Fractional Replicator Dynamics

This paper studies how the temporal structure of adjustment shapes evolutionary dynamics in symmetric games. We introduce a fractional replicator dynamic that modifies the classical replicator only through the time operator, replacing the ordinary derivative with a fractional derivative of order α ∈ (0, 1]. This formulation preserves payoff monotonicity, feasibility, and the equilibrium set, while allowing past payoff differences to affect current behavior through long memory with power-law decay. We show that fractional time fundamentally alters local stability and equilibrium selection. While evolutionarily stable strategies remain locally asymptotically stable, equilibria that are unstable under the classical replicator can become locally stable when memory is sufficiently persistent, generating stability switching and purely temporal bifurcations without any change in payoffs or strategic interaction. Moreover, convergence toward stable equilibria becomes polynomial rather than exponential, implying slow adjustment even in simple games. As a consequence, equilibrium selection may fail to be completed over economically relevant horizons despite being guaranteed asymptotically. Finally, we provide a microfoundation based on standard payoff-monotone revision processes with heterogeneous and asynchronous revision opportunities. Fractional dynamics thus offer a parsimonious way to incorporate persistent memory into evolutionary models while preserving their core economic structure.