Mixed strategy approach destabilizes cooperation in finite populations with clustering coefficient

Evolutionary game theory, encompassing discrete, continuous, and mixed strategies, is pivotal for understanding cooperation dynamics. Discrete strategies involve deterministic actions with a fixed probability of one, whereas continuous strategies employ intermediate probabilities to convey the extent of cooperation and emphasize expected payoffs. Mixed strategies, though akin to continuous ones, calculate immediate payoffs based on the action chosen at a given moment within intermediate probabilities. Although previous research has highlighted the distinct impacts of these strategic approaches on fostering cooperation, the reasons behind the differing levels of cooperation among these approaches have remained somewhat unclear. This study explores how these strategic approaches influence cooperation in the context of the prisoner's dilemma game, particularly in networked populations with varying clustering coefficients. Our research goes beyond existing studies by revealing that the differences in cooperation levels between these strategic approaches are not confined to finite populations; they also depend on the clustering coefficients of these populations. In populations with nonzero clustering coefficients, we observed varying degrees of stable cooperation for each strategic approach across multiple simulations, with mixed strategies showing the most variability, followed by continuous and discrete strategies. However, this variability in cooperation evolution decreased in populations with a clustering coefficient of zero, narrowing the differences in cooperation levels among the strategies. These findings suggest that in more realistic settings, the robustness of cooperation systems may be compromised, as the evolution of cooperation through mixed and continuous strategies introduces a degree of unpredictability.