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

Probabilistic Approach to Finite State Mean Field Games

We study mean field games and corresponding N-player games in continuous time over a finite time horizon where the position of each agent belongs to a finite state space. As opposed to previous works on finite state mean field games, we use a probabilistic representation of the system dynamics in terms of stochastic differential equations driven by Poisson random measures. Under mild assumptions, we prove existence of solutions to the mean field game in relaxed open-loop as well as relaxed feedback controls. Relying on the probabilistic representation and a coupling argument, we show that mean field game solutions provide symmetric \u3b5_N-Nash equilibria for the N-player game, both in open-loop and in feedback strategies (not relaxed), with \u3b5_N 64constant 1aN. Under stronger assumptions, we also find solutions of the mean field game in ordinary feedback controls and prove uniqueness either in case of a small time horizon or under monotonicity

Weak solutions to the master equation of potential mean field games

The purpose of this work is to introduce a notion of weak solution to the master equation of a potential mean field game and to prove that existence and uniqueness hold under quite general assumptions. Remarkably, this is achieved without any monotonicity constraint on the coefficients. The key point is to interpret the master equation in a conservative sense and then to adapt to the infinite dimensional setting earlier arguments for hyperbolic systems deriving from a Hamilton-Jacobi-Bellman equation. Here, the master equation is indeed regarded as an infinite dimensional system set on the space of probability measures and is formally written as the derivative of the Hamilton-Jacobi-Bellman equation associated with the mean field control problem lying above the mean field game. To make the analysis easier, we assume that the coefficients are periodic, which allows to represent probability measures through their Fourier coefficients. Most of the analysis then consists in rewriting the master equation and the corresponding Hamilton-Jacobi-Bellman equation for the mean field control problem as partial differential equations set on the Fourier coefficients themselves. In the end, we establish existence and uniqueness of functions that are displacement semi-concave in the measure argument and that solve the Hamilton-Jacobi-Bellman equation in a suitable generalized sense and, subsequently, we get existence and uniqueness of functions that solve the master equation in an appropriate weak sense and that satisfy a weak one-sided Lipschitz inequality. As another new result, we also prove that the optimal trajectories of the associated mean field control problem are unique for almost every starting point, for a suitable probability measure on the space of probability measures

Finite State N-player and Mean Field Games

Mean field games represent limit models for symmetric non-zero sum dynamic games when the number N of players tends to infinity. In this thesis, we study mean field games and corresponding N- player games in continuous time over a finite time horizon where the position of each agent belongs to a finite state space. As opposed to previous works on finite statemean field games, we use a probabilistic representation of the system dynamics in terms of stochastic differential equations driven by Poisson random measures. Firstly, under mild assumptions, we prove existence of solutions to the mean field game in relaxed open-loop as well as relaxed feedback controls. Relying on the probabilistic representation and a coupling argument, we show that mean field game solutions provide symmetric εN- Nash equilibria for the N-player game, both in open-loop and in feedback strategies (not relaxed), with εN≤ constant √N. Under stronger assumptions, we also find solutions of the mean field game in ordinary feedback controls and prove uniqueness either in case of a small time horizon or under monotonicity. Then, assuming that players control just their transition rates from state to state, we show the convergence, as N tends to infinity, of the N-player game 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. If the master equation possesses a unique regular solution, then such solution can be used to prove the convergence of the value functions of the N players and of the feedback Nash equilibria, and a propagation of chaos property for the associated optimal trajectories. A sufficient condition for the required regularity of the master equation is given by the monotonicity assumptions. Further, we employ the convergence results to establish a Central Limit Theorem and a Large Deviation Principle for the evolution of the N-player optimal empirical measures. Finally, we analyze an example with as state space and anti-monotonous cost,and show that the mean field game has exactly three solutions. The Nash equilibrium is always unique and we prove that the N-player game always admits a limit: it selects one mean field game solution, except in one critical case, so there is propagation of chaos. The value functions also converge and the limit is the entropy solution to the master equation, which for two state models can be written as a scalar conservation law. 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 optimum of this problem when it is unique

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

A mean field model for the development of renewable capacities

We propose a model based on a large number of small competitive producers of renewable energies, to study the effect of subventions on the aggregate level of capacity, taking into account a cannibalization effect. We first derive a model to explain how long-time equilibrium can be reached on the market of production of renewable electricity and compare this equilibrium to the case of monopoly. Then we consider the case in which other capacities of production adjust to the production of renewable energies. The analysis is based on a master equation and we get explicit formulae for the long-time equilibria. We also provide new numerical methods to simulate the master equation and the evolution of the capacities. Thus we find the optimal subventions to be given by a central planner to the installation and the production in order to reach a desired equilibrium capacity

Finite state mean field games with Wright Fisher common noise as limits of NN-player weighted games

Forcing finite state mean field games by a relevant form of common noise is a subtle issue, which has been addressed only recently. Among others, one possible way is to subject the simplex valued dynamics of an equilibrium by a so-called Wright-Fisher noise, very much in the spirit of stochastic models in population genetics. A key feature is that such a random forcing preserves the structure of the simplex, which is nothing but, in this setting, the probability space over the state space of the game. The purpose of this article is hence to elucidate the finite player version and, accordingly, to prove that $N$-player equilibria indeed converge towards the solution of such a kind of Wright-Fisher mean field game. Whilst part of the analysis is made easier by the fact that the corresponding master equation has already been proved to be uniquely solvable under the presence of the common noise, it becomes however more subtle than in the standard setting because the mean field interaction between the players now occurs through a weighted empirical measure. In other words, each player carries its own weight, which hence may differ from $1/N$ and which, most of all, evolves with the common noise

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