In yesterday’s hanging example of the Prisoner’s Paradox, we presented the case of Alice and Ben, who are interrogated for a crime they might have committed. They are held in separate cells with no means of communication. The prosecutor offers each a deal. Since he is an honest prosecutor, he discloses to each the same deal was offered to the other prisoner. The deal is

If you confess you two committed the crime and your partner denies it, we will let you go free and use your testimony to get your partner five years.

If you both plead not guilty, we have enough circumstantial evidence to convict at a lower charge and you’ll both get two years.

If both of you confess, you’ll both get the usual four year sentences.

With some thought, both come to the conclusion that they are better off to confess (i.e. cooperate with the authorities), but that action leads to more time served than if both deny committing the crime (i.e. defect - usual notation).

To see this better, consider the actions and the payoff:

R = Reward for mutual cooperation
S = Sucker’s payoff
T = Temptation to defect
P = Punishment for mutual defection

Then we can define the Prisoner’s Dilemma to be a situation with a preference ranking of the four payoffs from best to worst as, T, R, P, S with the conditions that R is greater than the average of T and S. Any situation with these conditions is called a “Prisoner’s Dilemma”. This model results in a condition in which the rational action of each individual is to not cooperate (that is, to defect), yet, if both parties act rationally, each party’s reward is less that it would have been if both acted irrationally and cooperated!

Obviously this model can appear in many real world situations from genetics to business to politics. However, in the real world, many things can happen to change the payoff matrix. For instance, suppose that Alice and Ben were lovers and she knows that he will sacrifice himself for her. So if she confesses, he will likely plead not guilty to save her.

Or suppose they are both familiar with this paradox and so both decide to plead not guilty.

Suppose that they know the prosecutor is famous for making this proposal and that most of the time one of the prisoners cracks. Does that change the decision?

This relatively simple paradox has many aspects, and not all of them have been resolved to everyone’s satisfaction. We could go on for days about this one example, but you can google on “Prisoner” and “Paradox” and get more analyses than you can shake a big stick at.

For those who wish to delve further into decision theory without wading through a lot of equations, I have posted a tutorial on elementary decision theory. It shows examples of faulty physicians’ diagnoses (important for those considering surgery) and how to evaluate anti-terrorist activities (important for everyone). That tutorial can be found here.