Decentralized Strategies for Finite Population Linear-Quadratic-Gaussian Games and Teams

TL;DR

This paper introduces a new class of finite population mean-field games involving multiple agents, providing conditions for decentralized Nash equilibria and solutions via Riccati equations, applicable to both finite and infinite horizons.

Contribution

It develops a framework for decentralized strategies in finite population LQG games, linking Nash equilibria to non-standard FBSDEs and Riccati equations, extending classical mean-field game results.

Findings

Decentralized strategies form a Nash equilibrium in finite population LQG games.

Conditions for solvability of Riccati equations in infinite-horizon problems.

Explicit solutions for social optimal control and costs under mild conditions.

Abstract

This paper is concerned with a new class of mean-field games which involve a finite number of agents. Necessary and sufficient conditions are obtained for the existence of the decentralized open-loop Nash equilibrium in terms of non-standard forward-backward stochastic differential equations (FBSDEs). By solving the FBSDEs, we design a set of decentralized strategies by virtue of two differential Riccati equations. Instead of the $\varepsilon$-Nash equilibrium in classical mean-field games, the set of decentralized strategies is shown to be a Nash equilibrium. For the infinite-horizon problem, a simple condition is given for the solvability of the algebraic Riccati equation arising from consensus. Furthermore, the social optimal control problem is studied. Under a mild condition, the decentralized social optimal control and the corresponding social cost are given.