Prohibiting repetitions makes playing games substantially harder

Abstract

If one prohibits repetition of moves (i.e., moving to a position the player has already moved to before) in combinatorial games, a won game cannot be successfully played using a strategy in the narrow sense only. In general, one has to regard the actual development of the play arbitrarily far back and process it in an arbitrarily complex way. The problem of best play in such a game whose graph of positions is explicitly given is, indeed, polynomial space complete.