# uses of {|} notation in combinatorial game theory

I am a geometer. but i have become interested in combinatoral game theory because of 2 things. 1)Go 2) proving mathematical theory's can be regarded as a 2 player game.

so i studied 100 pages of winning ways. and learnt about the {|} notation. at first i found it weird cause we always know who is going to make the first move. so why should i write it this way? then after learning several other weird things like {-2|+3} = 0 i kinda guessed that the whole theory is based on addition of games. we consider every players option cause we might choose a different game and then our opponent will have those options available to him or every game that won't change the result under addition is zero. so no surprise that the theory's examples usually come from games that can be broken to smaller games. even it's application to go is in the end game that the game is broken to pieces.

So the only way i know this notation helps is that if one can break a game to some sub games and understand those sub games well, then he can understand the whole game.

my question is: is there any other way this notation can help? for example is there an example that i break a game into sub games such that the product of those sub games is the whole game and this leads to understanding the whole game?