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

Abstract

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.

A unifying framework for submodular mean field games

“…Lastly, we mention that our finite state MFGs do not involve common noise. Choosing a common noise for finite state problems is in fact less straightforward than in the usual continuous space setting; see the recent works [6,7] for continuous time finite state problems. 1.2.2.…”

Mean Field Game Master Equations with Anti-monotonicity Conditions

“…In this case, one approach is to consider a special type of equilibria, see e.g. [24], Delarue-Foguen Tchuendom [26], Cecchin-Delarue [25], Bayraktar-Cecchin-Cohen-Delarue [4,5]. A larger literature is on the possible convergence of the equilibria for the N -player game, which is quite often unique because the corresponding Nash system is non-degenerate due to the presence of the individual noises, to the mean field equilibria (which may or may not be unique), see, e.g., [19,22,43], Delarue-Lacker-Ramanan [27,28], Djete [29], Lacker [35,36,37,38], Lacker-Flem [39], Nuts-San Martin-Tan [44].…”

Mean Field Games Master Equations with Non-separable Hamiltonians and Displacement Monotonicity

“…For a non-exhaustible list of results on the global in time well-posedness theory of mean field games master equations in various settings, we refer the reader to [15,17,18,19], and in the realm of potential mean field games, to [7,8,21]. We also refer to [30] for global existence and uniqueness of weak solutions and to [4,5,6,10] for classical solutions of finite state mean field games master equations. All the above global well-posedness results require the Hamiltonian H to be separable in µ and p, i.e.…”