On the existence and uniqueness of p-harmonious functions

TL;DR

This paper provides elementary proofs for the existence, uniqueness, and measurability of p-harmonious functions, also establishing these properties for value functions in noisy tug-of-war games.

Contribution

It offers a self-contained, elementary proof of fundamental properties of p-harmonious functions and their connection to game theory.

Findings

Existence of p-harmonious functions confirmed

Uniqueness of p-harmonious functions established

Measurability of value functions proven

Abstract

We give a self-contained and short proof for the existence, uniqueness and measurability of so called $p$-harmonious functions. The proofs only use elementary analytic tools. As a consequence, we obtain existence, uniqueness and measurability of value functions for the tug-of-war game with noise.