Why does a fixed ending make cooperation difficult in a repeated prisoner’s dilemma?
The normal-form table illustrates the concept of a saddlepoint, or entry, in a payoff matrix at which the expected gain of each participant (row or column) has the highest guaranteed payoff.
In a repeated prisoner’s dilemma, a fixed ending makes cooperation difficult due to a process of backward induction.1
When the number of rounds is known, players realize that on the final day, there is no future opportunity for their opponent to retaliate for noncooperation. Consequently, both players have a rational incentive to act selfishly (e.g., lower prices or confess) on that last day to maximize their individual gain.1
Because both players anticipate this outcome, the incentive to maintain cooperation collapses for the second-to-last day as well, as there is no reason to cooperate if the opponent is expected to defect on the final day. This logic cascades backward through every preceding round, leading rational players to choose noncooperation from the very beginning of the sequence.1
Cooperation can only be sustained when the game is repeated indefinitely or when the players do not know when the sequence will end, as this uncertainty preserves the threat of future retaliation and the potential for long-term mutual benefit.1