Set Values of Dynamic Nonzero Sum Games and Set Valued Hamiltonians

TL;DR

This paper extends classical PDE characterizations of stochastic control and zero-sum games to nonzero sum games by introducing set valued Hamiltonians to describe the set of all possible equilibrium values.

Contribution

It introduces set valued Hamiltonians and characterizes the set of equilibrium values in nonzero sum games via backward SDEs, generalizing existing PDE methods.

Findings

Set valued Hamiltonians can describe multiple Nash equilibria.

The set value of the game is characterized through backward SDEs.

Special case reduces to classical control and zero-sum game results.

Abstract

It is well known that the (unique) value of a stochastic control problem or a two person zero sum game under Isaacs condition can be characterized through a PDE driven by the Hamiltonian. Our goal of this paper is to extend this classical result to nonzero sum games, which typically have multiple Nash equilibria and multiple values. Our object is the set value of the game, which roughly speaking is the set of values over all equilibria and thus is by nature unique. We shall introduce set valued Hamiltonians and characterize the set value of the game through backward SDEs driven by appropriate selectors of the set valued Hamiltonians, where the selectors are typically path dependent. When the set valued Hamiltonian is a singleton, our result covers the standard control problem and two person zero sum game problem under Isaacs condition.