A Class of Hybrid LQG Mean Field Games with State-Invariant Switching and Stopping Strategies

TL;DR

This paper introduces a new hybrid LQG mean field game framework that models agents with state-invariant switching and stopping strategies, incorporating major and minor agents with stochastic linear dynamics and quadratic costs.

Contribution

It develops a hybrid systems approach to derive unique $psilon$-Nash equilibria in complex multi-agent games with switching and stopping behaviors.

Findings

Optimal switching and stopping times are state-invariant.

The framework provides explicit strategies for major and minor agents.

The model captures the influence of a major agent on a large population.

Abstract

A novel framework is presented that combines Mean Field Game (MFG) theory and Hybrid Optimal Control (HOC) theory to obtain a unique $\epsilon$-Nash equilibrium for a non-cooperative game with switching and stopping times. We consider the case where there exists one major agent with a significant influence on the system together with a large number of minor agents constituting two subpopulations, each agent with individually asymptotically negligible effect on the whole system. Each agent has stochastic linear dynamics with quadratic costs, and the agents are coupled in their dynamics and costs by the average state of minor agents (i.e. the empirical mean field). It is shown that for a class of Hybrid LQG MFGs, the optimal switching and stopping times are state-invariant and only depend on the dynamical parameters of each agent. Accordingly, a hybrid systems formulation of the game is presented via the indexing by discrete events: (i) the switching of the major agent between alternative dynamics or (ii) the termination of the agents' trajectories in one or both of the subpopulations of minor agents. Optimal switchings and stopping time strategies together with best response control actions for, respectively, the major agent and all minor agents are established with respect to their individual cost criteria by an application of Hybrid LQG MFG theory.