Title: Adaptive Mean Field (NCE) Methodology in Leader-Follower Stochastic Dynamic Games (½ hour)
Abstract: We consider an adaptive leader-follower dynamic games model for large population systems [1]. In this formulation the leaders track a convex combination of their overall average together with a certain reference trajectory which is unknown to the followers. The followers use a maximum likelihood estimator (based on a fixed ratio sample of the population of the leaders' trajectories) to identify the member of a given finite class of models which is generating the reference trajectory of the leaders. We approach this large population game problem by use of the so-called Mean Field (Nash Certainty Equivalence) methodology [2,3,4]. Subject to reasonable conditions the true reference trajectory model is identified in finite time with probability one as the leaders' population goes to infinity [5]. Moreover, the system performance for the leaders is almost surely optimal in the \epsilon-Nash sense and the system performance for the adaptive followers is almost surely \epsilon-optimal with respect to the leaders [6].
References:
[1] M. Nourian, P. E. Caines, R. P. Malhamé, and M. Huang,
"Leader-Follower maximum likelihood ratio adaptive NCE methodology,"
Technical Report, McGill University, April 2010.
[2] P. E. Caines, "Bode lecture: Mean Field stochastic control," Proceedings
of the 48th IEEE Conference on Decision and Control, Shanghai, China,
December, 2009. [Online]. Available:
http://www.ieeecss.org/CAB/conferences/cdc2009/shanghai2009.31.pdf
[3] M. Huang, P. E. Caines, and R. P. Malhamé, "Large-population
cost-coupled LQG problems with nonuniform agents: Individual-mass behavior
and decentralized e-Nash equilibria," IEEE Transactions on Automatic
Control, vol. 52, no. 9, pp. 1560~ 1571, 2007.
[4] M. Huang, P. E. Caines, and R. P. Malhamé, "The NCE (Mean Field)
principle with locality dependent cost interactions," IEEE Transactions on
Automatic Control, to appear.
[5] M. Nourian, R.P. Malhamé, M. Huang, and P.E. Caines. "A mean field
(NCE) formulation of estimation based leader-follower collective
motion". International Journal of Robotics and Automation. (submitted
Aug. 2009; provisionally accepted Jan. 2010).
[6] M. Nourian, R.P. Malhamé, M. Huang, and P.E. Caines. "Optimality of
Adaption Based Mean Field (NCE) Control Laws in Leader-Follower Stochastic
Dynamic Games". submitted to 49th IEEE Conference on Decision and Control,
December 15-17, 2010, Atlanta, Georgia USA.