Linear-quadratic risk-sensitive mean field games

In this paper, we consider linear-quadratic risk-sensitive mean field games (LQRSMFGs). Each agent strives to minimize an exponentiated integral quadratic cost or risk-sensitive cost function, which is coupled with other agents via a mean field term. By invoking the Nash certainty equivalence principle, we first obtain a robust decentralized control law for each agent to construct a mean field system. We then provide appropriate conditions under which the mean field system admits a unique deterministic function that approximates the mean field term with arbitrarily small error when the number of agents, say N, goes to infinity. We also show the closed-loop system stability, and prove that the set of N robust decentralized control laws possesses an ε-Nash equilibrium property. Moreover, we show that ε can be taken to be arbitrarily close to zero as N → ∞, but our ε bound is weaker than its linear-quadratic mean field game (LQMFG) counterpart due to risk-sensitivity in the present case. Finally, we discuss two different limiting cases, and show that one of these is equivalent to the corresponding LQMFG.