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We study a version of the stochastic "tug-of-war" game, played on graphs andsmooth domains, with the empty set of terminal states. We prove that, when therunning payoff function is shifted by an appropriate constant, the values ofthe game after n steps converge in the continuous case and the case of finitegraphs with loops. Using this we prove the existence of solutions to theinfinity Laplace equation with vanishing Neumann boundary condition.

Tug-of-war and infinity Laplace equation with vanishing Neumann boundary condition