Robust Mean Field Games

Abstract

Recently there has been renewed interest in large-scale games in several research disciplines, with diverse application domains as in the smart grid, cloud computing, financial markets, biochemical reaction networks, transportation science, and molecular biology. Prior works have provided rich mathematical foundations and equilibrium concepts but relatively little in terms of robustness in the presence of uncertainties. In this paper, we study mean field games with uncertainty in both states and payoffs. We consider a population of players with individual states driven by a standard Brownian motion and a disturbance term. The contribution is threefold: First, we establish a mean field system for such robust games. Second, we apply the methodology to production of an exhaustible resource. Third, we show that the dimension of the mean field system can be significantly reduced by considering a functional of the first moment of the mean field process.

Decomposition and Mean-Field Approach to Mixed Integer Optimal Compensation Problems

“…From a mathematical point of view, the mean-field approach leads to the study of a system of partial differential equations (PDEs), where the classical Hamilton-Jacobi-Bellman equation is coupled with a Fokker-Planck equation for the density of the players, in a forward-backward fashion. The decomposition method proposed here requires that each agent i computes in advance the time evolution of the local average (see, e.g., the Fokker-Planck-Kolmogorov equation in [23,[28][29][30][31][32][33]). However, since this is practically impossible, we use here the predictive control method to approximate the computation of the solution.…”

“…Example 6.1 In this specific example, dynamics (1) take the form given below in (29). Such dynamics are particularly significant as they reproduce the typical interaction between position and velocity in a sampled second-order system.…”

“…Second, high fixed costs lead to solutions with the fewest number of control actions and this facilitates the validation and interpretation of the simulated results. Next, we decompose dynamics (29) in scalar lot sizing form (13) which we rewrite below:…”

Consensus via multi-population robust mean-field games

“…some players may be irrational). The resulting game is then a robust mean-field game as the one in [1] and in the same spirit as [14]. The model involves a system of coupled partial differential equations (PDEs).…”

Crowd-Averse Cyber-Physical Systems: The Paradigm of Robust Mean-Field Games

“…Explicit solutions exist for the linear-quadratic structure, see [2], while in general a variety of numerical solution schemes are available in the literature [1]. More recently, robustness and risk-sensitivity have been brought into the picture of mean-field games [4,5,19]. The first PDE is then the Hamilton-Jacobi-Isaacs (HJI) equation.…”