This monograph introduces discrete-time mean field games with finite state and action spaces, emphasizing optimization formulations and learning algorithms. Starting from Markov decision processes and finite-player stochastic games, it develops the notion of mean field Nash equilibrium, establishes existence and uniqueness results under suitable assumptions, and explains how mean field equilibria yield approximate equilibria in large finite-player games. Building on the linear programming formulation of Markov decision processes, occupation measures and primal-dual formulations connect equilibrium computation with mathematical optimization. The monograph presents fixed-point iterations, fictitious play, online mirror descent, and optimization-based methods, discussing their convergence properties and limitations. Numerical examples illustrate crowd motion, traffic routing, epidemic management, and the computation of multiple equilibria. It combines examples, theoretical results, and implementable algorithms, with bibliographic notes providing connections to the broader literature.
This monograph provides an introduction to mean field reinforcement learning through the lens of Markov decision processes arising from large-population stochastic control with mean field interactions and common noise. Starting from the connection between multi-agent reinforcement learning and mean field control, it develops the probabilistic, mathematical, and control-theoretic framework needed to formulate representative-agent learning problems, analyze their relationship with finite-population systems, and study both general and linear-quadratic models. The presentation includes dynamic programming principles, propagation-of-chaos limits, and theoretical analyses of tabular Q-learning and policy-gradient methods. It also discusses numerical implementations, including tabular schemes and deep reinforcement learning methods such as deep deterministic policy gradient. The goal is to give readers a coherent bridge between mean field control theory and reinforcement learning methodology, emphasizing the mathematical structure of the problems and the design of tractable learning approaches for large stochastic populations.