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Given a zero-sum infinite game we examine the question if players have optimal memoryless deterministic strategies. It turns out that under some general conditions the problem for twoplayer games can be reduced to the same problem for one-player games which in turn can be reduced to a simpler related problem for multi-armed bandits.doi:10.4230/lipics.fsttcs.2010.65 dblp:conf/fsttcs/Zielonka10 fatcat:kwcsmza5grfafnd3eoqeuqvlhu