Mean field games of major-minor agents with recursive functionals

TL;DR

This paper introduces a new class of mean field games with a major agent and many minor agents, where the agents' recursive functionals are modeled via nonlinear BSDEs, and develops a unified analytical framework for these complex interactions.

Contribution

It proposes the recursive major-minor (RMM) framework with a novel structural scheme for analyzing complex couplings and derives equilibrium conditions, extending mean field game theory.

Findings

Established a general RMM modeling approach using empirical averages.

Developed a unified structural scheme for analyzing complex couplings.

Explored linear-quadratic RMM problems for illustrative insights.

Abstract

This paper investigates a novel class of mean field games involving a major agent and numerous minor agents, where the agents' functionals are recursive with nonlinear backward stochastic differential equation (BSDE) representations. We term these games "recursive major-minor" (RMM) problems. Our RMM modeling is quite general, as it employs empirical (state, control) averages to define the weak couplings in both the functionals and dynamics of all agents, regardless of their status as major or minor. We construct an auxiliary limiting problem of the RMM by a novel unified structural scheme combining a bilateral perturbation with a mixed hierarchical recomposition. This scheme has its own merits as it can be applied to analyze more complex coupling structures than those in the current RMM. Subsequently, we derive the corresponding consistency condition and explore asymptotic RMM equilibria. Additionally, we examine the RMM problem in specific linear-quadratic settings for illustrative purposes.