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.
Static stability in an economic model typically 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 offers a universal and intuitive approach to this link. We prove static stability assures dynamic stability as long as agents' decisions of switching strategies are rationalizable by revealing costs and constraints behind distortions from exact optimization. This idea guides us to track the remaining expected maximal payoff gain from switches, after deducting the costs and to be maximized subject to the constraints, as a disequilibrium index, namely, a Lyapunov function. While our analysis here is confined to myopic evolutionary dynamics in population games, our approach is applicable to more complex situations.
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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
Static stability in an economic model typically 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 offers a universal and intuitive approach to this link. We prove static stability assures dynamic stability as long as agents' decisions of switching strategies are rationalizable by revealing
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. 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
behind distortions from exact optimization. This idea guides us to track the remaining expected maximal payoff gain from switches, after deducting the costs and to be maximized subject to the constraints, as a disequilibrium index, namely,