Risk-Sensitive Mean Field Games

TL;DR

This paper investigates risk-sensitive mean-field stochastic differential games, deriving explicit solutions and characterizing equilibria through coupled PDEs, with numerical illustrations of the mean-field behavior.

Contribution

It introduces explicit solutions for risk-sensitive mean-field games with log-quadratic costs and affine dynamics, and characterizes equilibria via coupled PDEs.

Findings

Explicit solutions for specific cost and dynamics cases.

Equilibrium characterization through coupled PDEs.

Numerical examples illustrating mean-field behavior.

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.