Prisoner’s Dilemma

Russell and Norvig present the standard formulation. Two suspects, Alice and Bob, are interrogated separately. If one testifies against the other and the other refuses, the one who testifies goes free while the other serves ten years. If both testify, each serves five. If both refuse, each serves one year on a lesser charge. Each cares only about their own sentence.

The payoff matrix therefore has both testifying at −5 each, both refusing at −1 each, and the asymmetric cases at 0 for the one who testifies and −10 for the one who refuses. Reasoning case by case, testifying is better for Alice whether Bob testifies or refuses, so it dominates. The same holds for Bob.

Both act on that reasoning and receive five years, when refusing together would have given each one year. The equilibrium is Pareto dominated: an outcome exists that both would prefer. Nothing has gone wrong with the reasoning; each conclusion is individually correct, and the difficulty is that the jointly better outcome is not sustainable, since either player could improve their own position by defecting from it.

The structure recurs far beyond its narrative framing, wherever a shared benefit requires individual restraint that is individually costly. What changes the analysis is repetition. In a game played once, defection is unanswerable; when the same players interact repeatedly and care about future rounds, cooperation can be sustained because defection can be punished later, which is why the finite and indefinitely repeated versions behave so differently.

The payoff matrix, with each cell giving Alice’s outcome then Bob’s: both testify, (−5, −5); Alice testifies while Bob refuses, (0, −10); Alice refuses while Bob testifies, (−10, 0); both refuse, (−1, −1).

Alice compares columns. Against Bob testifying she scores −5 by testifying versus −10 by refusing. Against Bob refusing she scores 0 versus −1. Testifying wins in both, by 5 in the first case and by 1 in the second.

Bob’s matrix is symmetric, so he reasons identically. The result is (−5, −5), while (−1, −1) sat available and was better for both. Neither can reach it unilaterally: from mutual refusal, either could improve from −1 to 0 by defecting, which is exactly why it does not hold.