My paper "Cycles in War" addresses this question, too. Does any instance of Universal War have infinite expected length? Any strictly dominating card wins the trick otherwise, there is war amongst the players whose cards were not strictly dominated. 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. Is there a devious shuffle in War, which leads to an infinitely long game? Since there are only finitely many shuffles, this devious shuffle will contribute infinitely to the Expected Value. 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. On the Wikipedia page, they tabulate the results of 1 million simulated random games, reporting an average length game of 248 battles. 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). Let us assume that the cards are returned to the deck in a well-defined manner. If a player runs out of cards while dealing the face-down cards of a war, he may play the last card in his deck as his face-up card and still have a chance to stay in the game. A player wins by collecting all the cards.
In the case of another tie, the war process is repeated until there is no tie.
If the two cards played are of equal value, each player lays down three face-down cards and a fourth card face-up (a "war"), and the higher-valued card wins all of the cards on the table, which are then added to the bottom of the player's stack. In unison, each player reveals the top card on his stack (a "battle"), and the player with the higher card takes both the cards played and moves them to the bottom of his stack. (from (card_game)) The deck is divided evenly among the two players, giving each a face-down stack. The question is: Is the expected length of the game infinite? 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.
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