Tug-of-war games related to oblique derivative boundary value problems with the normalized $p$-Laplacian

In this paper, we are concerned with game-theoretic interpretations to the following oblique derivative boundary value problem \begin{align*} \left\{ \begin{array}{ll} \Delta_{p}^{N}u=0 & \textrm{in $\Omega$,}\\ \langle \beta , Du \rangle + \gamma u = \gamma G & \textrm{on $\partial \Omega$,}\\ \end{array} \right. \end{align*} where $\Delta_{p}^{N}$ is the normalized $p$-Laplacian. This problem can be regarded as a generalized version of the Robin boundary value problem for the Laplace equations. We construct several types of stochastic games associated with this problem by using `shrinking tug-of-war'. For the value functions of such games, we investigate the properties such as existence, uniqueness, regularity and convergence.