Mean Field Stackelberg Games: Aggregation of Delayed Instructions

In this paper, we consider an $N$-player interacting strategic game in the presence of a (endogenous) dominating player, who gives direct influence on individual agents, through its impact on their control in the sense of Stackelberg game, and then on the whole community. Each individual agent is subject to a delay effect on collecting information, specifically at a delay time, from the dominating player. The size of his delay is completely known by the agent, while to others, including the dominating player, his delay plays as a hidden random variable coming from a common fixed distribution. By invoking a noncanonical fixed point property, we show that for a general class of finite $N$-player games, each of them converges to the mean field counterpart which may possess an optimal solution that can serve as an $\epsilon$-Nash equilibrium for the corresponding finite $N$-player game. Second, we provide, with explicit solutions, a comprehensive study on the corresponding linear quadratic mean field games of...

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

[2]  Shige Peng,et al.  Anticipated backward stochastic differential equations , 2007, 0705.1822.

[3]  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..

[4]  Alain Bensoussan,et al.  Systems of Bellman equations to stochastic differential games with non-compact coupling , 2010 .

[5]  A. Bensoussan.  Stochastic control by functional analysis methods , 1982 .

[6]  A. Bensoussan,et al.  Stochastic Games for N Players , 2000 .

[7]  R. Elliott.  The Existence of Value in Stochastic Differential Games , 1976 .

[8]  P. Caines,et al.  Individual and mass behaviour in large population stochastic wireless power control problems: centralized and Nash equilibrium solutions , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[9]  J. Cruz,et al.  On the Stackelberg strategy in nonzero-sum games , 1973 .

[10]  A. Bensoussan,et al.  Mean Field Games and Mean Field Type Control Theory , 2013 .

[11]  Terry Lyons,et al.  Backward stochastic dynamics on a filtered probability space , 2009, 0904.0377.

[12]  Xiaoming Xu,et al.  Fully Coupled Forward-Backward Stochastic Functional Differential Equations and Applications to Quadratic Optimal Control , 2013, Journal of Systems Science and Complexity.

[13]  Alain Bensoussan,et al.  Linear Quadratic Differential Games with Mixed Leadership: The Open-Loop Solution , 2013 .

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

[15]  Mark H. A. Davis.  Linear estimation and stochastic control , 1977 .

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

[17]  A. Bensoussan,et al.  On diagonal elliptic and parabolic systems with super-quadratic Hamiltonians , 2008 .

[18]  A. Bensoussan,et al.  Mean Field Games with a Dominating Player , 2014, 1404.4148.

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

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

[21]  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..

[22]  Suresh P. Sethi,et al.  Differential Games with Mixed Leadership: The Open-Loop Solution , 2009, Appl. Math. Comput..

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