Asymptotic Lipschitz regularity for tug-of-war games with varying probabilities

TL;DR

This paper establishes an asymptotic Lipschitz regularity result for the value functions of tug-of-war games with variable probabilities, using a novel game-theoretic coupling approach.

Contribution

It introduces a new method to prove Lipschitz regularity for variable-probability tug-of-war games through a coupling-based analysis.

Findings

Proves asymptotic Lipschitz regularity of value functions.

Develops a game-theoretic coupling technique for analysis.

Extends regularity results to games with varying probabilities.

Abstract

We prove an asymptotic Lipschitz estimate for value functions of tug-of-war games with varying probabilities defined in $\Omega\subset \mathbb R^n$. The method of the proof is based on a game-theoretic idea to estimate the value of a related game defined in $\Omega\times \Omega$ via couplings.