Dynamic Set Values for Nonzero Sum Games with Multiple Equilibriums

TL;DR

This paper introduces the concept of the set value in nonzero sum games with multiple Nash equilibriums, establishing its properties and deriving a dynamic programming principle applicable in both discrete and continuous time models.

Contribution

It defines the set value for nonzero sum games, proves its properties, and develops a dynamic programming principle using closed-loop, path-dependent controls, with duality methods for PDE-based computation.

Findings

The set value is unique and always exists, possibly empty.

The dynamic programming principle holds for the set value under certain control conditions.

A duality approach via PDEs enables efficient numerical computation of the set value.

Abstract

Nonzero sum games typically have multiple Nash equilibriums (or no equilibrium), and unlike the zero sum case, they may have different values at different equilibriums. Instead of focusing on the existence of individual equilibriums, we study the set of values over all equilibriums, which we call the set value of the game. The set value is unique by nature and always exists (with possible value $\emptyset$). Similar to the standard value function in control literature, it enjoys many nice properties such as regularity, stability, and more importantly the dynamic programming principle. There are two main features in order to obtain the dynamic programming principle: (i) we must use closed-loop controls (instead of open-loop controls); (ii) we must allow for path dependent controls, even if the problem is in a state dependent (Markovian) setting. We shall consider both discrete and continuous time models with finite time horizon. For the latter we will also provide a duality approach through certain standard PDE (or path dependent PDE), which is quite efficient for numerically computing the set value of the game.