Pietro Grassi proves O(N^{-1/2}) convergence for evolutionary dynamics in finite games

Pietro Grassi has published a rigorous bachelor's thesis on arXiv (2606.11193) that establishes fundamental approximation properties of evolutionary dynamics in continuous-time finite state space games. The work bridges stochastic finite-population models with their deterministic mean-field limits, a critical area for multi-agent systems and game-theoretic AI.

The thesis first refines ergodic theorems for Markov chains with a single positive-recurrent class, guaranteeing unique invariant distributions and almost-sure convergence of time averages. It then proves that the mean-field model—described by Lipschitz-continuous ODEs—has a unique solution depending continuously on initial conditions, which serves as the almost-sure limit for empirical distributions under fixed policies. Most notably, Grassi shows that every mixed stationary Nash equilibrium of the mean-field game approximates a Nash equilibrium of the corresponding N-player game within error epsilon for sufficiently large N. Using Kurtz's theorem, he demonstrates convergence in probability of the empirical state-policy distribution to the mean-field trajectory. Numerical simulations in MATLAB confirm the theoretical O(N^{-1/2}) convergence rate across a range of population sizes.