Nash Equilibria for Exchangeable Team against Team Games, their Mean Field Limit, and Role of Common Randomness

TL;DR

This paper investigates Nash equilibria in large team-based stochastic games, demonstrating existence, structural properties, and the role of common randomness, especially highlighting differences between finite and infinite decision maker scenarios.

Contribution

It establishes the existence of exchangeable and symmetric Nash equilibria in team games, and shows that common randomness is unnecessary in large team games, extending mean-field game theory.

Findings

Existence of Nash equilibria in finite and infinite team games.

Exchangeability and symmetry properties of equilibria.

Common randomness is unnecessary for large team games.

Abstract

We study stochastic mean-field games among finite number of teams with large finite as well as infinite number of decision makers. For this class of games within static and dynamic settings, we establish the existence of a Nash equilibrium, and show that a Nash equilibrium exhibits exchangeability in the finite decision maker regime and symmetry in the infinite one. To arrive at these existence and structural theorems, we endow the set of randomized policies with a suitable topology under various decentralized information structures, which leads to the desired convexity and compactness of the set of randomized policies. Then, we establish the existence of a randomized Nash equilibrium that is exchangeable (not necessarily symmetric) among decision makers within each team for a general class of exchangeable stochastic games. As the number of decision makers within each team goes to infinity (that is for the mean-field game among teams), using a de Finetti representation theorem, we show existence of a randomized Nash equilibrium that is symmetric (i.e., identical) among decision makers within each team and also independently randomized. Finally, we establish that a Nash equilibrium for a class of mean-field games among teams (which is symmetric) constitutes an approximate Nash equilibrium for the corresponding pre-limit (exchangeable) game among teams with large but finite number of decision makers. We thus show that common randomness is not necessary for large team-against-team games, unlike the case with small sized teams.