||Not really. Without any other restrictions this would lead to a possibly infinite game. At any moment a player may force a "draw" by repeatedly undoing his opponent's moves.
Note that in the horses game you have a measure that causes the game to be finite: the sum of all horses' coordinates decreases. You wouldn't have any such measure in your game. Or, in other words, you need an additional constraint that would guarantee what you called "the condition that the game will converge at last".