Abstract:

A key, counterintuitive feature of the basic Mean Field Games (MFG) theory is that the mean field of a stochastic dynamic game between a set of asymptotically negligible (so-called Minor) agents is deterministic. However, when a Major player is present the mean field becomes stochastic. 

 In the non-linear case the infinite population mean field game problem is decomposed into: (i) two non-standard stochastic optimal control problems with random coefficient processes which yield forward adapted stochastic best response control processes and (ii) two stochastic coefficient McKean-Vlasov (SMV) FPK equations which characterize the state distribution measure of the Major agent and the measure determining the mean field behaviour of the Minor agents. Existence and uniqueness of the solution to this Stochastic Mean Field (SMF) system (SHJB + SMV FPK equations) is established by a fixed point argument in the Wasserstein space of random probability measures. It will then be shown that \epslion- Nash equilibria exist and are unique for this class of problems. 

To analyze the MM-MFG problem where the Major agent’s state is partially observed we develop Nonlinear Filtering Theory for partially observed stochastic dynamical systems with McKean-Vlasov (MV) type stochastic differential state equations. We then consider the MFG problem where (i) partial observations on the Major agent’s state provided to each Minor agents and (ii) complete observations on that state are provided to the Major agent. Applying the standard separation methodology of stochastic control, we construct the associated completely observed system via the application of the nonlinear filtering theory derived above. The existence and uniqueness of \epslion- Nash equilibria is then analyzed in this setting. 

Biographical Sketch:  

Peter E. Caines received the BA in mathematics from Oxford University in 1967 and the PhD in systems and control theory in 1970 from Imperial College, University of London. In 1980 he joined McGill University where he is James McGill Professor and Macdonald Chair in the Department of Electrical and Computer Engineering. He is a Life Fellow of the IEEE, Fellow of SIAM and IMA and is a Fellow of the Royal Society of Canada. In 2009 he received the IEEE Control Systems Society Bode Lecture Prize and in 2013 a Queen Elizabeth II Diamond Jubilee Medal. He is the author of Linear Stochastic Systems, John Wiley, 1988.