  We study the mean value properties of $\mathbf{p}$-harmonic functions on the
first Heisenberg group $\mathbb{H}$, in connection to the dynamic programming
principles of certain stochastic processes. We implement the approach of
Peres-Scheffield to provide the game-theoretical interpretation of the
sub-elliptic $\mathbf{p}$-Laplacian; and of Manfredi-Parviainen-Rossi to
characterize its viscosity solutions via the asymptotic mean value expansions.
