Thursday, March 26, 2009

prisoner's dilemma

This article was mentioned on the Freakonomics blog. DNA evidence collected from the scene of a jewelry robbery in Germany was matched to two people, identical twins with indistinguishable DNA. Because the evidence failed to incriminate one particular person, both brothers were released. Of course, between the two of them, they probably know which one did it (if either of them did).

The Freakonomics post about this situation alludes to the prisoner's dilemma, which I actually have been pondering lately. Not because I'm a prisoner, but because I read another chapter in my intro to game theory book while grading tests a few nights ago.

The prisoner's dilemma goes like this.

Two prisoners are held in connection with a crime. They are not able to communicate to one another. Each is given the opportunity to confess, with the understanding that the sentencing will occur as follows:

  • If one talks, he goes free and the other gets a maximum sentence (because there is evidence against him).
  • If both talk, they each get a usual sentence. (evidence against them, but less for cooperating).
  • If neither talks, they each get a minimum sentence. (not enough evidence for a conviction).

The outcomes can be arranged in a matrix as follows, showing how many years each prisoner gets as a result of each of the four outcomes:

B talksB doesn't talk
A talks A:10, B:10A:0, B:20
A doesn't talk A:20, B:0A:5, B:5

Here is the tricky part, as seen by Prisoner A (B's considerations are the same). Prisoner A is trying to make a rational decision with the sole intention of serving the least amount of jail time possible. Prisoner A knows that if B talks, A should talk (to get 10 years instead of 20). If B doesn't talk, A should talk (to get 0 years instead of 5). So while the outcome is uncertain, the strategy is clear. Prisoner A should talk, no matter what B decides.

Likewise, Prisoner B would be better off talking, no matter what A decides. Therefore, they will both be rational and talk. They will each serve 10 years.

The intriguing facet of this situation is that there exists an outcome which is better for both prisoners (they both stay quiet and each serves 5 years), but which would involve each of them acting irrationally. Of course, if their decision were made to avoid jail time, it could hardly be called irrational. That's the catch-22.

At a glance, it seems like a clever illusion and it is tempting to think that after considering the outcomes, each prisoner would be silent and thereby obtain the most favourable outcome available: 5 years and 5 years. But remember that the only reason a prisoner would be quiet is if he were confident that the other prisoner would also be quiet. And if he were confident of that, would he really pass up the chance of immediate freedom?

This leads to a mathematical discussion of why we do not see more cooperation in the world. The best outcome for everyone is usually necessarily artificially imposed or it will not happen.

My simplified rendition assumes that both prisoners are equally guilty and know that they would be convicted by the other's testimony. It also assumes that they do not like being in jail. Honor among thieves could certainly keep them quiet if the jail time were a matter of days, but as the sentence gets more substantial, the reward for informing on the other prisoner grows considerably, in addition to being the only rational choice. This means that the relative magnitudes of punishments and temptations come into play. Here is a more lengthy discussion of the dilemma that explains some of the variables.

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