Strength asymmetry and replacement bias shape the evolution of cooperation in finite populations.

Abstract

Heterogeneity is common in real populations, but many classic evolutionary game models assume identical individuals. Here, we study how two simple forms of heterogeneity affect the evolution of cooperation in a finite population. We construct a model of evolution in a finite population with two heterogeneous subpopulations, referred to as weak and strong subpopulations. The heterogeneity is reflected in two aspects: strength asymmetry-the probability that a weak individual defeats a strong one, and replacement bias, which renders one subpopulation more likely to be updated than the other. Under weak selection, we employ the expected fixation probability framework to derive an explicit threshold for the benefit-cost ratio that determines when cooperation is favoured. We find three main results. First, when replacement is symmetric, increasing the proportion of weak individuals can enhance cooperation, with the strongest effect at an intermediate size imbalance. Second, when only replacement bias varies, the cooperation threshold increases with the bias, but the reason differs on either side of the symmetric point. Finally, when strong individuals are more likely to be updated, there is a clear 'sweet spot', where stronger strength asymmetry and stronger replacement bias jointly and monotonically promote cooperation.