On the Solvability of Risk-Sensitive Linear-Quadratic Mean-Field Games

TL;DR

This paper introduces a direct method to solve risk-sensitive linear-quadratic mean-field games, providing explicit strategies and conditions for existence and uniqueness of equilibria without relying on traditional PDE or maximum principle methods.

Contribution

It presents a novel, straightforward approach to solving risk-sensitive mean-field games, avoiding complex PDEs and offering explicit solutions and equilibrium conditions.

Findings

Explicit solutions for optimal strategies are derived.

Unique equilibrium conditions are established under certain parameters.

The method extends to robust mean-field games with disturbances.

Abstract

In this paper we formulate and solve a mean-field game described by a linear stochastic dynamics and a quadratic or exponential-quadratic cost functional for each generic player. The optimal strategies for the players are given explicitly using a simple and direct method based on square completion and a Girsanov-type change of measure. This approach does not use the well-known solution methods such as the Stochastic Maximum Principle and the Dynamic Programming Principle with Hamilton-Jacobi-Bellman-Isaacs equation and Fokker-Planck-Kolmogorov equation. In the risk-neutral linear-quadratic mean-field game, we show that there is unique best response strategy to the mean of the state and provide a simple sufficient condition of existence and uniqueness of mean-field equilibrium. This approach gives a basic insight into the solution by providing a simple explanation for the additional term in the robust or risk-sensitive Riccati equation, compared to the risk-neutral Riccati equation. Sufficient conditions for existence and uniqueness of mean-field equilibria are obtained when the horizon length and risk-sensitivity index are small enough. The method is then extended to the linear-quadratic robust mean-field games under small disturbance, formulated as a minimax mean-field game.