Quantile-based Mean-Field Games with Common Noise

TL;DR

This paper investigates how quantiles influence optimal strategies in mean-field games with common noise and jumps, providing a new equilibrium characterization via stochastic PDEs and explicit solutions in specific cases.

Contribution

It introduces a novel class of mean-field games incorporating quantiles into the dynamics and payoffs, with a new optimality system and explicit solutions for the quantile process.

Findings

Quantile processes satisfy a stochastic PDE under common noise.

Explicit solutions are derived for Ornstein-Uhlenbeck type processes.

The framework extends mean-field game analysis to include quantile-dependent strategies.

Abstract

In this paper we explore the impact of quantiles on optimal strategies under state dynamics driven by both individual noise, common noise and Poisson jumps. We first establish an optimality system satisfied the quantile process under jump terms. We then turn to investigate a new class of finite horizon mean-field games with common noise in which the payoff functional and the state dynamics are dependent not only on the state-action pair but also on conditional quantiles. Based on the best-response of the decision-makers, it is shown that the equilibrium conditional quantile process satisfies a stochastic partial differential equation in the non-degenerate case. A closed-form expression of the quantile process is provided in a basic Ornstein-Uhlenbeck process with common noise.