Nonlinear stochastic differential games involving a major player and a large number of collectively acting minor agents

TL;DR

This paper analyzes a complex stochastic differential game involving one major player and many minor agents, establishing the existence of saddle points and characterizing their limit behavior as the number of minor agents grows large.

Contribution

It introduces a framework for 2-person zero-sum stochastic differential games with a major player and many minor agents, and characterizes the limit of saddle point controls as the number of minor agents increases.

Findings

Existence of saddle point feedback controls for the game with N minor agents.

Characterization of the limit of saddle point controls as N approaches infinity.

Convergence of the game’s Hamiltonian to a mean-field limit.

Abstract

The purpose of this paper is to study 2-person zero-sum stochastic differential games, in which one player is a major one and the other player is a group of $N$ minor agents which are collectively playing, statistically identical and have the same cost-functional. The game is studied in a weak formulation; this means in particular, we can study it as a game of the type "feedback control against feedback control". The payoff/cost functional is defined through a controlled backward stochastic differential equation, for which driving coefficient is assumed to satisfy strict concavity-convexity with respect to the control parameters. This ensures the existence of saddle point feedback controls for the game with $N$ minor agents. We study the limit behavior of these saddle point controls and of the associated Hamiltonian, and we characterize the limit of the saddle point controls as the unique saddle point control of the limit mean-field stochastic differential game.