Multi-player games on the cycle

In multi-player games n individuals interact in any one encounter and derive a payoff from that interaction. We assume that individuals adopt one of two strategies, and we consider symmetric games, which means the payoff depends only on the number of players using either strategy, but not on any particular configuration of the encounter. On the cycle we assume that any string of n neighbouring players interacts. We study fixation probabilities of stochastic evolutionary dynamics. We derive analytical results on the cycle both for linear and exponential fitness for any intensity of selection, and compare those to results for the well-mixed population. As particular examples we study multi-player public goods games, stag hunt games and snowdrift games.