## Friday, May 11, 2012

### Solving Prisoner's Dilemma

Many of us probably first encountered game theory from the 2001 film A Beautiful Mind where the Nobel-economist John Nash, played by Russel Crowe, showed us that "every man for himself" may not be the best strategy in picking up girls in a bar.

Game theory has a couple of advantages over other approaches in studying decision making. First, it recognizes that the consequences (e.g., payoffs) of the decision of one player are affected by the decisions of other players. And second, it acknowledges that in the real world, players often have to make decisions simultaneously and that waiting for other players to make their move and just react is not possible (or practical).

In this post we will take another look at the "prisoner's dilemma" game that I introduced in the previous post and try to come up with a methodological solution that you may or may not agree with.

Two men are arrested, but the police do not possess enough information for a conviction. Following the separation of the two men, the police offer both a similar deal—if one testifies against his partner (defects/betrays), and the other remains silent (cooperates/assists), the betrayer goes free and the cooperator receives the full one-year sentence. If both remain silent, both are sentenced to only one month in jail for a minor charge. If each 'rats out' the other, each receives a three-month sentence. Each prisoner must choose either to betray or remain silent; the decision of each is kept quiet. What should they do?

We can summarize the payoffs for two prisoners A and B as follows:

One approach in arriving at a solution to this game is to identify dominant and dominated strategies. A dominant strategy is one which dominates all other strategies: that is, it is the best move for a particular player regardless of what the other player does. For example, in the prisoner's dilemma game, if Prisoner B keeps silent, the best move for A is to betray Prisoner B; if Prisoner B betrays A, then it would be best if Prisoner A betrays B. These arguments show us that "betray" is a dominant strategy for A, and following the same reasoning, also for B. By eliminating the dominated strategies (keeping silent for each prisoner), we are left with one solution: for the two prisoners to betray each other and be sentenced to three months each.

Do you agree with this solution? I'm sure that a lot of you don't. The problem with the dominant strategy approach is that it (implicitly) excludes the option to cooperate or collude. But as John Nash (in the movie) and that player from Golden Balls showed us, if players work together, each will be able to receive the best payoffs for himself and the group: everyone will get laid!