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

We prove that if U subset of R-n is an open domain whose closure (U) over bar is compact in the path metric, and F is a Lipschitz function on partial derivative U, then for each beta is an element of R there exists a unique viscosity solution to the beta-biased infinity Laplacian equation beta vertical bar del u vertical bar + Delta(infinity)u = 0 on U that extends F, where Delta(infinity)u = |del u|(-2)Sigma(i, j) u(xi)u(xixj)u(xj). In the proof, we extend the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the beta-biased epsilon-game as follows. The starting position is x(0) is an element of U. At the kth step the two players toss a suitably biased coin (in our key example, player I wins with odds of exp(beta epsilon) to 1), and the winner chooses x(k) with d(x(k), x(k-1)) < epsilon. The game ends when x(k) is an element of partial derivative U, and player II pays the amount F(x(k)) to player I. We prove that the value u(epsilon) (x(0)) of this game exists, and that parallel to u(epsilon) - u parallel to(infinity) -> 0 as epsilon -> 0, where u is the unique extension of F to <(U)over bar> 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.