On the coexistence of cooperators, defectors and conditional cooperators in the multiplayer iterated Prisoner's Dilemma

TL;DR

This paper analyzes the coexistence of cooperators, defectors, and conditional cooperators in multiplayer iterated Prisoner's Dilemma using replicator dynamics, revealing stable coexistence for small groups but not for larger ones.

Contribution

It introduces a replicator dynamics model for multiplayer Prisoner's Dilemma, showing coexistence of strategies in small groups and the absence of such points in large groups.

Findings

Stable coexistence in groups up to 4 players

No interior coexistence points for groups of 5 or more

Analytical proof that no coexistence exists as group size approaches infinity

Abstract

Recent experimental evidence [Gruji\'c et al., PLoS ONE 5, e13749 (2010)] on the spatial Prisoner's Dilemma suggests that players choosing to cooperate or not on the basis of their previous action and the actions of their neighbors coexist with steady defectors and cooperators. We here study the coexistence of these three strategies in the multiplayer iterated Prisoner's Dilemma by means of the replicator dynamics. We consider groups with n = 2, 3, 4 and 5 players and compute the payoffs to every type of player as the limit of a Markov chain where the transition probabilities between actions are found from the corresponding strategies. We show that for group sizes up to n = 4 there exists an interior point in which the three strategies coexist, the corresponding basin of attraction decreasing with increasing number of players, whereas we have not been able to locate such a point for n = 5. We analytically show that in the infinite n limit no interior points can arise. We conclude by discussing the implications of this theoretical approach on the behavior observed in experiments.