Convergence, Fluctuations and Large Deviations for finite state Mean Field Games via the Master Equation

We show the convergence of finite state symmetric N-player differential games, where players control their transition rates from state to state, to a limiting dynamics given by a finite state Mean Field Game system made of two coupled forward-backward ODEs. We exploit the so-called Master Equation, which in this finite-dimensional framework is a first order PDE in the simplex of probability measures, obtaining the convergence of the feedback Nash equilibria, the value functions and the optimal trajectories. The convergence argument requires only the regularity of a solution to the Master equation. Moreover, we employ the convergence method to prove a Central Limit Theorem and a Large Deviation Principle for the evolution of the N-player empirical measures. The well-posedness and regularity of solution to the Master Equation are also studied

On the convergence problem in Mean Field Games: a two state model without uniqueness

We consider N-player and mean field games in continuous time over a finite horizon, where the position of each agent belongs to {-1,1}. If there is uniqueness of mean field game solutions, e.g. under monotonicity assumptions, then the master equation possesses a smooth solution which can be used to prove convergence of the value functions and of the feedback Nash equilibria of the N-player game, as well as a propagation of chaos property for the associated optimal trajectories. We study here an example with anti-monotonous costs, and show that the mean field game has exactly three solutions. We prove that the value functions converge to the entropy solution of the master equation, which in this case can be written as a scalar conservation law in one space dimension, and that the optimal trajectories admit a limit: they select one mean field game soution, so there is propagation of chaos. Moreover, viewing the mean field game system as the necessary conditions for optimality of a deterministic control problem, we show that the N-player game selects the optimizer of this problem

On the convergence problem in mean field games: A two state model without uniqueness

We consider $N$-player and mean field games in continuous time over a finite horizon, where the position of each agent belongs to ${-1,1}$. If there is uniqueness of mean field game solutions, e.g., under monotonicity assumptions, then the master equation possesses a smooth solution which can be used to prove convergence of the value functions and of the feedback Nash equilibria of the $N$-player game, as well as a propagation of chaos property for the associated optimal trajectories. We study here an example with antimonotonous costs and show that the mean field game has exactly three solutions. We prove that the value functions converge to the entropy solution of the master equation, which in this case can be written as a scalar conservation law in one space dimension, and that the optimal trajectories admit a limit: they select one mean field game solution, so there is propagation of chaos. Moreover, viewing the mean field game system as the necessary conditions for optimality of a deterministic control problem, we show that the $N$-player game selects the optimizer of this problem

Topics in finite state mean field games and non-Markovian interacting spin systems

This Dissertation is devoted to the study of large stochastic systems of small interacting individuals and their macroscopic limit formulations, under symmetric properties of the interactions. The examples we consider belong to two separate contexts, depending on whether the individuals can control their dynamics or not. In the first case, treated in Part 1, we fall into the framework of N-player and mean field games, while in the latter, analyzed in Part 2, the resulting models are examples of interacting particle systems. More specifically, in the first part (Chapters 1-2) we focus on the convergence problem in mean field games, i.e. on the rigorous justification of mean field games as limits, when the number of players tends to infinity, of Nash equilibria of symmetric non-zero sum non-cooperative N-player games. In particular, we study finite state mean field games, where the state of each player belongs to a discrete finite space, analyzing separately the uniqueness case (Chapter 1) and a scenario with non-uniqueness of solutions to the mean field game (Chapter 2). In the second part of the Dissertation (Chapters 3-4) we study some examples of interacting spin systems, with non-Markovian individual dynamics, arising as proper modifications of classical ferromagnetic mean field spin systems dynamics. In particular, we focus on two mechanisms for relaxing the Markovianity: a state augmentation procedure, and the insertion of memory effects in the evolution. While one of the goals is still to rigorously justify the passage to a macroscopic description, the models of Part 2 present some features of independent interest, including phase transitions (Chapters 3-4), the emergence of self-sustained oscillations (Chapter 3), and the presence of multiple spatio-temporal scales phenomena (Chapter 4)

Topics in finite state mean field games and non-Markovian interacting spin systems

This Dissertation is devoted to the study of large stochastic systems of small interacting individuals and their macroscopic limit formulations, under symmetric properties of the interactions. The examples we consider belong to two separate contexts, depending on whether the individuals can control their dynamics or not. In the first case, treated in Part 1, we fall into the framework of N-player and mean field games, while in the latter, analyzed in Part 2, the resulting models are examples of interacting particle systems. More specifically, in the first part (Chapters 1-2) we focus on the convergence problem in mean field games, i.e. on the rigorous justification of mean field games as limits, when the number of players tends to infinity, of Nash equilibria of symmetric non-zero sum non-cooperative N-player games. In particular, we study finite state mean field games, where the state of each player belongs to a discrete finite space, analyzing separately the uniqueness case (Chapter 1) and a scenario with non-uniqueness of solutions to the mean field game (Chapter 2). In the second part of the Dissertation (Chapters 3-4) we study some examples of interacting spin systems, with non-Markovian individual dynamics, arising as proper modifications of classical ferromagnetic mean field spin systems dynamics. In particular, we focus on two mechanisms for relaxing the Markovianity: a state augmentation procedure, and the insertion of memory effects in the evolution. While one of the goals is still to rigorously justify the passage to a macroscopic description, the models of Part 2 present some features of independent interest, including phase transitions (Chapters 3-4), the emergence of self-sustained oscillations (Chapter 3), and the presence of multiple spatio-temporal scales phenomena (Chapter 4)