Title: Simple property of heterogeneous aspiration dynamics: Beyond weak selection,
Abstract: 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 multi-player 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.