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Register a new userRestoring Uniqueness to Mean-Field Games by Randomizing the Equilibria
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Authors
Francois Delarue
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We here address the question of restoration of uniqueness in mean-field games deriving from deterministic differential games with a large number of players. The general strategy for restoring uniqueness is inspired from earlier similar results on ordinary and stochastic differential equations. It consists in randomizing the equilibria through an external noise. As a main feature, we choose the external noise as an infinite dimensional Ornstein-Uhlenbeck process. We first investigate existence and uniqueness of a solution to the noisy system made of the mean-field game forced by the Ornstein-Uhlenbeck process. We also show how such a noisy system can be interpreted as the limit version of a stochastic differential game with a large number of players.
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In the context of simple finite-state discrete time systems, we introduce a generalization of mean field game solution, called correlated solution, which can be seen as the mean field game analogue of a correlated equilibrium. Our notion of solution is justified in two ways: We prove that correlated solutions arise as limits of exchangeable correlated equilibria in restricted (Markov open-loop) strategies for the underlying $N$-player games, and we show how to construct approximate $N$-player correlated equilibria starting from a correlated solution to the mean field game.
The purpose of this paper is to provide a complete probabilistic analysis of a large class of stochastic differential games for which the interaction between the players is of mean-field type. We implement the Mean-Field Games strategy developed analytically by Lasry and Lions in a purely probabilistic framework, relying on tailor-made forms of the stochastic maximum principle. While we assume that the state dynamics are affine in the states and the controls, our assumptions on the nature of the costs are rather weak, and surprisingly, the dependence of all the coefficients upon the statistical distribution of the states remains of a rather general nature. Our probabilistic approach calls for the solution of systems of forward-backward stochastic differential equations of a McKean-Vlasov type for which no existence result is known, and for which we prove existence and regularity of the corresponding value function. Finally, we prove that solutions of the mean-field game as formulated by Lasry and Lions do indeed provide approximate Nash equilibriums for games with a large number of players, and we quantify the nature of the approximation.
A theory of existence and uniqueness is developed for general stochastic differential mean field games with common noise. The concepts of strong and weak solutions are introduced in analogy with the theory of stochastic differential equations, and existence of weak solutions for mean field games is shown to hold under very general assumptions. Examples and counter-examples are provided to enlighten the underpinnings of the existence theory. Finally, an analog of the famous result of Yamada and Watanabe is derived, and it is used to prove existence and uniqueness of a strong solution under additional assumptions.
In this paper, we develop a PDE approach to consider the optimal strategy of mean field controlled stochastic system. Firstly, we discuss mean field SDEs and associated Fokker-Plank eqautions. Secondly, we consider a fully-coupled system of forward-backward PDEs. The backward one is the Hamilton-Jacobi-Bellman equation while the forward one is the Fokker-Planck equation. Our main result is to show the existence of classical solutions of the forward-backward PDEs in the class $H^{1+frac{1}{4},2+frac{1}{2}}([0,T]timesmathbb{R}^n)$ by use of the Schauder fixed point theorem. Then, we use the solution to give the optimal strategy of the mean field stochastic control problem. Finally, we give an example to illustrate the role of our main result.
This paper continues the study of the mean field game (MFG) convergence problem: In what sense do the Nash equilibria of $n$-player stochastic differential games converge to the mean field game as $nrightarrowinfty$? Previous work on this problem took two forms. First, when the $n$-player equilibria are open-loop, compactness arguments permit a characterization of all limit points of $n$-player equilibria as weak MFG equilibria, which contain additional randomness compared to the standard (strong) equilibrium concept. On the other hand, when the $n$-player equilibria are closed-loop, the convergence to the MFG equilibrium is known only when the MFG equilibrium is unique and the associated master equation is solvable and sufficiently smooth. This paper adapts the compactness arguments to the closed-loop case, proving a convergence theorem that holds even when the MFG equilibrium is non-unique. Every limit point of $n$-player equilibria is shown to be the same kind of weak MFG equilibrium as in the open-loop case. Some partial results and examples are discussed for the converse question, regarding which of the weak MFG equilibria can arise as the limit of $n$-player (approximate) equilibria.
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