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— CH. 1 · INTRODUCTION —

Game theory

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  • Game theory is the study of mathematical models of strategic interactions, and it began with a question about what a rational person should do when someone else is trying to outmaneuver them. The field started narrow. At first it addressed two-person zero-sum games, where one participant's gains are exactly balanced by the other's losses. By the 1950s it had stretched far past that, into non zero-sum situations, and eventually into a wide range of behavioral relations. Today it is an umbrella term for the science of rational decision making in humans, animals, and computers. How did a tool built to solve card games end up explaining the stability of monarchies, the sex ratios of animals, and the logic of nuclear deterrence? Why did fifteen of its practitioners win the Nobel Prize in economics? And why, according to one of its pioneers, does it fit biology better than the economics it was designed for?

  • Cardano wrote on games of chance in Liber de ludo aleae, the Book on Games of Chance, composed around 1564 but only published posthumously in 1663. Long before anyone spoke of modern game theory, the mathematics of games drew careful minds toward the structure of luck and choice. Influenced by Fermat and Pascal working on the problem of points, Huygens developed the concept of expectation by reasoning about games of chance. He published his gambling calculus in De ratiociniis in ludo aleae, On Reasoning in Games of Chance, in 1657. In 1713, a letter attributed to Charles Waldegrave, an active Jacobite and uncle to the British diplomat James Waldegrave, analyzed a card game called le her. Waldegrave gave a minimax mixed strategy solution to a two-person version of the game. That puzzle is now known as the Waldegrave problem. In 1838, Antoine Augustin Cournot built a model of competition in oligopolies in his Recherches sur les principes mathematiques de la theorie des richesses. He did not use the term, but his solution is what we now call the Nash equilibrium. In 1883, Joseph Bertrand attacked Cournot's model as unrealistic and offered an alternative model of price competition, later formalized by Francis Ysidro Edgeworth. In 1913, Ernst Zermelo published a paper on set theory applied to chess, proving that optimal chess strategy is strictly determined.

  • John von Neumann published On the Theory of Games of Strategy in 1928, and with it game theory became its own independent field. His original proof used Brouwer's fixed-point theorem on continuous mappings into compact convex sets. That technique became a standard method in game theory and mathematical economics. Emile Borel had circled the same territory. In his 1938 book Applications aux Jeux de Hasard and earlier notes, Borel proved a minimax theorem for two-person zero-sum matrix games, but only when the pay-off matrix is symmetric. He even conjectured that mixed-strategy equilibria do not exist in finite two-person zero-sum games. Von Neumann proved that conjecture false. Von Neumann's work culminated in the 1944 book Theory of Games and Economic Behavior, co-authored with Oskar Morgenstern, which considered cooperative games of several players. The second edition supplied an axiomatic theory of utility, reviving Daniel Bernoulli's old theory of utility of money as an independent discipline. That step let mathematical statisticians and economists treat decision-making under uncertainty as a formal problem rather than a guess.

  • In 1950, John Nash proposed a criterion for the mutual consistency of players' strategies, now called the Nash equilibrium. It applies to a far wider variety of games than the earlier von Neumann and Morgenstern criterion. Nash proved that every finite n-player, non-zero-sum, non-cooperative game has a Nash equilibrium in mixed strategies. A Nash equilibrium is a set of strategies, one for each player, such that no player can improve their payoff by unilaterally changing strategy. The 1950s brought a flurry of new ideas. Game theorists developed the core, the extensive form game, fictitious play, repeated games, and the Shapley value. The decade also saw the first applications of game theory to philosophy and political science. The first mathematical discussion of the prisoner's dilemma appeared during this period. Mathematicians Merrill M. Flood and Melvin Dresher ran an experiment on it as part of the RAND Corporation's investigations. RAND pursued these studies because of their possible applications to global nuclear strategy.

  • Reinhard Selten introduced the solution concept of subgame perfect equilibria in 1965, a refinement of the Nash equilibrium, and he later added trembling hand perfection. In 1994, Nash, Selten, and John Harsanyi shared the Nobel Memorial Prize in the Economic Sciences for their contributions to economic game theory. The honors kept coming as the field matured. In 2005, Thomas Schelling and Robert Aumann became laureates. Schelling worked on dynamic models, early examples of evolutionary game theory, while Aumann introduced correlated equilibria and built a formal analysis of common knowledge and its consequences. In 2007, Leonid Hurwicz, Eric Maskin, and Roger Myerson won for laying the foundations of mechanism design theory. Myerson contributed the notion of proper equilibrium and an influential graduate text, Game Theory, Analysis of Conflict, while Hurwicz formalized the concept of incentive compatibility. In 2012, Alvin E. Roth and Lloyd S. Shapley won for the theory of stable allocations and the practice of market design. In 2014, the prize went to Jean Tirole. Outside economics, John Maynard Smith received the Crafoord Prize in 1999 for applying evolutionary game theory.

  • Eric Rasmusen names four essential elements of any game with the acronym PAPI: the players, the actions, the payoffs, and the information. A game must specify the players, the information and actions available to each one at every decision point, and the payoffs for each outcome. Games split along several axes. A game is cooperative when players can form binding commitments enforced externally, for instance through contract law, and non-cooperative when all agreements must be self-enforcing through credible threats. Zero-sum games are those in which choices can neither increase nor decrease available resources, so a player benefits only at the equal expense of others. Poker exemplifies a zero-sum game if you ignore the house's cut, since one wins exactly what opponents lose. Other zero-sum games include matching pennies, Go, and chess. Many studied games, including the prisoner's dilemma, are non-zero-sum, because outcomes can net more or less than zero. Symmetric games pay each player the same for the same choice, and the standard versions of chicken, the prisoner's dilemma, and the stag hunt are all symmetric. Asymmetric games, like the ultimatum game and the dictator game, give players different strategies. Games also divide into simultaneous and sequential, and into perfect and imperfect information. Tic-tac-toe, checkers, chess, and Go are perfect-information games. Poker and bridge are imperfect-information games, because players do not know all the moves already made.

  • The extensive form represents games with a time sequencing of moves, drawn as a game tree where each vertex is a point of choice for a player. The lines out of a vertex are possible actions, and the payoffs sit at the bottom of the tree. To solve an extensive form game, you use backward induction, working up from the last vertex to decide what a rational player would do, then what the prior player would do given that, and so on to the first vertex. Consider a two-player tree where Player 1 moves first, choosing fair or unfair, and Player 2 then chooses accept or reject. If Player 1 chooses one branch and Player 2 the other, Player 1 might get a payoff of eight and Player 2 a payoff of two, where eight could mean money, days of vacation, or even countries conquered. The normal form, by contrast, is usually a matrix showing players, strategies, and payoffs. One player picks the row and the other picks the column, with payoffs in the interior. Suppose Player 1 plays Up and Player 2 plays Left, giving Player 1 a payoff of 4 and Player 2 a payoff of 3. Cooperative games are usually written in characteristic function form, which lists the payoff of each coalition and traces its origin to the von Neumann and Morgenstern book. Every extensive-form game has an equivalent normal-form game, but the transformation can cause an exponential blowup in the size of the representation. Newer representations exist too, including congestion games from 1973, graphical games from 2001, and action graph games from 2012.

  • John Maynard Smith wrote in the preface to Evolution and the Theory of Games that, paradoxically, game theory turned out to be more readily applied to biology than to the economic behavior for which it was designed. In biology, payoffs are interpreted as fitness, and the central concept is the evolutionarily stable strategy, or ESS. Every ESS is a Nash equilibrium, yet its original motivation involved none of the mental requirements of rationality. The reach is extraordinary. Game theory has been used to explain the roughly 1:1 sex ratios in many species, the mobbing behavior in which prey animals swarm a larger predator, and the evolution of animal communication through signaling games. It even explains biological altruism through kin selection. Hamilton's rule states that the cost to the altruist must be less than the benefit to the recipient multiplied by the coefficient of relatedness. Vampire bats regurgitate blood for group members who failed to feed, worker bees tend the queen for life and never mate, and vervet monkeys warn the group of a predator even at risk to themselves. In political science, Anthony Downs applied the Hotelling firm location model to elections in his 1957 book An Economic Theory of Democracy, showing candidates converging on the median voter. Game theory was applied in 1962 to the Cuban Missile Crisis during the presidency of John F. Kennedy. It has even been used to explain why a king, who cannot physically control his subjects, retains authority through a coordination problem resembling the prisoner's dilemma. Peter John Wood's 2013 research found that climate treaties to cut greenhouse gas emissions could falter because they create a prisoner's dilemma for nations. The same ideas reached cinema. The 1964 film Dr. Strangelove satirizes deterrence by having the Soviet Union irrevocably commit to a catastrophic nuclear response without making the threat public.

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Common questions

What is game theory and what is it used for?

Game theory is the study of mathematical models of strategic interactions. It has applications across the social sciences and is used extensively in economics, logic, systems science, and computer science. It is now an umbrella term for the science of rational decision making in humans, animals, and computers.

Who founded modern game theory?

John von Neumann founded modern game theory, publishing On the Theory of Games of Strategy in 1928 and using Brouwer's fixed-point theorem in his proof. His work culminated in the 1944 book Theory of Games and Economic Behavior, co-authored with Oskar Morgenstern.

What is the Nash equilibrium in game theory?

The Nash equilibrium is a set of strategies, one for each player, such that no player can improve their payoff by unilaterally changing their strategy. John Nash developed it in 1950 and proved that every finite n-player, non-zero-sum, non-cooperative game has a Nash equilibrium in mixed strategies.

How many game theorists have won the Nobel Prize in economics?

Fifteen game theorists had won the Nobel Prize in economics as of 2020. They include John Nash, Reinhard Selten, and John Harsanyi in 1994, and most recently Paul Milgrom and Robert B. Wilson.

What is the prisoner's dilemma in game theory?

The prisoner's dilemma is a non-zero-sum game in which two arrested gang members each face the choice to stay silent or betray the other. The dominant strategy is to betray, yet both staying silent would yield a greater reward for both than mutual betrayal.

How is game theory applied to biology?

In biology, game theory interprets payoffs as fitness and centers on the evolutionarily stable strategy, where every ESS is a Nash equilibrium. It has been used to explain roughly 1:1 sex ratios, animal communication, and biological altruism through kin selection and Hamilton's rule. John Maynard Smith applied it in his 1982 book Evolution and the Theory of Games.

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All sources

146 references cited across the entry

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