Prisoner's Dilemma

The prisoner's dilemma is a classic game used to illustrate the potential suboptimality arising from lack of cooperation. Its name refers to a situation in which two suspects are arrested and charged with a crime. The police, who lack sufficient evidence to convict the suspects unless at least one of them confesses, hold the suspects in separate cells and explain the consequences of the actions they may take. If neither confesses, both will be convicted for minor offences and sentenced to 1 month in jail. If both confess, both will be sentenced to jail for 6 months. Finally, if one confesses [Page 535]but the other does not, the confessor is immediately released and the other is sentenced to 9 months—6 for the crime and 3 more for obstructing justice. This situation can be represented in the following game theoretical format:

The payoffs are all negative because they represent months in jail. It can be seen that, regardless of what Prisoner 2 does, Prisoner 1 is always better off confessing, so confessing is his dominant strategy. Because the game is symmetric, confessing is likewise dominant for Prisoner 2. Hence, we should expect both players to confess and spend 6 months in jail each. The strategy profile both confess is an equilibrium in dominant strategies and is the only Nash equilibrium of this game.

Nevertheless, both confess is Pareto-dominated by the neither confesses outcome, so the two prisoners end up spending 6 months in jail, while a better alternative for both (1 month in jail) was available.

Notice that both prisoners are perfectly informed about all the elements of the game and that even knowing in advance what the other player is doing does not lead to the superior outcome. Even if Prisoner 1 chooses first and Prisoner 2 observes his or her choice, Prisoner 2 is still be better off confessing, regardless of Prisoner 1's choice. Anticipating this, Prisoner 1 still confesses. The problem is that Prisoner 2 cannot commit to not confessing if Prisoner 1 does likewise.

Hence, solutions to the problem have little to do with information: the roots of the dilemma lie in the particular nature of the interaction coupled with the lack of coordinating and commitment devices, that is, without appropriate institutional arrangements. The prisoner's dilemma is therefore a metaphor of the collective action problem, whereby all agents have an incentive to free ride on the behavior of other agents and, by doing so, generate inefficient (i.e., Pareto-dominated) outcomes.

Hence, rational outcome-oriented individuals playing individually optimal strategies may fail to realize commons interests. The prisoner's dilemma (and its multiplayer variant) illustrate that Adam Smith's invisible hand does not operate in all strategic interactions. Sometimes explicit coordination, backed up by the ability to make credible commitments or by sanctions of some kind, is required to realize common interests.

However, tacit cooperation may be achieved in the iterated prisoner's dilemma, that is, if the same prisoners play the game many times. If the prisoner's dilemma is played a fixed number of times known to both players, both confess in each period is the only Nash equilibrium of the iterated game. If, however, the game between the same players is repeated an infinite or unknown or random number of times, the possibility of cooperation to realize common interests emerges. Cooperation becomes possible because the players benefit from reciprocity and have therefore an incentive to be “nice” in order to obtain a nice behavior in return. One possible strategy in the iterated prisoner's dilemma is to cooperate in the first round and continue to cooperate as long the opponent cooperates too. Only after the opponent fails to cooperate does the player switch to defection, and then does so forever. More forgiving versions of this strategy require retaliation for defection only for a given number of periods. The most forgiving version is tit for tat, according to which a player cooperates in the first period and thereafter reciprocates whatever the other player did in the previous period. If both players play any such strategy (or more generally any “nice” strategy, i.e., in which one is never the first to defect), the result is a Nash equilibrium in the iterated prisoner's dilemma, producing mutual cooperation in each period.

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