Uniform measure density condition and game regularity for tug-of-war games

TL;DR

This paper demonstrates that a uniform measure density condition guarantees game regularity in 'tug-of-war with noise' for all p between 2 and infinity, using strategies and probabilistic estimates.

Contribution

It establishes a link between measure density conditions and game regularity in stochastic tug-of-war games, extending understanding across a range of p-values.

Findings

Uniform measure density implies game regularity for all 2<p<∞.

The proof combines strategy selection with estimates of stopping times.

Density estimates for sums of i.i.d. random vectors are utilized.

Abstract

We show that a uniform measure density condition implies game regularity for all $2<p<\infty$ in a stochastic game called 'tug-of-war with noise'. The proof utilizes suitable choices of strategies combined with estimates for the associated stopping times and density estimates for the sum of independent and identically distributed random vectors.