The goal of this paper is to present a universal and  intuitive approach to link static stability with dynamic stability, i. e., asymptotic stability of equilibrium, through economic principles behind evolutionary dynamics. In economics, static stability is traditionally defined as a notion about how incentives respond to choices : for a population game, it means that, on average, the payoff of a strategy responds negatively to an increase in the share of the strategy. This is characterized by negative definiteness of the Jacobian matrix of the payoff function and satisfied globally in a contractive game and locally at a regular ESS. We say an evolutionary dynamic is economically reasonable , if an agent' s choice of a new strategy can be justified as an optimal choice possibly by introducing additional costs and constraints ; for example, inertia can be justified by switching costs, and switches to suboptimal strategies can be by restriction to available strategies . This class of dynamics includes not only the best response dynamic, pairwise payoff comparison dynamicsand excess payoff dynamics . The key in our proof is a net gain from revision of the strategy , i.e., the payoff improvement by the revision minus the switching cost . Static stability implies that the aggregate net gain monotonically decreases over time under an economic reasonable dynamic and thus can be used as a Lyapunov function. Our Lyapunov function allows us to extend the dynamic stability to mixture of heterogeneous populations, who may follow different dynamics or different payoff functions. While our analysis here is confined to normal form gamesfor clear illustration of our approach, we argue that our approach is promising for further applications to more complex situations.
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.
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The goal of this paper is to present
This paper presents
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economically
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link static stability with dynamic stability, i. e., asymptotic stability of equilibrium, through economic principles behind evolutionary dynamics.
linking static and dynamic stability.
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as a notion about how incentives respond to choices
by negative relationship between endogenous and exogenous variables in a model
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it means that, on average, the payoff of a strategy responds negatively to an increase in the share of the strategy. This
this
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definiteness
semidefiniteness
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and satisfied globally in a contractive game and locally at a regular ESS. We say an evolutionary dynamic is economically reasonable , if an agent' s choice of a new strategy can be justified as an optimal choice
. We consider economically reasonable dynamics, in which we can justify agents' choices of new strategies as optimal choices
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; for example, inertia can be justified by switching costs, and switches to suboptimal strategies can be by restriction to available strategies
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includes not only the best response dynamic, pairwise payoff comparison dynamicsand excess payoff dynamics
covers major payoff-based (non-imitative) evolutionary dynamics
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in our proof is a net gain from revision of the strategy , i.e., the payoff improvement by the revision minus the switching cost
is expected net gains (payoff improvements) from strategy revisions after paying switching costs
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monotonically decreases
diminishes
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an economic reasonable dynamic
economic reasonable dynamics
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Our Lyapunov function allows us to extend the dynamic stability to mixture of heterogeneous populations, who may follow different dynamics or different payoff functions.
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normal form gamesfor clear illustration of our approach, we argue that
myopic evolutionary dynamics in population games,
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promising for further applications
applicable