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Should the prisoners confess?

Two prisoners, X and Y, are held in separate cells for a serious crime that they did, in fact, commit. The prosecutor, however, has only enough hard evidence to convict them of a minor offense, for which the penalty is a year in jail. Each prisoner is told that if one confesses while the other remains silent, the confessor will go scot free while the other spends 20 years in prison. If both confess, they will get an intermediate sentence, say five years. These payoffs are summarized in the payoff matrix below. The two prisoners are not allowed to communicate with one another. If the prisoners are rational and narrowly self-interested, what will they do?

Any game has 3 basic elements: (1) The Players; (2) The Strategies/Actions available to the Players; and (3) The Payoffs that each Player receives for each possible combination of Strategies. We can represent these elements in a game matrix or game table.

So in this game, the two players are: Prisoner X and Prisoner Y. Each has the same two strategies/actions available: confess or remain silent. Each receives a payoff in terms of prison time, and this payoff depends on the action taken by the prisoner as well as the action taken by the other prisoner. So their marginal benefits and marginal costs are interdependent. Note how this game resembles the experiment in which you had to play the role of Player X or Player Y.

Definition: A dominant strategy is a strategy that yields a higher payoff no matter what the other player chooses (antonym: a dominated strategy is a strategy that yields a lower payoff no matter what the other player chooses).

Each prisoner has a dominant strategy to confess. No matter what Y does, X gets a lighter sentence by confessing -- if Y too confesses, X gets five years instead of 20; and if Y remains silent, X goes free instead of spending a year in jail. The payoffs are symmetric, so Y also does better to confess, no matter what X does.

Definition: An equilibrium outcome (or solution) is one in which no player would have an incentive to change their strategy if they found themselves about to realize this outcome and were given the opportunity to change their strategy.

In the game above, both prisoners choosing to confess is an equilibrium outcome (or solution). No Player has an incentive to change their action if they found themselves in this cell of the game matrix.

Note, however, that when each prisoner behaves in his or her own self-interest, they both do worse than if each had shown restraint. Thus, when both confess, they get five years, instead of the one year they could have gotten by remaining silent. A game with these structure is a called a Prisoner's Dilemma. As we will see, there are Prisoner Dilemmas inherent in many decisions that people must make, not just ones in which prisoners make a decision to confess or remain silent.

Now let's look at the outcomes for the experiment in which you each had to choose "C" or "S" to earn extra credit points. Show results. Did everyone play the dominant strategy? Why not? If people bring up "feeling guilty/bad about deviating from socially optimal strategy" point out that you will return to this issue later: culture/social preferences as a commitment device.

The pattern you see has been observed in a high payoff version of the Prisoner's Dilemma. The Game Show Network ran a show called "Friend or Foe" (based on a previously aired UK show called "Shaft or Don't Shaft"). After a two-member team earns money through various tasks, each has to choose "friend" or "foe" without knowing what the other person chose. If both choose "friend," they split the earnings. If both choose "foe," they lose the money. If one chooses "friend" and the other chooses "foe," the "foe" receives all the money and the "friend" earns nothing. An analysis of the data shows that less than 23% of the pairs achieved a "friend/friend" outcome and over 50% of the participants chose "foe" (you can see the data yourself at http://www.amstat.org/publications/jse/v12n3/datasets.kalist.html)

 
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