The Master Equation for Large Population Equilibriums

Abstract

We use a simple N -player stochastic game with idiosyncratic and common noises to introduce the concept of Master Equation originally proposed by Lions in his lectures at the Collège de France. Controlling the limit N Ñ 8 of the explicit solution of the N -player game, we highlight the stochastic nature of the limit distributions of the states of the players due to the fact that the random environment does not average out in the limit, and we recast the Mean Field Game (MFG) paradigm in a set of coupled Stochastic Partial Differential Equations (SPDEs). The first one is a forward stochastic Kolmogorov equation giving the evolution of the conditional distributions of the states of the players given the common noise. The second is a form of stochastic Hamilton Jacobi Bellman (HJB) equation providing the solution of the optimization problem when the flow of conditional distributions is given. Being highly coupled, the system reads as an infinite dimensional Forward Backward Stochastic Differential Equation (FBSDE). Uniqueness of a solution and its Markov property lead to the representation of the solution of the backward equation (i.e. the value function of the stochastic HJB equation) as a deterministic function of the solution of the forward Kolmogorov equation, function which is usually called the decoupling field of the FBSDE. The (infinite dimensional) PDE satisfied by this decoupling field is identified with the master equation. We also show that this equation can be derived for other large populations equilibriums like those given by the optimal control of McKean-Vlasov stochastic differential equations.The paper is written more in the style of a review than a technical paper, and we spend more time and energy motivating and explaining the probabilistic interpretation of the Master Equation, than identifying the most general set of assumptions under which our claims are true.

Dynamic Programming for Optimal Control of Stochastic McKean--Vlasov Dynamics

“…Now, by writing 14) and noting thatf andĝ are continuous on P 2 (R d )×A, resp. on P 2 (R d ), under the continuity assumption in (H2)(ii), we conclude by the same arguments as in [29] using (3.13) (see Chapter 3, Sec.…”

“…, a) ds 14) with N ∈ L 2 (G; R n ) of zero mean, and unit variance, and independent of (B, ξ). Since the…”

Mean Field Games with Singular Controls of Bounded Velocity

“…(14). For this MFG, [17] derives a solution as N → ∞ with a mean information process defined as m t = xµ t (dx), where µ t is a probability measure of x i t .…”

“…In this approach, the MFG is essentially analyzed by studying two coupled PDEs, the backward Hamilton-Jacobi-Bellman (HJB) equation and the forward McKeanVlasov SDE. Buckdahn et al [8,9] and Carmona, Delarue, and Lacker [13,14,16] propose alternative probabilistic approaches to directly analyze the combined (forward-)backward stochastic differential equations. Recently, Pham and Wei [41,42] suggest using the stochastic McKean-Vlasov equation and the dynamic programming principle to solve the MFG.…”

Mean-Field Games and Ambiguity Aversion

“…For mean-field games, a more general result holds in the form of a master equation, which is a second-order PDE on the space of probability measures. Although this thesis does not discuss the master equation, we nonetheless refer interested readers to Cardaliaguet et al (2015), Carmona and Delarue (2014), Bensoussan et al (2015) for details. However, in order to obtain explicit solutions, it is preferable to work with forwards and backwards equations.…”