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Prisoner's Dilemma

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The prisoner's dilemma is a well-known game in non-cooperative game theory that highlights how the individually rational behaviour of 'players' can lead to an outcome that is suboptimal for all involved. It is a game that is often used to analyze the possibility of cooperation in situations where self-interested individuals interact strategically.

The story of the prisoner’s dilemma is as follows: two criminals are arrested on suspicion of having committed a crime, let's say a bank robbery. The police do not have enough evidence to convict them for the robbery, but they have smaller charges on each of the two. The prisoners are being held captive in separate cells of a prison without any possibility of communication. The general attorney offers both prisoners the same deal: if they betray their accomplice (or 'defect' as it is called in the table below) by testifying as a witness they can go free and will not have to serve any prison sentence whatsoever.

However, it is not that simple. Should the other also testify, meaning that they both implicate each other, each will serve 2 years in prison. If only one testifies while the other remains silent, the one who does not testify (the silent one) will be sentenced to 4 years in prison, while the one who does testify will go free. In the case that they do not betray each other, that is, that both remain silent and cooperate, they cannot be proven guilty of the bank robbery and both will be sentenced to 1 year in prison for the minor crimes.

In this two-player game, each player has two options: cooperate with their partner in crime and remain silent, or testify and betray their accomplice, i.e., defect. The prison sentences represent the payoffs of each decision and these payoffs of course depend on whether each accomplice accepts the deal offered and defects, or remains silent and cooperates. The following payoff matrix describes the possible outcomes of the game.

 

Player 2

Cooperate

Defect

Player 1

Cooperate

-1,-1

-4,0

Defect

0,-4

-2,-2


What is the equilibrium of this game? To answer this question, we first have to find the best responses of the two players. Suppose player 2 cooperates and remains silent. In this case the best option for player 1 is to defect and betray his partner as he would receive a sentence of 0 years, instead of the 1 year he would get if he remained silent. Now suppose that player 2 defects and betrays her partner. The best response of player 1 is again to defect as this would lead to a sentence of 2 years for each of them, far preferable to the 4 years he would have to serve if he kept quiet. For player 2 the situation is identical (the game is symmetric), which means that her best response is to defect and betray player 1. Thus, in the unique Nash equilibrium of this game both prisoners will defect and accept the deal they were offered by the general attorney. As both implicate each other, each of them has to serve a 2-year prison sentence. 

In this game “defect” is the dominant strategy and “cooperate” is the dominated strategy. A strategy is dominant if it leads to a higher payoff no matter what the other player(s) do. A strategy is said to be dominated if under no circumstances it is optimal for a player to use it, as in it yields a lower payoff than any other strategy regardless of the other players’ strategies. In the prisoner’s dilemma it does not matter whether the accomplice decides to remain silent or betray his or her partner. In both cases the other player maximizes their payoffs by defecting.  

This result is interesting in so far as both prisoners would be better off if they cooperated and remained silent, as in this case each would only be sentenced to 1 year. Yet, the fact that our players are fully rational and only think about maximizing their own payoff (or utility) this outcome cannot be reached. Even if the accomplice remains silent, each prisoner has an incentive to deviate and betray the partner to reduce their sentence.  

A lot of research in game theory has focused on finding ways in which cooperation can be established in the prisoner’s dilemma. One way out of the dilemma is to consider a repeated game in which the players meet not only once, as in, there are multiple rounds. In the repeated prisoner´s dilemma where the final round is not specified, betraying an accomplice will have a negative impact on a criminal´s reputation and will make future cooperation harder. For example, if the involved criminals are part of a gang, going free by betraying a fellow gang member may no longer be a dominant strategy given the future consequences. Possible strategies in the repeated or iterated prisoner´s dilemma include tit-for-tat and grim-trigger strategies. If players play a grim-trigger strategy, they start with cooperation in the first round of the game and then cooperate until one player defects. If someone defects, all players will defect from this point in time onwards. Following this strategy, cooperation can be sustained as long as players care sufficiently about the future. If players play tit-for-tat, they also start with cooperation in the first instance. A player cooperates if the other player cooperated in the previous period and defects if the other player defected in the previous period.

Further reading

Theoretically cooperation is not possible if the players know exactly how many rounds will be played, because everyone will want to defect in the last round and by backward induction this also means that in the other periods all players will defect. In experiments cooperation occurs even in the finitely repeated prisoner´s dilemma. Embrey et al. (The Quarterly Journal of Economics, 2018) show that in experiments a player cooperates as long as the others cooperate until a certain threshold round. The more experienced the players, i.e. the more often they have already played the game, the earlier they start defecting.

Good to know

Game theory in general and the prisoner´s dilemma more specifically have been used in the 1950s and 1960s to analyze conflict during the period of the cold war. The US and the USSR were the two players, who could choose between not attacking the other nation (to cooperate) and being the first to strike (to defect). That the conflict never escalated can be (partially) explained by the negative consequences an attack would have had on potential future cooperation, because none of the two parties wanted to start a war.