Narrowing the field of a combitorial game
You have discovered the Calkin–Wilf tree, which is the game tree of your game. There is a unique way to reach any pair of coprime positive integers. Given a pair of coprime positive integers, repeatedly subtract the smaller one from the larger one, and you will eventually reach $(1,1)$. Reversing your moves, you obtain the unique way to generate the given (ordered) pair.