  We propose a new finite difference approximation to the Dirichlet problem for
the homogeneous $\mathbf{p}$-Laplace equation posed on an $N$-dimensional
domain, in connection with the Tug of War games with noise. Our game and the
related mean-value expansion that we develop, superposes the ``deterministic
averages'' ``$\frac{1}{2}(\inf +\sup)$'' taken over balls, with the
``stochastic averages'' ``$\fint$'', taken over $N$-dimensional ellipsoids
whose aspect ratio depends on $N,\mathbf{p}$ and whose orientations span all
directions while determining $\inf / \sup$. We show that the unique solutions
$u_\epsilon$ of the related dynamic programming principle are automatically
continuous for continuous boundary data, and coincide with the well-defined
game values. Our game has thus the min-max property: the order of supremizing
the outcomes over strategies of one player and infimizing over strategies of
their opponent, is immaterial. We further show that domains satisfying the
exterior corkscrew condition are game regular in this context, i.e. the family
$\{u_\epsilon\}_{\epsilon\to 0}$ converges uniformly to the unique viscosity
solution of the Dirichlet problem.
