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Dynamical aspects of evolutionary stability. (English) Zbl 0734.92020

The paper is a pretty exercise in functional analysis, and generalizes results on stability and (strong) uninvadability of evolutionary equilibria to fairly general settings. The set of possible strategies is an arbitrary measure space S, and a state P is a probability distribution on S. \(F_ p(x)\) denotes the mean payoff of an individual playing \(x\in S\) against a population in state P (frequency dependent selection).
Assuming that the map \(Q\to F_ Q\) has a Fréchet derivative \(DF_ P\) at P, Theorem 1 says that P is evolutionarily stable if \[ \int DF_ P(Q-P)d(Q-P)<0\text{ for all } states\quad Q; \] and P is strongly uninvadable if there exists \(c>0\) s.t. \[ \int DF_ P(Q-P)d(Q-P)\leq - c.\| Q-P\|^ 2\text{ for all states } Q. \] Replicator dynamics on the state space is introduced by properly formalizing the idea that “successful” strategies should become relatively more frequent over time. Theorem 2 says that if P is strongly uninvadable, then every trajectory of the replicator dynamics starting close enough to P (in the Kullback-Leibler distance) converges to P. The last section gives a dynamical characterization of evolutionary stability resp. uninvadability for populations in which the individuals play mixed strategies.
Reviewer: M.Nermuth (Wien)

MSC:

92D15 Problems related to evolution
91A40 Other game-theoretic models
91A80 Applications of game theory
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References:

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