Re-parametrizing the dilemmas: Comment on “Universal scaling for the dilemma strength in evolutionary games”, by Z. Wang et al.

Abstract

Understanding the mechanisms at the origin of cooperation is a major challenge to Darwin’s natural selection theory [1]: If only the fittest individuals survive, how to explain the ubiquity of cooperation? Among many examples of altruistic behaviors, one can mention insects that coordinate their efforts for the benefit of their “queen” [2]. In this context, Evolutionary Game Theory (EGT), see e.g. [3–7], provides a suitable framework to address this question by means of parsimonious models in which each agent’s reproductive potential (expected payoff or fitness) depends on the others’ fitness. In infinitely large populations, the frequency of the competing species/strategies evolves according to the underlying “replicator (nonlinear differential) equations”. In such a mean field setting, the evolutionary stable strategies of symmetric two-strategy games can readily be found in terms of two parameters directly obtained by rescaling the entries of the payoff matrix. In what follows, we will refer to this operation as the “replicator parametrization”. Interestingly, the fitness optimization at an individual level may cause the reduction of the population’s average fitness yielding a “social dilemma”, as in the celebrated two-strategy prisoner’s dilemma, snowdrift and stag-hunt games [3–7]. At mean field level, these cooperation dilemmas can easily and comprehensively be analyzed by using the replicator parametrization. However, when the size of the population is finite, stochastic effects play an important role and the notion of evolutionary stability needs to be refined: In the simple case of two-strategy games, a strategy is evolutionary stable (ESSN ) in a well-mixed finite population if it has a higher fitness than the alternative strategy, and if a single player adopting the mutant-strategy has a lower fixation probability than in the absence of selection. The latter condition accounts for the selection to oppose the replacement by the alternative strategy [9]. Due to these two conditions, whether a strategy is an ESSN cannot be inferred by simply rescaling the payoff matrix and a comprehensive analysis of the dilemmas can no longer be carried out by using only the replicator parametrization.