Stochastic nonzero-sum games: a new connection between singular control and optimal stopping

TL;DR

This paper reveals a novel relationship between nonzero-sum optimal stopping games and singular control games, showing how Nash equilibria in one correspond to equilibria in the other through boundary hitting times.

Contribution

It establishes a new theoretical connection linking Nash equilibria in stopping games with those in singular control games via boundary conditions.

Findings

Nash equilibria in stopping games correspond to boundary-triggered equilibria in control games.

A differential relationship exists between players' value functions across the two game types.

The results provide a unified framework for analyzing nonzero-sum stochastic games.

Abstract

In this paper we establish a new connection between a class of 2-player nonzero-sum games of optimal stopping and certain $2$-player nonzero-sum games of singular control. We show that whenever a Nash equilibrium in the game of stopping is attained by hitting times at two separate boundaries, then such boundaries also trigger a Nash equilibrium in the game of singular control. Moreover a differential link between the players' value functions holds across the two games.