A mixed problem for the infinity laplacian via Tug-of-War games

In this paper we prove that a function $u\in \mathcal{C}(\bar{\mathrm{\Omega }})$ 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 $\bar{\mathrm{\Omega }}$ (in the sense of \cite{Aronsson} and \cite{PSSW}) of the boundary data $F:{\mathrm{\Gamma }}_D\to \mathbb{R}$.