We consider general evolutionary dynamics under persistent payoff heterogeneity and study the dynamic relation between the strategy composition over different types and the aggregate strategy distribution of the entire population. It is rigorously proven that continuity of either the revision protocol or the type distribution guarantees the existence of a unique solution trajectory . In many major evolutionary dynamics, an agent's switching rate between actions increases with the payoff gain from this switch, which causes nonaggregability: the current strategy composition must be identified to predict the transition of the aggregate strategy. Looking at the strategy composition, we retain equilibrium stationarity in general and  stability in potential games under admissible dynamics. Local stability of an equilibrium composition under any admissible dynamic can be tested by local stability of the corresponding aggregate equilibrium under the best response dynamic with i.i.d.~payoff perturbation. All the results are maintained under heterogeneity in revision protocols .
A general framework of evolutionary dynamics under heterogeneous populations is presented. The framework allows continuously many types of heterogeneous agents, heterogeneity both in payoff functions and in revision protocols and the entire joint distribution of strategies and types to influence the payoffs of agents. We clarify regularity conditions for the unique existence of a solution trajectory and for the existence of equilibrium. We confirm that equilibrium stationarity in general and equilibrium stability in potential games are extended from the homogeneous setting to the heterogeneous setting. In particular, a wide class of admissible dynamics share the same set of locally stable equilibria in a potential game through local maximization of the potential .
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We consider general
A general framework of
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persistent payoff heterogeneity and study the dynamic relation between the strategy composition over different types and the aggregate strategy distribution of the entire population. It is rigorously proven that continuity of either the revision protocol or the type distribution guarantees the
heterogeneous populations is presented. The framework allows continuously many types of heterogeneous agents, heterogeneity both in payoff functions and in revision protocols and the entire joint distribution of strategies and types to influence the payoffs of agents. We clarify regularity conditions for the unique
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unique solution trajectory . In many major evolutionary dynamics, an agent's switching rate between actions increases with the payoff gain from this switch, which causes nonaggregability: the current strategy composition must be identified to predict the transition of the aggregate strategy. Looking at the strategy composition, we retain
solution trajectory and for the existence of equilibrium. We confirm that
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equilibrium
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under admissible dynamics. Local stability of an equilibrium composition under any admissible dynamic can be tested by local stability of the corresponding aggregate equilibrium under the best response dynamic with i.i.d.~payoff perturbation. All the results are maintained under heterogeneity in revision protocols
are extended from the homogeneous setting to the heterogeneous setting. In particular, a wide class of admissible dynamics share the same set of locally stable equilibria in a potential game through local maximization of the potential