  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.
