  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.
