More Players, Less Cooperation? Evolution of Conditional Cooperation in the N-player Iterated Prisoner’s Dilemma Game

The N-player iterated prisoner’s dilemma (IPD) game well manifests conflicts of interest in social dilemma, such as contribution vs. free-riding in digital commons and sharing data or not in federated learning. This paper explores the effects of the benefit-to-cost ratio and the number of players on the emergence of cooperation in the N-player IPD game. Theoretical conditions on the benefit-to-cost ratio are first derived to ensure the optimality of simultaneous cooperation for any player subset, thereby precluding partial defection and promoting conditional cooperation among the players. A conditional-cooperation mechanism based on social comparison is then proposed to evolve the expected cooperation. Simulation results demonstrate that the proposed mechanism effectively facilitates the evolution of conditional cooperation. Interestingly, a larger number of players leads to a higher cooperation level when the value of the benefit-to-cost ratio over the number of players is kept constant, in that the players’ incentive to cooperate scales with the number of players in the game while the per capita benefit of cooperation remains invariant. This holds in both well-mixed and network-structured interactions. Furthermore, breaking down boundaries of clusters of players is observed to further promote the conditional cooperation, contrasting with prior findings that clustering fosters cooperation. This occurs because low clustering facilitates the spread of conditional cooperation among players under social comparison. Finally, the obtained conditional-cooperation results prove quite robust against uncertain payoffs and noise occurrences.