  In this paper we prove that a function $ u\in\mathcal{C}(\bar{\Omega})$ is
the continuous value of the Tug-of-War game described in \cite{PSSW} if and
only if it is the unique viscosity solution to the infinity laplacian with
mixed boundary conditions
  {-\Delta_{\infty}u(x)=0\quad & \text{in} \Omega,
  \frac{\partial u}{\partial n}(x)=0\quad & \text{on} \Gamma_N, u(x)=F(x)\quad
& \text{on} \Gamma_D. By using the results in \cite{PSSW}, it follows that this
viscous PDE problem has a unique solution, which is the unique {\it absolutely
minimizing Lipschitz extension} to the whole $\bar{\Omega}$ (in the sense of
\cite{Aronsson} and \cite{PSSW}) of the boundary data $ F:\Gamma_D\to\R $.
