Game Theory: The Prisoner’s Dilemma

By Mary Jane Sterling

The classic prisoner’s dilemma is a popular problem in game theory, and so you may encounter it in a finite math course. The prisoner’s dilemma has many other applications, but it is probably best described with the following situation.

Two gang members are arrested and put into two separate rooms for questioning. The prosecutor has enough evidence to convict both of them on a minor charge but not enough to convict them on a major felony. The gang members hope to get away with just being convicted of the minor charge and get a short sentence.

During questioning, the prosecutor makes each gang member an offer. The offer is that the prisoner being questioned would be set free if he testifies that the other committed the major felony. But there are consequences. Here are the consequences, naming the two prisoners Ron and Cal.

• If Ron betrays Cal and Cal remains silent, then Ron will be set free, and Cal will serve ten years.
• If Cal betrays Ron and Ron remains silent, then Cal will be set free, and Ron will serve ten years.
• If both betray the other, then both will serve five years.
• If both remain silent, both will serve just one year for the minor charge.

How does this play out?

FNTMATH_1501

Now, putting this in a payoff matrix with the “consequences” in terms of what happens to Ron, you get the following (the negative numbers indicate years served):

FNTMATH_1502

So how is this played? Do you see the optimum play? The saddle point is the in the first row, first column. Also, the first column dominates the second column, and the first row dominates the second row. According to the game, the best option is for both prisoners to “sing.” Doesn’t say much for loyalty.