A Probabilistic Approach to Mean Field Games with Major and Minor Players

We propose a new approach to mean field games with major and minor players. Our formulation involves a two player game where the optimization of the representative minor player is standard while the major player faces an optimization over conditional McKean-Vlasov stochastic differential equations. The definition of this limiting game is justified by proving that its solution provides approximate Nash equilibriums for large finite player games. This proof depends upon the generalization of standard results on the propagation of chaos to conditional dynamics. Because it is on independent interest, we prove this generalization in full detail. Using a conditional form of the Pontryagin stochastic maximum principle (proven in the appendix), we reduce the solution of the mean field game to a forward-backward system of stochastic differential equations of the conditional McKean-Vlasov type, which we solve in the Linear Quadratic setting. We use this class of models to show that Nash equilibriums in our formulation can be different from those of the formulations contemplated so far in the literature

LQG Mean Field Games with a Major Agent: Nash Certainty Equivalence versus Probabilistic Approach

Mean field game systems consisting of a major agent and a large population of minor agents were introduced in (Huang, 2010) in an LQG setup. In the past years several approaches towards major-minor mean field games have been developed, principally (i) the Nash certainty equivalence (Huang, 2010), (ii) master equations, (iii) asymptotic solvability, and (iv) the probabilistic approach. In a recent work (Huang, 2020), for the LQG case the equivalence of the solutions obtained via approaches (i)-(iii) was established. In this work we demonstrate that the closed-loop Nash equilibrium derived in the infinite-population limit through approaches (i) and (iv) are identical

A Class of Mean-field LQG Games with Partial Information

The large-population system consists of considerable small agents whose individual behavior and mass effect are interrelated via their state-average. The mean-field game provides an efficient way to get the decentralized strategies of large-population system when studying its dynamic optimizations. Unlike other large-population literature, this current paper possesses the following distinctive features. First, our setting includes the partial information structure of large-population system which is practical from real application standpoint. Specially, two cases of partial information structure are considered here: the partial filtration case (see Section 2, 3) where the available information to agents is the filtration generated by an observable component of underlying Brownian motion; the noisy observation case (Section 4) where the individual agent can access an additive white-noise observation on its own state. Also, it is new in filtering modeling that our sensor function may depend on the state-average. Second, in both cases, the limiting state-averages become random and the filtering equations to individual state should be formalized to get the decentralized strategies. Moreover, it is also new that the limit average of state filters should be analyzed here. This makes our analysis very different to the full information arguments of large-population system. Third, the consistency conditions are equivalent to the wellposedness of some Riccati equations, and do not involve the fixed-point analysis as in other mean-field games. The $\epsilon$-Nash equilibrium properties are also presented.Comment: 19 page