A class of stochastic games and moving free boundary problems

TL;DR

This paper introduces a new class of N-player stochastic games with moving free boundaries, providing conditions for Nash equilibria and linking strategies to controlled stochastic differential equations.

Contribution

It develops a framework for analyzing stochastic games with moving free boundaries, including a verification theorem and connections to rank-dependent SDEs.

Findings

Derived sufficient conditions for Nash equilibria.

Formulated a multi-dimensional free boundary problem.

Linked NE strategies to controlled rank-dependent SDEs.

Abstract

In this paper we propose and analyze a class of $N$-player stochastic games that include finite fuel stochastic games as a special case. We first derive sufficient conditions for the Nash equilibrium (NE) in the form of a verification theorem. The associated Quasi-Variational-Inequalities include an essential game component regarding the interactions among players, which may be interpreted as the analytical representation of the conditional optimality for NEs. The derivation of NEs involves solving first a multi-dimensional free boundary problem and then a Skorokhod problem. We call it a "moving free boundary" to highlight the difference between standard control problems and stochastic games. Finally, we present an intriguing connection between these NE strategies and controlled rank-dependent stochastic differential equations.