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Game Theory

Prisoner's Dilemma



Game theorists and social scientists have been fascinated by Prisoner's Dilemma, a two-by-two game (two players, each with two possible pure strategies) with a particular payoff matrix (Rapoport and Chammah; Poundstone). The game's nickname and the accompanying story were provided by A. W. Tucker. Suppose that two prisoners, accused of jointly committing a serious crime, are interrogated separately. The prosecutor has sufficient evidence to convict them of a lesser crime without any confession, but can get a conviction on the more serious charge only with a confession. If neither prisoner confesses, they will each be sentenced to two years for the lesser charge. If both confess, each will receive a sentence of five years. However, if only one prisoner confesses, that prisoner will be sentenced to only one year, while the other prisoner will get ten years. In the absence of any external authority to enforce an agreement to deny the charges, each player has a dominant strategy of confessing (given that the other player has denied the charges, one year is a lighter sentence than two years; given that the other player has confessed, five years is a lighter sentence than ten years). The unique Nash equilibrium is for both players to confess (defect from any agreement to cooperate) and receive sentences of five years, even though both would be better off if both denied the charges (cooperated).



This game has been used as an explanation of how individually rational behavior can lead to undesirable outcomes ranging from arms races to overuse of natural resources ("the tragedy of the commons," a generalization to more than two players). If the game is repeated a known finite number of times, however large, the predicted result is the same: both players will confess (defect) on the last play, since there would be no opportunity of future punishment for defection or reward for cooperation; therefore both will also confess (defect) on the next-to-last play, since the last play is determined, and so on, with mutual defection on each round as the only sub-game perfect Nash equilibrium. However, the "folk theorem" states that for infinitely repeated games, even with discounting of future benefits or a constant probability of the game ending on any particular round (provided that the discount rate and the probability of the game ending on the next round are sufficiently small and that the dimensionality of payoffs allows for the possibility of retaliation), any sequence of actions can be rationalized as a subgame perfect Nash equilibrium. (The folk theorem owes its name to its untraceable origin.)

However, players do not generally behave in accordance with Nash's prediction. Frequent cooperation in one-shot or finitely repeated Prisoner's Dilemma has been observed ever since it was first played. The first Prisoner's Dilemma experiment, conducted at RAND by Merrill Flood and Melvin Drescher in January 1950, involved one hundred repetitions with two sophisticated players, the economist Armen Alchian from the University of California, Los Angeles, and the game theorist John Williams, head of RAND's mathematics department. Alchian and Williams succeeded in cooperating on sixty plays, and mutual defection, the Nash equilibrium, occurred only fourteen times (Poundstone, pp. 107–116). Robert Axelrod (1984) conducted a computer tournament for iterated Prisoner's Dilemma, finding that Rapoport's simple "tit for tat" strategy (cooperate on the first round, then do whatever the other player did on the previous round) yielded the highest payoff.

One way to explain the observed extent of cooperation in experimental games and in life is to recognize that humans are only boundedly rational, relying on rules of thumb and conventions, and making choices about what to know because information is costly to acquire and process. Assumptions about rationality in game theory, such as common knowledge, can be very strong: "An event is common knowledge among a group of agents if each one knows it, if each one knows the others know it, if each one knows that each one knows that the others know it, and so on … the limit of a potentially infinite chain of reasoning about knowledge" (Geanakoplos, p. 54). Ariel Rubinstein (1998) sketches techniques for explicitly incorporating computability constraints and the process of choice in models of procedural rationality. Alternatively, evolutionary game theory, surveyed by Larry Samuelson (2002), emphasizes adaptation and evolution to explain behavior, rather than fully conscious rational choice, returning to human behavior the extension of game theory to evolutionarily stable strategies for animal behavior (Maynard Smith; Dugatkin and Reeve).

Additional topics

Science EncyclopediaScience & Philosophy: Formate to GastropodaGame Theory - The Origins Of Game Theory, Nash Equilibrium, The Nash Bargaining Solution, And The Shapley Value