead of 10), and the same holds for B. So both testify and each serves 5 years, although both staying silent would have given each just 1 year. The game shows how individually sensible choices can produce a collectively worse result, a pattern behind price wars, arms races, overfishing and the tragedy of the commons. When the game is repeated many times, cooperation can be sustained by strategies that reward cooperation and punish defection, an idea tested in Robert Axelrod's computer tournaments around 1980.

• Market structure. Models of duopoly and oligopoly, going back to Antoine Augustin Cournot (1838) and Joseph Bertrand (1883), explain how a few firms set prices or output, and why a cartel tends to break down when each member is tempted to cheat. Competition authorities use these models in antitrust cases.
• Auctions and market design. Auction theory, recognized by the 2020 prize, guided the design of radio-spectrum auctions run by the US Federal Communications Commission from 1994; matching theory, recognized in 2012, is used for school choice and kidney exchange. See also auctions in economics.
• Central banking. Finn Kydland and Edward Prescott's 1977 work on time inconsistency, a game between a central bank and the public, helped make the case for independent central banks and inflation targeting.
• Bank runs. The Diamond–Dybvig model (1983) treats a bank run as a coordination game with two equilibria, calm and panic, which supports the case for deposit insurance; Douglas Diamond and Philip Dybvig shared the 2022 prize with Ben Bernanke.
• Crypto networks. Consensus systems such as proof of work are designed so that honest behavior is the best response for miners or validators, a branch sometimes called mechanism design or cryptoeconomics.
• Politics and public policy. Voting, lobbying and international agreements are studied as games, an approach shared with public choice theory.

Game theory's predictions depend on strong assumptions: players are rational, know the rules and payoffs, and expect others to be rational too. Laboratory experiments often find more cooperation and fairness than theory predicts (in the ultimatum game, for instance, many people reject low offers even at a cost to themselves), which led to behavioral game theory and to models of bounded rationality. Many games have several equilibria, and theory alone may not say which one will occur. Critics also note that results can be sensitive to small modelling choices. Supporters reply that game theory is a language for structuring strategic problems rather than a crystal ball, and that its applications in auction design and market regulation have proved useful in practice.

• Stanford Encyclopedia of Philosophy: Game Theory
• Stanford Encyclopedia of Philosophy: Prisoner's Dilemma
• Nobel Prize in Economic Sciences 1994