Risk-Sensitive Mean Field Games
Abstract
In this paper, we study a class of risk-sensitive mean-field stochastic differential games. We show that under appropriate regularity conditions, the mean-field value of the stochastic differential game with exponentiated integral cost functional coincides with the value function described by a Hamilton-Jacobi-Bellman (HJB) equation with an additional quadratic term. We provide an explicit solution of the mean-field best response when the instantaneous cost functions are log-quadratic and the state dynamics are affine in the control. An equivalent mean-field risk-neutral problem is formulated and the corresponding mean-field equilibria are characterized in terms of backward-forward macroscopic McKean-Vlasov equations, Fokker-Planck-Kolmogorov equations, and HJB equations. We provide numerical examples on the mean field behavior to illustrate both linear and McKean-Vlasov dynamics.
Keywords
Cite
@article{arxiv.1210.2806,
title = {Risk-Sensitive Mean Field Games},
author = {Hamidou Tembine and Quanyan Zhu and Tamer Basar},
journal= {arXiv preprint arXiv:1210.2806},
year = {2012}
}