Abstract
Mean field games (MFGs) describe the limit, as $n$ tends to infinity, of stochastic differential games with $n$ players interacting with one another through their common empirical distribution. Under suitable smoothness assumptions that guarantee uniqueness of the MFG equilibrium, a form of law of large of numbers (LLN), also known as propagation of chaos, has been established to show that the MFG equilibrium arises as the limit of the sequence of empirical measures of the $n$-player game Nash equilibria, including the case when player dynamics are driven by both idiosyncratic and common sources of noise. The proof of convergence relies on the so-called master equation for the value function of the MFG, a partial differential equation on the space of probability measures. In this work, under additional assumptions, we establish a functional central limit theorem (CLT) that characterizes the limiting fluctuations around the LLN limit as the unique solution of a linear stochastic PDE. The key idea is to use the solution to the master equation to construct an associated McKean-Vlasov interacting $n$-particle system that is sufficiently close to the Nash equilibrium dynamics of the $n$-player game for large $n$. We then derive the CLT for the latter from the CLT for the former. Along the way, we obtain a new multidimensional CLT for McKean-Vlasov systems. We also illustrate the broader applicability of our methodology by applying it to establish a CLT for a specific linear-quadratic example that does not satisfy our main assumptions, and we explicitly solve the resulting stochastic PDE in this case.
Highlights
Equilibria of Mean field games (MFGs) were introduced as potentially more tractable approximations of Nash equilibria of large finite systems of agents interacting with one another through their common empirical distribution
While much of the theoretical work of the past decade has focused on questions of existence and uniqueness of MFG equilibria, the probabilistic limit theory is less well understood
A law of large numbers has only recently come into focus [6, 31, 30, 19], clarifying how a MFG arises as the limit of a suitable sequence of n-player games as n → ∞, in the presence of both idiosyncratic and common sources of noise
Summary
McKean-Vlasov theory (see, e.g., [22, 41, 44] for the case without common noise, or [11, 15, 17, 28] and [9, Chapter 2, Section 2.1] for the common noise case), we conclude that both empirical measure sequences converge in law to the unique solution (μt)t∈[0,T ] of the (conditional) McKean-Vlasov SDE dXt = b Xt, μt, α Xt, μt, DxU (t, Xt, μt) dt + σdBt + σ0dWt, μt = Law(Xt | (Ws)s≤t), where (X0, B) is a copy of (X01, B1), independent of W It is known from early work on the master equation that this limit (μt)t∈[0,T ] is the unique equilibrium of the MFG, which is to say that:. Relevant properties of derivatives of functionals on the Wasserstein space of probability measures are collected in Appendix A
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