  In this work we discuss a random Tug-of-War game in graphs where one of the
players has the power to decide at each turn whether to play a round of
classical random Tug-of-War, or let the other player choose the new game
position in exchange of a fixed payoff. We prove that this game has a value
using a discrete comparison principle and viscosity tools, as well as
probabilistic arguments. This game is related to Jensen's extremal equations,
which have a key role in Jensen's celebrated proof of uniqueness of infinity
harmonic functions.
