  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 pick
respectively a direction and then, instead of flipping a coin in order to
decide 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 which
chooses at the same time both the direction and the distance to travel.
Starting from this game, we heuristically we derive a deterministic non-local
integro-differential equation that we call "infinity fractional Laplacian". We
study existence, uniqueness, and regularity, both for the Dirichlet problem and
for a double obstacle problem, both problems having a natural interpretation as
"tug-of-war" games.
