Linear Quadratic Mean Field Games with Quantile-Dependent Cost Coefficients

TL;DR

This paper introduces a novel class of linear quadratic mean field games where cost coefficients depend on population mean and variance via quantiles, providing insights into agents sensitive to both average and variability.

Contribution

It develops a framework for mean field games with quantile-dependent coefficients, deriving equilibrium conditions and establishing existence and uniqueness results.

Findings

Derived coupled Riccati and variance equations for equilibrium

Proved conditions for existence and uniqueness of solutions

Numerical simulations illustrating solution behavior

Abstract

This paper studies a class of linear quadratic mean field games where the coefficients of quadratic cost functions depend on both the mean and the variance of the population's state distribution through its quantile function. Such a formulation allows for modelling agents that are sensitive to not only the population average but also the population variance. The corresponding mean field game equilibrium is identified, which involves solving two coupled differential equations: one is a Riccati equation and the other the variance evolution equation. Furthermore, the conditions for the existence and uniqueness of the mean field equilibrium are established. Finally, numerical results are presented to illustrate the behavior of two coupled differential equations and the performance of the mean field game solution.