Instability of cooperation in finite populations

Abstract

Evolutionary game theory has been developed primarily under the implicit assumption of an infinite population. We rigorously analyze a standard model for the evolution of cooperation (the multi-player snowdrift game) and show that in many situations in which there is a cooperative evolutionarily stable strategy (ESS) if the population is infinite, there is no cooperative ESS if the population is finite (no matter how large). In these cases, contributing nothing is a globally convergently stable finite-population ESS, implying that apparent evolution of cooperation in such games is an artifact of the infinite population approximation. The key issue is that if the size of groups that play the game exceeds a critical proportion of the population then the infinite-population approximation predicts the wrong evolutionary outcome (in addition, the critical proportion itself depends on the population size). Our results are robust to the underlying selection process.