Does War have infinite expected length?
My question concerns the (completely deterministic) card game known as War, played by seven-year-olds everywhere, such as my son Horatio, and sometimes also by others, such as their fathers.
The question is: Is the expected length of the game infinite?
Let us assume that the cards are returned to the deck in a well-defined manner. For example, in the order that the cards are played, with the previous round's winner's cards going first (and a first player selected for the opening battle).
On the Wikipedia page, they tabulate the results of 1 million simulated random games, reporting an average length game of 248 battles. But this does not actually answer the question, because it could be that there is a devious initial arrangement of the cards leading to a periodic game lasting forever. Since there are only finitely many shuffles, this devious shuffle will contribute infinitely to the Expected Value. Thus, the question really amounts to:
Question. Is there a devious shuffle in War, which leads to an infinitely long game?
Of course, the game described above is merely a special case of the more general game that might be called Universal War, played with N players using a deck of cards representing elements of a finite partial pre-order. Any strictly dominating card wins the trick; otherwise, there is war amongst the players whose cards were not strictly dominated. Does any instance of Universal War have infinite expected length?