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Within the literature on noncooperative game theory, there have been a number of algorithms which will compute Nash equilibria. We show that the family of algorithms known as Markov chain Monte Carlo (MCMC) can be used to calculate Nash equilibria. MCMC is a type of Monte Carlo simulation that relies on Markov chains to ensure its regularity conditions. MCMC has been widely used throughout the statistics and optimization literature, where variants of this algorithm are known as simulated annealing. We show that there is interesting connection between the trembles that underlie the functioning of this algorithm and the type of Nash refinement known as trembling hand perfection. We show that it is possible to use simulated annealing to compute this refinement.