We study the Dirichlet problem of the following discrete infinity Laplace equation on a subgraph with finite width \(\begin{aligned} \Delta _{\infty }u(x):=\inf _{y\sim x}u(y)+\sup _{y\sim x}u(y)-2u(x)=f(x). \end{aligned}\) We say that a subgraph has finite width if the distances from all vertices to the boundary are uniformly bounded. Using Perron’s method, we demonstrate the existence of bounded solutions. We also prove the uniqueness if \(f\ge 0\) or \(f\le 0\) by establishing a comparison result, and hence obtain the existence of game values for corresponding tug-of-war games introduced by Peres et al. (J Am Math Soc 22(1):167–210, 2009). As an application we show a strong Liouville property for infinity harmonic functions. By an argument of Arzelà–Ascoli, we prove the convergence of solutions of \(\varepsilon \) -tug-of-war games as \(\varepsilon \rightarrow 0\) . Correspondingly, we obtain the existence of bounded solutions to normalized infinity Laplace equations on Euclidean domains with finite width.
