Totalitarian random Tug-of-War games in graphs

TL;DR

This paper introduces a novel variant of Tug-of-War games on graphs where one player can influence the game's rules, proving the existence of a game value and connecting it to Jensen's extremal equations and infinity harmonic functions.

Contribution

It develops a new game model with strategic rule choices, proving the existence of a value using advanced mathematical tools and linking it to key equations in harmonic analysis.

Findings

The game has a well-defined value.

The value is proven using comparison principles and viscosity solutions.

Connections to Jensen's extremal equations and infinity harmonic functions.

Abstract

In this work we discuss a random Tug-of-War game in graphs where one of the players has the power to decide at each turn whether to play a round of classical random Tug-of-War, or let the other player choose the new game position in exchange of a fixed payoff. We prove that this game has a value using a discrete comparison principle and viscosity tools, as well as probabilistic arguments. This game is related to Jensen's extremal equations, which have a key role in Jensen's celebrated proof of uniqueness of infinity harmonic functions.