Tug-of-war and infinity Laplace equation with vanishing Neumann boundary condition

TL;DR

This paper investigates a stochastic tug-of-war game on graphs and domains, demonstrating convergence of game values and establishing solutions to the infinity Laplace equation with Neumann boundary conditions.

Contribution

It introduces a new analysis of the tug-of-war game with no terminal states and proves existence of solutions to a specific PDE with boundary conditions.

Findings

Game values converge after n steps in continuous and graph cases

Existence of solutions to the infinity Laplace equation with Neumann boundary

Shifted payoff functions lead to convergence results

Abstract

We study a version of the stochastic "tug-of-war" game, played on graphs and smooth domains, with the empty set of terminal states. We prove that, when the running payoff function is shifted by an appropriate constant, the values of the game after n steps converge in the continuous case and the case of finite graphs with loops. Using this we prove the existence of solutions to the infinity Laplace equation with vanishing Neumann boundary condition.