Evolutionary stability in continuous nonlinear public goods games

Abstract

We investigate a type of public goods games played in groups of individuals who choose how much to contribute towards the production of a common good, at a cost to themselves. In these games, the common good is produced based on the sum of contributions from all group members, then equally distributed among them. In applications, the dependence of the common good on the total contribution is often nonlinear (e.g., exhibiting synergy or diminishing returns). To date, most theoretical and experimental studies have addressed scenarios in which the set of possible contributions is discrete. However, in many real-world situations, contributions are continuous (e.g., individuals volunteering their time). The "n-player snowdrift games" that we analyze involve continuously varying contributions. We establish under what conditions populations of contributing (or "cooperating") individuals can evolve and persist. Previous work on snowdrift games, using adaptive dynamics, has found that what we term an "equally cooperative" strategy is locally convergently and evolutionarily stable. Using static evolutionary game theory, we find conditions under which this strategy is actually globally evolutionarily stable. All these results refer to stability to invasion by a single mutant. We broaden the scope of existing stability results by showing that the equally cooperative strategy is locally stable to potentially large population perturbations, i.e., allowing for the possibility that mutants make up a non-negligible proportion of the population (due, for example, to genetic drift, environmental variability or dispersal).

Instability of cooperation in finite populations

“…As we have previously shown (30), if an NSG ( Methods §5.1) is played in an infinite population then there are always two (and only two) ESSs: Both ESSs are global, and both are locally convergently stable [theorem 4.1 of (30)]. At the cooperative ESS, everyone contributes an equal share of the amount that maximizes individual fitness given that everyone contributes equally.…”

“…Since all real populations are finite, it is important to understand how inferences based on infinitepopulation analyses of the multi-player snowdrift game ( e.g. , (3, 30, 42)) might be affected. More generally, under what circumstances are infinite-population analyses of the evolution of cooperation likely to lead to invalid inferences about real populations?…”

“…This biologically motivated version of the continuous snowdrift game (§2) was introduced in (30). We consider a population of individuals that are identical except (possibly) with respect to the strategy (contribution level) adopted when playing the snowdrift game.…”

“…To avoid mathematical complexities that are not relevant to the biological issues that concern us, we impose a few natural conditions on the cost and benefit functions and refer to the natural snowdrift game (NSG; see Methods §5.1 ). The NSG was introduced in (30), where it was shown that—when played in infinite populations—the game always has a cooperative ESS. Cost, benefit and fitness functions for an NSG example are shown in figure 1.…”

“…We show that evolutionary outcomes in finite and infinite populations can be dramatically different. In particular, for a class of snowdrift games for which a cooperative ESS exists in infinite populations (30), we find conditions under which there is no cooperative ESS when played in finite populations. This qualitative difference in predictions for finite and infinite populations can occur no matter how large the finite population is, and is universal in the sense that it is independent of the selection process (31).…”