A game in combinatorial game theory [closed]

I have been reading the Winning Ways, the bible (?) of Combinatorial Game Theory.

I tried to calculate some games of the form {L|R}. But it is not easy to me.

For example, I don't know what {$\uparrow$,$\ast$ | $\downarrow$,$\ast$} is.

The game, say $G$, is fuzzy and $G+G=0$.

So I thought that the game might be $\ast$.

But $G+\ast$ is still fuzzy. Moreover $G+*n ~||~ 0$. Thus G is not an impartial game.

I think $G$ can be simplified to the comibination of several symbols like $\ast$ or nimbers. But I have no idea.