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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Abstract

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.

Article information

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

Dates
First available: 27 July 2009

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1248700620

Digital Object Identifier
doi:10.1214/08-AAP577

Zentralblatt MATH identifier
05599227

Mathematical Reviews number (MathSciNet)
MR2538073

Subjects
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)

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

Citation

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. http://projecteuclid.org/euclid.aoap/1248700620.


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