The mean-field game (MFG) studies the Nash equilibrium of a non-cooperative game involving a continuum of players, with many applications in economics, epidemics, crowd motion, data science, and more. In an MFG, each player seeks to optimize a strategy that minimizes their individual cost in response to a given state distribution of the entire player population. The individual best strategies collectively shape a new state distribution of the population, and the equilibrium of an MFG is the fixed point of this interaction. However, simple fixed-point iterations do not always guarantee convergence. Fictitious play is a very simple iterative algorithm that leverages a best-response mapping combined with a weighted average of the best response and previous responses. In this talk, I will first present a simple and unified convergence analysis with an explicit convergence rate for the fictitious play algorithm in MFGs of general types, especially non-potential MFGs. Based on this analysis, we propose several numerical strategies to accelerate fictitious play. Then, I will present a simple and effective iterative strategy, Equilibrium Correction Iteration (ECI), to solve a class of inverse MFG problems, where simple Nash equilibrium state measurements can be used to infer the unknown ambient potential, such as obstacles.
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