Gains in evolutionary dynamics: A unifying and intuitive approach to linking static and dynamic stability

This paper presents a universal and economically intuitive approach to linking static and dynamic stability. In economics, static stability is traditionally defined by negative relationship between endogenous and exogenous variables in a model: for a population game, this is characterized by negative semidefiniteness of the Jacobian matrix of the payoff function. We consider economically reasonable dynamics, in which we can justify agents' choices of new strategies as optimal choices possibly by introducing additional costs and constraints. This class of dynamics covers major payoff-based (non-imitative) evolutionary dynamics. The key is expected net gains (payoff improvements) from strategy revisions after paying switching costs. Static stability implies that the aggregate net gain diminishes over time under economic reasonable dynamics and thus can be used as a Lyapunov function. While our analysis here is confined to myopic evolutionary dynamics in population games, our approach is applicable to more complex situations.