He has played this game with me twice, and both times got bored and gave up before there was any sign of a conclusion. Which raises the question: is there any guarantee this won't go on forever? One can imagine that it's very difficult to make a player give up an ace (being the highest rank), because it will take either another ace followed by a lost battle, or a lost battle in which the ace falls as one of the two cards paid. And on the other hand, the deuces will continue to be passed from player to player, because the only time you will not have to give it up is if the other player also plays a deuce and you win the following battle.
So I knocked up a quick computer simulation to find out. And it turns out that the short answer is no (it can go on forever). And the long answer is it depends.
And what it depends on is what the winner of each pair of cards does with them.
So the simplest way of doing this by computer is to let player 1 play first. Then player 2 plays a card on top. If they are equal, they stay there, forming a stack; otherwise, the loser pays the appropriate number of cards (if any) on to the stack and then the winner turns the stack over and adds it to the bottom of their pile.
With these rules, roughly half of all possible starting positions lead to the game going on forever. If the game does reach a conclusion, each player will have played an average of 1500–2000 cards by the time the game ends.
But if you play the pairs of cards in a random order, or if you mix up the stack before adding it to the winner's pile, the game always reaches a conclusion, and does so in an average of 300–400 plays. In fact, there doesn't need to be any randomness to achieve this same effect: if you make the winner (or the loser — it doesn't seem to matter) of the previous pair play the first card of the next pair, the game never seems to get into an infinite loop and its average length is again 300–400 plays.
So there are two similar, deterministic versions of this game, one of which frequently goes on forever and the other of which always finishes in reasonable time. I find this curious. It appears that when smallclanger and I played it we were playing the former when we should have been playing the latter.
And now for something completely different. Earlier this month YouGov sent me a poll (on I forget what subject) which randomly and irrelevantly ended on the following question, so now I'm doing the same.
Which, if any, of the following super powers would you MOST like to have?