Instability of cooperation in finite populations

Abstract

7Evolutionary game theory has been developed primarily under the implicit assump-8 tion of an infinite population. We rigorously analyze a standard model for the evolution 9 of cooperation (the multi-player snowdrift game) and show that in many situations in 10 which there is a cooperative evolutionarily stable strategy (ESS) if the population is 11 infinite, there is no cooperative ESS if the population is finite (no matter how large). 12 In these cases, contributing nothing is a globally convergently stable finite-population 13 ESS, implying that apparent evolution of cooperation in such games is an artifact of the 14 infinite population approximation. The key issue is that if the size of groups that play 15 the game exceeds a critical proportion of the population then the infinite-population 16 approximation predicts the wrong evolutionary outcome (in addition, the critical pro-17 portion itself depends on the population size). Our results are robust to the underlying 18 selection process. 19cooperation | public goods games | evolutionary game theory ORCID IDs of authors: CM Many evolutionary games assume-for mathematical convenience-that populations are in-21 finitely large (e.g., (1-7)). This assumption is sometimes justified on the grounds that 22 "[p]opulations which stay numerically small quickly go extinct by chance fluctuations" (8, 23 §2.1). Of course, all real populations are finite, and important differences in evolutionary 24 dynamics between finite and infinite populations have been demonstrated (9-15). In spite 25 of the technical challenges of working with finite populations, some exact analytical results 26 have been obtained for two-player games with discrete strategy sets (9, 12,(14)(15)(16). How-27 ever, most existing finite-population results rely on approximation methods and simulations 28 (11, 15,(17)(18)(19)(20)(21). Notably, almost all finite-population results involve discrete strategy sets, 29 such as when individuals must choose between making a fixed positive contribution to a 30 public good, or nothing at all (e.g., (9, 12,(14)(15)(16)). Yet, evolutionary games involving con-31 tinuous strategy sets (e.g., allocating time or effort to a communal task) are both widely 32 applicable and extensively studied using infinite-population models (22). Moreover, to our 33 knowledge, all existing results for finite populations depend on a choice of selection process 34 (e.g., Moran or Wright-Fisher (23, 24)). 35Here, we present mathematically rigorous results that identify critical differences in the 36 predictions of evolutionary games in finite and infinite populations. We focus on a standard 37 model for exploring the evolution of cooperation-the continuous multi-player snowdrift 38 game (3)-which has previously been studied in infinite populations using exact analysis 39 and simulations (3, 7,(25)(26)(27) and in finite populations using approximations and simulations 40 (11, 21, 28, 29). 41 We show that evolutionary outcomes in finite and infinite populations can be dramatically 4...