Nonaggregable evolutionary dynamics under payoff heterogeneity

We consider general evolutionary dynamics under idiosyncratic but persistent payoff heterogeneity and study the dynamic relation between the strategy composition over different payoff types and the aggregate strategy distribution of the entire population. It is rigorously proven that continuity of the switching rate function or the type distribution guarantees the existence of a unique trajectory. In major evolutionary dynamics except the standard best response dynamic, an agent's switching rate from the current action to a new action increases with the payoff gain from this switch. This payoff sensitivity makes a heterogeneous dynamic nonaggregable: the transition of the aggregate strategy generically depends not only on the current aggregate strategy but also on the current strategy composition. However, if we look at the strategy composition, stationarity of equilibrium in general and stability in potential games hold under any admissible dynamics. In particular, local stability of each individual equilibrium composition under an admissible dynamic is equivalent to that of the corresponding aggregate equilibrium in the aggregate dynamic induced from the standard best response dynamic, though the basin of attraction may differ over different dynamics.