# Nonzero-Sum Submodular Monotone-Follower Games: Existence and Approximation of Nash Equilibria

## TL;DR

This paper proves the existence of Nash equilibria in complex N-player stochastic games with singular controls, introduces approximation methods for these equilibria, and provides algorithms for their computation.

## Contribution

It establishes Nash equilibrium existence in non-Markovian monotone-follower games and introduces a convergent approximation scheme with practical algorithms.

## Key findings

- Existence of Nash equilibria in non-Markovian settings.
- Convergence of approximate Nash equilibria to the original game.
- Algorithm for computing Nash equilibria in monotone-follower games.

## Abstract

We consider a class of N-player stochastic games of multi-dimensional singular control, in which each player faces a minimization problem of monotone-follower type with submodular costs. We call these games "monotone-follower games". In a not necessarily Markovian setting, we establish the existence of Nash equilibria. Moreover, we introduce a sequence of approximating games by restricting, for each natural number n, the players' admissible strategies to the set of Lipschitz processes with Lipschitz constant bounded by n. We prove that, for each n, there exists a Nash equilibrium of the approximating game and that the sequence of Nash equilibria converges, in the Meyer-Zheng sense, to a weak (distributional) Nash equilibrium of the original game of singular control. As a byproduct, such a convergence also provides approximation results of the equilibrium values across the two classes of games. We finally show how our findings can be employed to prove existence of open-loop Nash equilibria in an N-player stochastic differential game with singular controls, and we propose an algorithm to determine a Nash equilibrium for the monotone-follower game.