Discrete infinity Laplace equations on graphs and tug-of-war games

We study the Dirichlet problem of the following discrete infinity Laplace equation on a subgraph with finite width $$\Delta_{\infty} u(x) = \inf_{y \sim x}u(y)+\sup_{y \sim x}u(y)-2u(x) = f(x).$$ We say that a subgraph has finite width if the distances from all vertices to the boundary are uniformly bounded. By Perron's method, we show the existence of bounded solutions. We also prove the uniqueness if $f$ is nonnegative by establishing a comparison result, and hence obtain the existence of game values for corresponding tug-of-war games introduced by Peres, Schramm, Sheffield, and Wilson (2009). As an application we show a strong Liouville property for infinity harmonic functions. By an argument of Arzela-Ascoli, we prove the convergence of solutions of $\varepsilon$-tug-of-war games as $\varepsilon$ tends to 0. Correspondingly, we obtain the existence of bounded solutions to normalized infinity Laplace equations on Euclidean domains with finite width.