  We introduce a new class of strongly degenerate nonlinear parabolic PDEs
$$((p-2)\Delta_{\infty,X}^N+\Delta_X)u(X,Y,t)+(m+p)(X\cdot\nabla_Yu(X,Y,t)-\partial_tu(X,Y,t))=0,$$
$(X,Y,t)\in\mathbb R^m\times \mathbb R^m\times \mathbb R$, $p\in (1,\infty)$,
combining the classical PDE of Kolmogorov and the normalized $p$-Laplace
operator. We characterize solutions in terms of an asymptotic mean value
property and the results are connected to the analysis of certain tug-of-war
games with noise. The value functions for the games introduced approximate
solutions to the stated PDE when the parameter that controls the size of the
possible steps goes to zero. Existence and uniqueness of viscosity solutions to
the Dirichlet problem is established. The asymptotic mean value property, the
associated games and the geometry underlying the Dirichlet problem, all reflect
the family of dilation and the Lie group underlying operators of Kolmogorov
type and this makes our setting different from the context of standard
parabolic dilations and Euclidean translations applicable in the context of the
heat operator and the normalized parabolic infinity Laplace operator.
