Large Deviations and Stochastic Stability in the Small Noise Double Limit, I: Theory

Sandholm WH, Staudigl M. Large Deviations and Stochastic Stability in the Small Noise Double Limit, I: Theory. Center for Mathematical Economics Working Papers. Vol 505. Bielefeld: Center for Mathematical Economics; 2014.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. The control problems are tractable in certain interesting cases, allowing analytical descriptions of the escape dynamics and long run behavior of the stochastic evolutionary process