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

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 track the remaining expected payoff \textit{net }gains from switches, after deducting the costs and to be maximized subject to the constraints 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.