On the iterated prisoner's dilemma in a finite population

Two standard assumptions in analytical work on the iterated prisoner's dilemma are that the population is infinite, and that opponents—though randomly selected—are fixed for the duration of the game. This paper explores the consequences of relaxing both assumptions. It is shown in particular that if opponents are drawn at random throughout the game, then stable cooperation via reciprocity requires both that the probability of a further interaction be sufficiently high—higher than when opponents are fixed—and that the population not exceed a certain critical size, which depends on the probability of further interaction.