18.9 : Prisoner's Dilemma II

The prisoner's dilemma is a classic game theory model where two crime suspects must decide whether to betray each other or cooperatively remain silent. The choices they make determine their respective sentences. The Nash equilibrium occurs when each suspect chooses the best option based on the other's likely decision.

For Suspect A:

1. If Suspect B stays silent, Suspect A benefits most by betraying. This is because betrayal results in no prison time versus one year if A were to also remain silent.
2. If Suspect B betrays, Suspect A's best option is still to betray, as betrayal leads to two years in prison instead of three if A were to remain silent.

For Suspect B:

1. If Suspect A stays silent, B should choose to betray. This is because betrayal allows B to walk free rather than serving a one-year sentence if B were to also remain silent.
2. If Suspect A betrays, B's best move is also to betray, as betrayal results in two years in prison instead of three if B were to remain silent.

The Nash equilibrium in this situation is when both suspects choose to betray each other. This outcome is stable because neither suspect can improve their situation by changing their decision unilaterally, as doing so would lead to a worse outcome if the other chooses betrayal. The fear of receiving a harsher sentence encourages both to betray rather than cooperate.

This equilibrium is also a dominant strategy equilibrium because, for both suspects, betrayal is the best choice regardless of what the other chooses. The dilemma highlights how rational decision-making based on self-interest can lead to a worse collective outcome, as both suspects would have received shorter sentences if they had trusted each other and remained silent.