Theoretical Economics 11 (2016), 279–355

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### Large deviations and stochastic stability in the small noise double limit

*William H. Sandholm, Mathias Staudigl*

#### Abstract

We consider a model of stochastic evolution under general noisy best response protocols, allowing the probabilities of suboptimal choices to depend on their payoff consequences. Our analysis focuses on behavior in the small noise double limit: we first take the noise level in agents' decisions to zero, and then take the population size to infinity. We show that in this double limit, escape from and transitions between equilibria can be described in terms of solutions to continuous optimal control problems. These are used in turn to characterize the asymptotics of the the stationary distribution, and so to determine the stochastically stable states.
We use these results to perform a complete analysis of evolution in three-strategy coordination games that satisfy the marginal bandwagon property and that have an interior equilibrium, with agents following the logit choice rule.

Keywords: Evolutionary game theory, equilibrium breakdown, stochastic stability, large deviations

JEL classification: C72, C73

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