You can also not define anything up front; then take the proof that Bob can enforce that there's no convergence to any rational number.
Next step, look at the process of playing the game, and show that you can define operations like adding any two games or comparing them for equality etc.
(In some sense, that would be similar to showing that Dedekind cuts define a sensible structure. With the added complication that the game doesn't have to converge to arbitrary small intervals, because the players could agree to leave a finite interval like [1/4, 3/4] untouched, eg Alice plays 1/4-1/n and Bob plays 3/4+1/n.
You'd just be defining interval arithmetic on real numbers.)
Next step, look at the process of playing the game, and show that you can define operations like adding any two games or comparing them for equality etc.
(In some sense, that would be similar to showing that Dedekind cuts define a sensible structure. With the added complication that the game doesn't have to converge to arbitrary small intervals, because the players could agree to leave a finite interval like [1/4, 3/4] untouched, eg Alice plays 1/4-1/n and Bob plays 3/4+1/n.
You'd just be defining interval arithmetic on real numbers.)