The paper analyzes an evolutionary model with noise or mutations to study long-run equilibria in games. It introduces a discrete framework to determine the most likely long-run equilibrium, which is essentially the stochastically stable equilibrium. For symmetric 2x2 games with two strict Nash equilibria, the long-run equilibrium satisfies Harsanyi and Selten's risk-dominance criterion. The model considers bounded rationality, where players learn through trial and error, and mutations introduce randomness. The analysis shows that in coordination games, the long-run equilibrium coincides with the risk-dominant equilibrium. The paper also discusses the implications of mutations on equilibrium selection, showing that the long-run equilibrium depends on the likelihood of large jumps in strategy distribution. The results are robust to the specifics of the deterministic dynamics and highlight the importance of risk dominance in evolutionary game theory. The study provides a framework for understanding how players adapt over time and how equilibrium selection is influenced by stochastic shocks.The paper analyzes an evolutionary model with noise or mutations to study long-run equilibria in games. It introduces a discrete framework to determine the most likely long-run equilibrium, which is essentially the stochastically stable equilibrium. For symmetric 2x2 games with two strict Nash equilibria, the long-run equilibrium satisfies Harsanyi and Selten's risk-dominance criterion. The model considers bounded rationality, where players learn through trial and error, and mutations introduce randomness. The analysis shows that in coordination games, the long-run equilibrium coincides with the risk-dominant equilibrium. The paper also discusses the implications of mutations on equilibrium selection, showing that the long-run equilibrium depends on the likelihood of large jumps in strategy distribution. The results are robust to the specifics of the deterministic dynamics and highlight the importance of risk dominance in evolutionary game theory. The study provides a framework for understanding how players adapt over time and how equilibrium selection is influenced by stochastic shocks.