Chapter 7: Prisoners Dilemma

Overview

Claw me, and I’ll claw thee.

16th century proverb^[1]

The most famous paradigm in game theory is the prisoners’ dilemma. Here’s how it goes: The police arrest two criminals guilty of both murder and illegal weapons’ possession. The police can easily prove that both men violated weapons’ laws and could consequently imprison each criminal for one year. If the police could also establish that the criminals committed murder, however, they could send the men to fry in the electric chair. Unfortunately, the police can’t establish that either criminal committed murder unless at least one of them confesses.

• If both criminals keep quiet, the worst punishment they could get is one year in jail.

• If either criminal confesses, both men could be sentenced to death.

The captured criminals realize the game they’re in, so you might think that neither would ever confess. In the prisoners’ dilemma story, however, the police use game theory to induce the men to turn on each other.

The police put the criminals, whom I will name Adam and Ben, in separate rooms. They tell Adam the following:

If Ben confesses, then:

• Adam gets the death penalty if he doesn’t confess.

• Adam gets life in prison if he does confess.

The police need only one player to confess to convict either criminal of murder. Their threat to execute Adam if he does not confess and Ben does is credible. If Adam believes that Ben would confess, then Adam would himself benefit from confessing.^[2] If Ben confesses, then by confessing himself, Adam gets life in prison rather than the electric chair. In their efforts to induce a confession the police have now made some progress. If Adam believes that Ben will confess, then it will be in Adam’s self-interest also to confess.

The police then remind Adam that they already have enough evidence to convict him on weapons charges even if neither confesses. The police tell Adam that:

If Ben does not confess, then:

• Adam gets 1 year in prison if he doesn’t confess.

• Adam goes free if he does confess.

If Ben does not cooperate, Adam still benefits from confessing. Adam, therefore, should always confess, since regardless of what Ben does, Adam benefits from confessing. Indeed, confessing is a dominant strategy for Adam. Recall that a dominant strategy is one you should play no matter what you think that other players might do.

Having been so successful with Adam, the police use the same strategy on Ben. Ben consequently finds confessing to be a dominant strategy. As a result of their clever implementation of game theory, the police induce both men to confess and put them in jail for life.

Figure 34 illustrates this prisoners’ dilemma game. The result seems very counterintuitive. If both men had kept quiet, they would have gotten only one year in jail. By talking, both criminals get life. Shouldn’t the men understand the game they are in and adopt different strategies? No! If Adam thinks that Ben is not going to confess, Adam is still better off talking. Even if Adam could somehow convince Ben to stay silent, Adam would still want to confess. Of course, when Adam does confess, he increases the punishment that Ben receives. Remember, however, in game theory land people care only about themselves. In the context of a prisoners’ dilemma this assumption seems particularly realistic as someone who has just committed murder is probably not too interested in self-sacrifice.

Figure 34

Wouldn’t Adam fear that his confessing would cause Ben to confess? No! The police separate the criminals. When Ben decides whether to cooperate, he has no way of knowing if Adam confessed. The police are certainly not going to tell Ben that he should not confess because his brave partner stayed silent. Whether Ben confesses will have nothing to do with whether Adam talks. Consequently, each player profits from cooperating with the police, even though this causes them to both receive life sentences. The key result in prisoners’ dilemma is that even though all the players realize that the outcome is going to be bad, they still adopt strategies guaranteeing the bad outcome is achieved. Of course, the police are pleased with the results of the prisoners’ dilemma.

What if the two criminals made an agreement never to confess if caught by the police? If you’re about to commit a murder, you should always make such an agreement. This agreement, of course, shouldn’t prevent you from cooperating if caught. Rather, you should make the agreement to keep your naive co-criminal quiet and then confess to escape punishment. True, this will mean that your partner in crime dies. But so what? He is, after all, a murderer.

If the police interviewed both men in the same room, then the logic behind prisoners’ dilemma would collapse. Each man would suspect that if he confessed then his friend would too. If the men were interviewed together, then whether one player confessed would influence the probability of the other man confessing. This is why on television shows, at least, the police always separate suspected criminals when interviewing them.

Two rational criminals playing prisoners’ dilemma would always confess. Two irrational criminals might both not talk and get a far lower prison term. Does this mean that rational people do worse in prisoner’s dilemma games? No. You are always better off being rational. You are hurt, however, when your opponent is rational. While it’s true that if both of you are rational, you might be worse off than if both of you were irrational. The best outcome would be if you were rational, and your opponent was not.

Joining the Mafia would allow the prisoners to overcome their dilemma. The depiction of the Mafia in movies suggests that they harshly punish those who cooperate with the police. This added punishment changes the prisoners’ game by altering the payoffs. Figure 35 illustrates this Mafia modification of the prisoners’ dilemma. In an effort to avoid the Mafia death penalty, both criminals should now not cooperate and will get only one year in jail. Interestingly, belonging to an organization that threatens to kill you if you break the rules can increase your payoff, so Mafia membership clearly has its privileges. Since Mafia membership will effectively lower your prison term if caught, we would expect members of the Mafia to commit more murders. The Mafia thus has a competitive advantage in the market for crime.

Figure 35

^[1]Browning (1989), 364.

^[2]I am assuming that both criminals would rather receive a punishment of life in prison than be executed.