Finite State Mean Field Games with Wright–Fisher Common Noise as Limits ofN-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, prove that N-player equilibria indeed converge toward the solution of such a kind of Wright–Fisher mean field game. Whereas 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 [Formula: see text] and which, most of all, evolves with the common noise.

Propagation of Chaos of Forward-Backward Stochastic Differential Equations with Graphon Interactions

“…The theory of mean field games was pioneered independently by Lasry, Lions (see [33], [34], [35]) and Caines, Huang, Malhamé (see [28], [29]). It is the study of strategic decision making by small interacting agents in large populations, and we refer the readers to [3,4,7,21,22] for finite state mean field games, to [9,16,23,25] for uniqueness of mean field game solutions, and to [19,20] for a nice survey. Since then, the convergence of n-player game Nash equilibrium to the solution of mean field game has attracted lots of attention.…”

Control theory approach to continuous-time finite state mean field games

“…Here Φ i (s, µ) can be interpreted as the optimal outcome of the representative player who starts at the time s from the state i under assumptions that the initial distribution of agents is µ and all players use an optimal strategy realizing a solution of the MFG; ∂Φ/∂m stands for the derivative of Φ w.r.t. to m. Notice that formally this inclusion can be deduced from the mean field game system (2), ( 4), (5). Indeed, by construction Φ(s, m(s)) is solves Bellman equation ( 4), when m(•) satisfies (2) for the strategy û obeying (5).…”

“…to m. Notice that formally this inclusion can be deduced from the mean field game system (2), ( 4), (5). Indeed, by construction Φ(s, m(s)) is solves Bellman equation ( 4), when m(•) satisfies (2) for the strategy û obeying (5). The last assumption is equivalent to the inclusion…”

“…The further progress of the theory of finite state mean field games is due to the probabilistic approach [9] and master equation [4], [5], [6], [10]. The master equation first proposed by Lions [23] for the second-order mean field games is used to justify the convergence of feedback equilibria in the finite player games to the solution of the mean field game and to study the dependence of the solution of the mean field game on the initial distribution of players.…”

“…The master equation first proposed by Lions [23] for the second-order mean field games is used to justify the convergence of feedback equilibria in the finite player games to the solution of the mean field game and to study the dependence of the solution of the mean field game on the initial distribution of players. However, the theory of master equation for the mean filed games is developed in the case of unique solution to the mean field game [4], [5], [6], [10].…”