We give a self-contained and elementary proof for boundedness, existence, and uniqueness of solutions to dynamic programming principles (DPP) for biased tug-of-war games with running costs. The domain we work in is very general, and as a special case contains metric spaces. Technically, we introduce game-trees and show that a discretized flow converges uniformly, from which we obtain not only the existence, but also the unique-ness. Our arguments are entirely deterministic, and also do not rely on (semi-)continuity in any way; in particular, we do not need to mollify the DPP at the boundary for well-posedness.

We prove existence and uniqueness of viscosity solutions for the following problem: max {−∆p1u(x), −∆p2u(x)} = f(x) in a bounded smooth domain Ω ⊂ RN with u = g on ∂Ω. Here −∆pu = (N + p)−1|Du|2−pdiv(|Du|p−2Du) is the 1-homogeneous p−Laplacian and we assume that 2 ≤ p1, p2 ≤ ∞. This equation appears naturally when one considers a tug-of-war game in which one of the players (the one who seeks to maximize the payoff) can choose at every step which are the pa-rameters of the game that regulate the probability of playing a usual Tug-of-War game (without noise) or to play at random. Moreover, the opera-tor max {−∆p1u(x), −∆p2u(x)} provides a natural analogous with respect to p−Laplacians to the Pucci maximal operator for uniformly elliptic operators. We provide two different proofs of existence and uniqueness for this prob-lem. The first one is based in pure PDE methods (in the framework of viscosity solutions) while the second one is more connected to probability and uses game theory.