The Prisoner's Dilemma

The prisoner's dilemma, as it has classically been told, goes like this:

Two suspects are arrested for a certain crime (though it is not clear which, if either, are actually guilty). The two are separated and allowed no contact with each other, and each is offered an identical deal, with three different possible outcomes:

1. Each is offered the opportunity to testify against the other. If one agrees to testify but the other does not, the one who testifies will be released from prison, while the one who is testified against will be sent to prison for a 10 year sentence.
2. If neither testifies against the other, but both remain silent, then neither will be convicted, but both will have to serve a 6 month sentence anyway.
3. If both testify against each other, then they will both be forced to serve 6 year sentences for the crime.

The Question: What Will they Choose?

This is typical of the conundrum faced with game theorists, and it is certainly a tricky one. The more one reads through this list of options, the more one realizes just how difficult making a prediction of what a prisoner might choose (or should choose, even more importantly) in this situation must be.

One might be reminded, in fact, of the scene in The Princess Bride, where Vizzinni, the clever Sicilian, is attempting to figure out which glass of wine is poisoned by utilizing a hilariously skewed version of logical game theory.

But the question remains, and, despite the fact that it borders on the philosophical and psychological, there is still an element of mathematics to it. And what have the mathematicians decided?

Well that depends, of course, upon a great many conditions regarding the motives and morals of the two suspects, but given the notion that each of them is only looking to better themselves, caring not at all about the other person, statistics show that this would generally lead to both men turning on each other, testifying, and receiving the "lessened" penalty of 6 years in prison.

After all, it doesn't seem likely that either would reject the opportunity to lessen their own sentence, if by remaining quiet they retain the very distinct possibility of serving the full 10 years in prison. Testifying is the only way for them to ensure that they will not serve this full term. That, then, is the statistical answer.

Iterated Prisoner's Dilemma

An alternate version of this question is the iterated Prisoner's Dilemma. That is, a scenario where this situation is not just performed once, but is repeated multiple times.

Mathematically, what happens in this scenario is that at first, the prisoners do indeed both testify against each other in a selfish attempt to lessen their own penalties, though after several tries, they begin to learn that this is not serving them well, and eventually they will both learn that the best choice is to refuse to testify, thus lessening their sentence to only 6 months.

Here, in this reiterated version, the problem reaches a state of equilibrium, and it is here that this problem surely finds its greatest analogy to game theory in the "real world," where decisions are not made in isolated incidents only, but in the context of many others. Game theorists can thus use these principles to predict how decisions might progress over time as they perhaps approach a state of "equilibrium" - the point which is most beneficial to all sides involved.

This is just another example of the usefulness and complexity of game theory.

Painting of a Prisoner

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