Abstract
This thesis studies the convergence of finite-population stochastic evolutionary dynamics to their deterministic mean-field limit in continuous-time finite state space games. We first develop refined ergodic theorems for Markov chains with a single positive-recurrent class, guaranteeing the existence of a unique invariant distribution and almost-sure convergence of time averages. Next, we prove that the mean-field model, described by a system of Lipschitz-continuous ordinary differential equations, admits a unique solution that depends continuously on its initial condition and that constitutes the almost-sure limit for the empirical distributions with fixed policy. Furthermore, we show that every Mixed Stationary Nash Equilibrium of the mean-field game is approximated by a Nash equilibrium of the corresponding N-player game within an error ε for sufficiently large N. We finally demonstrate, by Kurtz's theorem, that the empirical state-policy distribution converges in probability to the mean-field trajectory. Numerical simulations conducted in MATLAB confirm the theoretical O(N^-1/2) convergence rate in both models across a range of population sizes.