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