The Prisoners Dilemma

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Originator:

Albert W. Tucker

Publication:

Tucker, Albert (1950) "A two-person dilemma", Stanford University Press

The Paradox:

Two men are found near the scene of a bank robbery. Both men are found to be carrying concealed weapons and are taken into custody. Each man is interrogated separately and because of a lack of evidence, the investigating officer offers to cut each man a deal:

The officer to the first man: "If you confess to the robbery then we'll cut you a deal. Implicate your partner in the robbery and we'll let you go free, dropping the firearms charges we have on you."

The man responds: "What will happen to my partner if I do?"

The officer: "He'll be prosecuted to the full extent of the law. We'll have him up for the robbery and on firearms charges. But you will go free."

The man responds: "And if my partner confesses too?"

The officer: "Then you'll both be prosecuted but since you've both cooperated we'll only push for a lenient penalty"

A similar conversation is carried out with the second man. Both men have a choice, to confess or stay silent. They are kept in isolation and can't discuss what they might do.

We are to assume that both men will act rationally to get the least sentence for themselves.

Firearms charges receive one year, bank robbery receives ten years while the reduced charges will receive five years. The outcomes can be summarized as follows:

Argument 1:

Looking at his options, the first man surmises: "If I confess then I'll either get five years or go free. If I stay silent then I'll either get eleven years or one year. Both outcomes for confessing are better than their respective outcomes for staying silent, therefore I will confess."

But then it dawns on him "My partner is also going to act rationally, so he'll also confess and we'll both go away for five years!"

Argument 2:

Having followed the same line of reasoning, the second man ponders "Since we are both rational, we will both make the same decision. Therefore we can either both confess or both stay silent. If we both confess then we go away for five years but if we both stay silent then we'll both only go away for one year. We'll be staying silent."

Thus two apparently acceptable arguments lead to a paradox.

The Solution:

The problem with the argument 1 is that it assumes that acting rationally requires the two men to make the same decision but it doesn't take that into account when summing up the possible outcomes. If both men are to act the same then the possibility of the first man confessing and the second man staying silent can't exist. Likewise the possibility of the first man staying silent and the second man confessing can't exist.

Argument 2 correctly takes this into account, giving the only two possible outcomes.

Footnote:

If it is not known whether the other man will act rationally or not then it is valid to sum up as in Argument 1. This strategy may be better suited to the real world (where men are seldom rational).

Another point is highlighted. Is it rational to seek the least penalty for yourself? What if the goal were to seek the least penalty for your partner? In that case, staying silent is the dominant strategy regardless of your partner's rationality.