Simple property of heterogeneous aspiration dynamics: Beyond weak selection

How individuals adapt their behavior in cultural evolution remains elusive. Theoretical studies have shown that the update rules chosen to model individual decision making can dramatically modify the evolutionary outcome of the population as a whole. This hints at the complexities of considering the personality of individuals in a population, where each one uses its own rule. Here, we investigate whether and how heterogeneity in the rules of behavior update alters the evolutionary outcome. We assume that individuals update behaviors by aspiration-based self-evaluation and they do so in their own ways. Under weak selection, we analytically reveal a simple property that holds for any two-strategy multiplayer games in well-mixed populations and on regular graphs: the evolutionary outcome in a population with heterogeneous update rules is the weighted average of the outcomes in the corresponding homogeneous populations, and the associated weights are the frequencies of each update rule in the heterogeneous population. Beyond weak selection, we show that this property holds for public goods games. Our finding implies that heterogeneous aspiration dynamics is additive. This additivity greatly reduces the complexity induced by the underlying individual heterogeneity. Our work thus provides an efficient method to calculate evolutionary outcomes under heterogeneous update rules.