Tug-of-war games with varying probabilities and the normalized p(x)-Laplacian

We study a tug-of-war game with varying probabilities. In particular, we show that the value of the game is locally asymptotically H\"{o}lder continuous. We also show the existence and uniqueness of values of the game. As an application, we prove that the value function of the game converges to a solution of the normalized $p(x)$-Laplacian.