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

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