One neat thing I've realized is that if the game does terminate, then the player with the highest card has to win. That is, if two players split a deck of 20 cards, then the player with the "20" card in their hand must win. This is for the obvious reason that no other card can beat it. Indeed, even if this player happened to have one card left, it must necessarily be the "20" card, which would beat anything the other player has. Consequently, the player can in fact make a comeback from having only one card and win the game. Of course, the game might also repeat with some period. But again, if the game can terminate, then the winner must necessarily be the player with the highest card.
Similar arguments can be made regarding the player who has the highest successive cards. For instance, again playing with 20 cards, if a player happens to have card "19" and "20," then this player will always have at least two cards. Of course from before, since this player has the highest card, they'll necessarily win. But having a 19 means if winning is possible, then they're likely to win earlier.
This last idea then suggests looking at sequences of permutations of consecutive cards. That is, if a player has in their hand some permutation of cards, say, 17, ..., 20, then again they'll necessarily win, but even faster than just having 19 and 20 because the other player cannot beat anything above 16 as this player has them all. This then got me thinking about assigning "weights" to a player's hand based off subsets of the previous kind. For instance, having 17, ..., 20 has a greater "pull of winning" then say 5, ..., 9. Of course, the player with the highest card must win if it is possible, but this idea of weights might capture the rate at which the game is played and perhaps then also indicate if the game will be periodic. Depending on the notion of weights introduced, perhaps "playing a game with hands of equal weights will never terminate."
I have been exploring these ideas and will update you accordingly of any new developments. If possible, I would like to work on this project on my own like I have been, and would thus appreciate you not forwarding this message to the class. Of course, you may inform them about the player with the highest card having to win. I believe it is an obvious point, but it leads to some interesting ideas.