Nonlinear Stochastic Differential Games Involving a Major Player and a Large Number of Collectively Acting Minor Agents

The purpose of this paper is to study two-person zero-sum stochastic differential games, in which one player is a major one and the other player is a group of $N$ minor agents which are collectively playing, are statistically identical, and have the same cost functional. The game is studied in a weak formulation; this means in particular that we can study it as a game of the type “feedback control against feedback control." The payoff/cost functional is defined through a controlled backward stochastic differential equation, for which the driving coefficient is assumed to satisfy strict concavity-convexity with respect to the control parameters. This ensures the existence of saddle point feedback controls for the game with $N$ minor agents. We study the limit behavior of these saddle point controls and of the associated Hamiltonian, and we characterize the limit of the saddle point controls as the unique saddle point control of the limit mean-field stochastic differential game.

[1]  Peter E. Caines,et al.  Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle , 2006, Commun. Inf. Syst..

[2]  Minyi Huang,et al.  Large-Population LQG Games Involving a Major Player: The Nash Certainty Equivalence Principle , 2009, SIAM J. Control. Optim..

[3]  Peter E. Caines,et al.  ε-Nash Mean Field Game theory for nonlinear stochastic dynamical systems with mixed agents , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[4]  Guo Hancheng,et al.  Mean-field backward stochastic differential equations with uniformly continuous generators , 2015 .

[5]  Alain Bensoussan,et al.  Linear-Quadratic Mean Field Games , 2014, Journal of Optimization Theory and Applications.

[6]  A. Sznitman.  Topics in propagation of chaos , 1991 .

[7]  S. Hamadène,et al.  Nonzero sum linear–quadratic stochastic differential games and backward–forward equations , 1999 .

[8]  Boualem Djehiche,et al.  Mean-Field Backward Stochastic Differential Equations . A Limit Approach ∗ , 2007 .

[9]  Said Hamadène,et al.  Backward–forward SDE’s and stochastic differential games , 1998 .

[10]  R. Carmona,et al.  Control of McKean–Vlasov dynamics versus mean field games , 2012, 1210.5771.

[11]  René Carmona,et al.  Probabilistic Analysis of Mean-field Games , 2013 .

[12]  Peter E. Caines,et al.  Epsilon-Nash Mean Field Game Theory for Nonlinear Stochastic Dynamical Systems with Major and Minor Agents , 2012, SIAM J. Control. Optim..

[13]  P. Lions,et al.  Jeux à champ moyen. II – Horizon fini et contrôle optimal , 2006 .

[14]  P. Lions,et al.  Mean field games , 2007 .

[15]  Said Hamadène,et al.  Backward equations, stochastic control and zero-sum stochastic differential games , 1995 .

[16]  Peter E. Caines,et al.  Nash, Social and Centralized Solutions to Consensus Problems via Mean Field Control Theory , 2013, IEEE Transactions on Automatic Control.

[17]  P. Lions,et al.  Jeux à champ moyen. I – Le cas stationnaire , 2006 .

[18]  P. Cardaliaguet,et al.  Mean Field Games , 2020, Lecture Notes in Mathematics.

[19]  S. Peng,et al.  Mean-field backward stochastic differential equations and related partial differential equations , 2007, 0711.2167.