Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones

We prove that if U\subset\R^n is an open domain whose closure \overline{U} is compact in the path metric, and F is a Lipschitz function on \partial{U}, then for each \beta\in\R there exists a unique viscosity solution to the \beta-biased infinity Laplacian equation \beta |\nabla u| + \Delta_\infty u=0 on U that extends F, where \Delta_\infty u= |\nabla u|^{-2} \sum_{i,j} u_{x_i}u_{x_ix_j} u_{x_j}. In the proof, we extend the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the \beta-biased \eps-game as follows. The starting position is x_0 \in U. At the k^\text{th} step the two players toss a suitably biased coin (in our key example, player I wins with odds of \exp(\beta\eps) to 1), and the winner chooses x_k with d(x_k,x_{k-1}) < \eps. The game ends when x_k \in \partial{U}, and player II pays the amount F(x_k) to player I. We prove that the value u^{\eps}(x_0) of this game exists, and that \|u^\eps - u\|_\infty \to 0 as \eps \to 0, where u is the unique extension of F to \overline{U} that satisfies comparison with \beta-exponential cones. Comparison with exponential cones is a notion that we introduce here, and generalizing a theorem of Crandall, Evans and Gariepy regarding comparison with linear cones, we show that a continuous function satisfies comparison with \beta-exponential cones if and only if it is a viscosity solution to the \beta-biased infinity Laplacian equation.