# Gradient and Lipschitz estimates for tug-of-war type games

## TL;DR

This paper introduces a random step size tug-of-war game and demonstrates that its value function's gradient exists almost everywhere, is uniformly bounded, and converges to the gradient of a p-harmonic function, with improved Lipschitz estimates near planar boundaries.

## Contribution

It extends tug-of-war game analysis by establishing gradient existence, boundedness, convergence, and improved Lipschitz estimates, advancing regularity theory for related PDEs.

## Key findings

- Gradient of value function exists almost everywhere
- Gradients are uniformly bounded and converge weakly to p-harmonic gradients
- Improved Lipschitz estimates near planar boundary values

## Abstract

We define a random step size tug-of-war game, and show that the gradient of a value function exists almost everywhere. We also prove that the gradients of value functions are uniformly bounded and converge weakly to the gradient of the corresponding $p$-harmonic function. Moreover, we establish an improved Lipschitz estimate when boundary values are close to a plane. Such estimates are known to play a key role in higher regularity theory of partial differential equations. The proofs are based on cancellation and coupling methods as well as improved version of the cylinder walk argument.