Net gains in evolutionary dynamics: A unifying and intuitive approach to dynamic stability

TL;DR

This paper introduces a universal approach linking static and dynamic stability in evolutionary game theory, showing static stability guarantees dynamic stability when agents rationally account for switching costs, with net gains serving as a Lyapunov function.

Contribution

It provides a unifying, intuitive framework proving static stability implies dynamic stability under rationalizable switching costs, applicable beyond simple models.

Findings

Static stability guarantees dynamic stability with rationalizable switching costs.

Net gains from switches act as a Lyapunov function for stability analysis.

The approach explains previous negative results in evolutionary dynamics.

Abstract

Static stability in economic models means negative incentives for deviation from equilibrium strategies, which we expect to assure a return to equilibrium, i.e., dynamic stability, as long as agents respond to incentives. There have been many attempts to prove this link, especially in evolutionary game theory, yielding both negative and positive results. This paper presents a universal and intuitive approach to this link. We prove that static stability assures dynamic stability if agents' choices of switching strategies are rationalizable by introducing costs and constraints in those switching decisions. This idea guides us to define \textit{net }gains from switches as the payoff improvement after deducting the costs. Under rationalizable dynamics, an agent maximizes the expected net gain subject to the constraints. We prove that the aggregate maximized expected net gain works as a Lyapunov function. It also explains reasons behind the known negative results. While our analysis here is confined to myopic evolutionary dynamics in population games, our approach is applicable to more complex situations.