Statistics of the number of equilibria in random social dilemma evolutionary games with mutation

Abstract

In this paper, we study analytically the statistics of the number of equilibria in pairwise social dilemma evolutionary games with mutation where a game's payoff entries are random variables. Using the replicator-mutator equations, we provide explicit formulas for the probability distributions of the number of equilibria as well as other statistical quantities. This analysis is highly relevant assuming that one might know the nature of a social dilemma game at hand (e.g., cooperation vs coordination vs anti-coordination), but measuring the exact values of its payoff entries is difficult. Our delicate analysis shows clearly the influence of the mutation probability on these probability distributions, providing insights into how varying this important factor impacts the overall behavioural or biological diversity of the underlying evolutionary systems.

Cost-efficient interventions for promoting fairness in the ultimatum game

“…Note however that, for simplicity, previous works have either omitted mutation (Han andTran-Thanh, 2018, Wang et al, 2019), or assumed that it is infinitely small (for analytical treatment) Han, 2021a, Han and. Mutation (behavioural exploration), where agents can freely experiment with new behaviours, is usually non-negligible in real populations and has been shown to play an important role in enabling cooperation in the context of social dilemmas (Antal et al, 2009, Duong and Han, 2019, 2021b, Han et al, 2012, Rand et al, 2013. Thus, the present work will also advance the state-of-the-art in this respect, where we will closely examine how different regimes of mutation, or agents' propensity for behavioural exploration, influence the manner in which external interference should be carried out.…”