Evolutionary dynamics of collective-risk social dilemma game with risk uncertainty and adaptive strategy

Abstract

The collective-risk dilemma game constitutes the leading metaphor for studying the evolution of cooperation in populations of selfish individuals. Although most theoretical studies often consider the risk level as a fixed parameter, this may not fully reflect the uncertain nature of risk detected in real-world scenarios. Motivated by the observation that individuals can infer their game environment and accordingly adjust their actual behavior, here we investigate this game with uncertain risk. In particular, we introduce a type of adaptive agents who incur an observation cost to infer the risk level during game interactions, and subsequently decide whether to cooperate or defect based on their tolerance thresholds. We first calculate the fixation probabilities of three different strategies in a mutation-free regime. Our results show that adaptive agents are favored by natural selection at low observation costs and moderate tolerance thresholds. Moreover, the fixation of adaptive agents results in higher levels of cooperation when both parameters are relatively low. We then consider behavioral mutation and systematically explore how average cooperation level and group achievement vary with observation costs, tolerance thresholds, and mutation rates. Importantly, the main results remain intact when the mutation is considered. Furthermore, when the tolerance threshold is small, the presence of adaptive agents results in a higher cooperation level and group achievement compared to the traditional setup when only unconditional cooperators and defectors are present. This advantage, however, gradually erodes with an increasing tolerance threshold and these two quantities fall below the levels observed in the classic two-strategy case.