Prisoner's Dilemma -- from Wolfram MathWorld

The prisoner's dilemma is a problem in game theory framed by A. Tucker. Suppose each of two prisoners and , who are not allowed to communicate with each other, is offered to be set free if he implicates the other. If neither implicates the other, both will receive the usual sentence. However, if the prisoners implicate each other, then both are presumed guilty and granted harsh sentences.

A dilemma arises in deciding the best course of action in the absence of knowledge of the other prisoner's decision. Each prisoner's best strategy is to turn the other in, regardless of the other's choice. However, if the prisoners turn each other in, both fare worse than if both remain silent.

Writing cooperation for remaining silent and defection for implicating the other prisoner, a general payoff matrix for the game is

where and, in the usual repeated-game formulation, . Here , , , and are conventionally called the temptation, reward, punishment, and sucker's payoff. Mutual defection is the unique Nash equilibrium, although mutual cooperation gives both players the larger payoff .

The underlying game was developed and experimentally tested by M. Flood and M. Dresher at RAND in 1949-1950. Tucker subsequently supplied the prisoner story and the name "prisoner's dilemma" (Poundstone 1992).

A repeated version of the game, in which strategies can depend on the history of previous moves, is known as the iterated prisoner's dilemma.

Mosteller (1987) describes a different problem he terms "the prisoner's dilemma." In this problem, three prisoners , , and with apparently equally good records have applied for parole, and the parole board has decided to release two, but not all three. A warder knows which two are to be released, and one of the prisoners () asks the warder for the name of the one prisoner other than himself who is to be released. While his chances of being released before asking are 2/3, he thinks his chances after asking and being told " will be released" are reduced to 1/2, since now either and or and are to be released. However, he is mistaken since his chances remain 2/3.