On finiteness in the card game of war

The game of war is a popular international children's card game. In the beginning of the game, the deck is split into two parts, then each player reveals their top card. The player having the highest card collects both and returns them to the bottom of their hand. The player left with no cards loses. It is often wrongly assumed that this game is deterministic and the result is set once the cards have been dealt. However, this is not so; the rules of the game do not prescribe the order in which the winning player will place their cards on the bottom of the hand. First, we provide an example of a cycling game with fixed rules and then assume that each player can seldom but regularly change the returning order. We have proved that in this case the mathematical expectation of the length of the game is finite. In principle it is equivalent to the graph of the game, which has edges corresponding to all acceptable transitions, having the following property: from each initial configuration there is at least one path to the end of the game