User blog comment:Alemagno12/Ordinals in CGT/@comment-5529393-20171007161637

You are making a common mistake in CGT: it's not just a matter of playing the game out and making the optimal moves every time. If that were the case, then there would only be four possible game values: * (first player win), 0 (second player win), 1 (left player win) or -1 (right player win). CGT is more sophisticated than that; it incorporates the notion if addition and subtraction of games. In particular, games G and H are only considered equivalent if G-H is a second player win.

So, let's look at the game ω, which is {0,*|0,*}. You might think it is going to have game value *, since the best move for the first player is at 0. But, this is not the case. If it were, then {0,*|0,*} - * would be a second player win. But if we play the game out, the best move for first player is to choose * for the first game. Then we have * - *. Then the second player well turn one of the *'s to a 0, and the first player will turn the other, and we end with a first player win. So in fact these games are not the same. So, {0,*|0,*} is a game that is distinct from 0 and *; we call this game *2. More generally, *0 is just 0, and *n is the game {*0, *1, ... ,*(n-1) | *0, *1, ..., *(n-1)}. ω thus has value *2.