Prisoners' Dilemma

The prisoners' dilemma (PD) is usually presented with the following narrative. Two men are arrested after committing a crime. Only a confession by one or both of them can lead to a conviction for the crime. If both of them remain silent, each will be charged with a lesser offense and serve a light sentence. If one confesses while the other remains silent, the one who confesses will be set free in exchange for his testimony against the other, and the one who remains silent will be convicted of the crime and receive a full sentence. If both confess, both of them will be convicted of the crime but will receive a reduced sentence. The dilemma here is that regardless of what the other chooses to do, it will be better for each of them to confess. However, if they both confess, they will be worse off than they would have been if both of them had remained silent. In this entry, various game-theoretical strategies and possible applications in political science are discussed.

The basic structure of the game is presented in Table 1, a 2 × 2 matrix that specifies the payoff each player receives, where T > R > P > S.

The game, thus, has a unique equilibrium in which both players defect and receive payoff P. They would have received a higher payoff, R, had each of them chosen “cooperate,” but this is not possible since each of them has a dominant strategy to “defect.”

In a repeated PD, the choices made today not only determine today's outcome but can also influence the choices made in the future. If the players were to repeat the game for a finite and known number of times, backward induction would imply that both players defect in every round. [Page 2131]In the last round, neither player fears the consequences of current choices, so they will both choose “defect.” As a result, both players end up choosing “defect” in each round, expecting the other player to “defect” in the following round. However, if the players were to play an infinitely repeated PD or a finite repeated PD for an unknown number of times, backward induction no longer applies. When a game is repeated infinitely or repeated for a finite but unknown number of times, players use weighted-average payoffs to determine which strategy to choose. To calculate the weighted-average payoffs in a given round, future payoffs are multiplied by a discount factor q, either to reflect that future payoffs are valued less than present payoffs or as a probability that there will be another round in the future. The value of cooperation in a given round depends on the value of q, increasing as q approaches one. As long as q is not zero, both players are better off if they both choose “cooperate” in every round.

Table 1 Prisoners' Dilemma

There is an equilibrium in which both players defect in every round because each player chooses a strategy to “always defect” and such an equilibrium exists for any value of q. However, it is no longer the only possible equilibrium outcome for both players to defect in every round because players can play reciprocal strategies that condition their choices in a given round on the course of the game. For instance, players can choose to play a grim trigger strategy, which punishes the defector by playing defect for all future rounds of the game. This reciprocal strategy can produce an equilibrium in which both players cooperate in every round for certain values of q. In fact, according to the Folk theorem, a wide range of payoffs can be supported in infinitely repeated games or repeated games with finite but unknown number of rounds as long as q is not zero, and consequently, the set of strategies that support mutual cooperation in every round can be quite large depending on the value of q. There exists as yet no consensus with respect to the reasons that may be invoked for favoring one reciprocal strategy over the other when any of them can support mutual cooperation in every round.

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