The Annals of Applied Probability

Time averages, recurrence and transience in the stochastic replicator dynamics

Josef Hofbauer and Lorens A. Imhof

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We investigate the long-run behavior of a stochastic replicator process, which describes game dynamics for a symmetric two-player game under aggregate shocks. We establish an averaging principle that relates time averages of the process and Nash equilibria of a suitably modified game. Furthermore, a sufficient condition for transience is given in terms of mixed equilibria and definiteness of the payoff matrix. We also present necessary and sufficient conditions for stochastic stability of pure equilibria.

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Ann. Appl. Probab. Volume 19, Number 4 (2009), 1347-1368.

First available: 27 July 2009

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Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 91A22: Evolutionary games 92D25: Population dynamics (general)

Averaging principle Dirichlet distribution exclusion principle invariant distribution Lyapunov function Nash equilibrium stochastic asymptotic stability stochastic differential equation


Hofbauer, Josef; Imhof, Lorens A. Time averages, recurrence and transience in the stochastic replicator dynamics. The Annals of Applied Probability 19 (2009), no. 4, 1347--1368. doi:10.1214/08-AAP577.

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