Tug-of-war games related to $p$-Laplace type equations with zeroth order terms

TL;DR

This paper studies tug-of-war games with discounting, linking them to $p$-Laplace PDEs with zeroth-order terms, and establishes fundamental properties of their solutions.

Contribution

It introduces a new class of discounted tug-of-war games and analyzes their connection to $p$-Laplace equations with zeroth-order terms, providing existence, uniqueness, and regularity results.

Findings

Existence and uniqueness of game value functions

Regularity properties of solutions to the PDEs

Insights into the behavior of solutions influenced by game dynamics

Abstract

In this paper, we investigate a class of tug-of-war games that incorporate a constant payoff discount rate at each turn. The associated model problems are $p$-Laplace type partial differential equations with zeroth-order terms. We establish existence, uniqueness, and regularity results for the corresponding game value functions. Furthermore, we explore properties of the solutions to the model PDEs, informed by the analysis of the underlying games.