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Motivated by the "tug-of-war" game studied in [12], we consider a "non-local"version of the game which goes as follows: at every step two players pickrespectively a direction and then, instead of flipping a coin in order todecide which direction to choose and then moving of a fixed amount $\epsilon>0$(as is done in the classical case), it is a $s$-stable Levy process whichchooses at the same time both the direction and the distance to travel.Starting from this game, we heuristically we derive a deterministic non-localintegro-differential equation that we call "infinity fractional Laplacian". Westudy existence, uniqueness, and regularity, both for the Dirichlet problem andfor a double obstacle problem, both problems having a natural interpretation as"tug-of-war" games.

Non-Local Tug-of-War and the Infinity Fractional Laplacian