A game in combinatorial game theory

Question

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.

Teach me, please.

Answer 1

Games $H$ such that $H=\{G_L|-G_L\}$ form a closed set under addition, satisfy $H+H=0$, and must be either equal to $0$ or fuzzy with $0$. Some of these are nimbers: $*=\{0|0\}, *2=\{0,*|0,*\}$, etc. Those that are not nimbers and not zero seem to be typically written as $\pm(G_L)$, such as $\pm 1 = \{1|-1\}$, so your game is $$G=\{\uparrow, * \big\vert\downarrow,*\} = \pm(\uparrow,*).$$ Your game and the result of adding $*$ to it, $$G+* = \pm(0,\uparrow*),$$ are two of the simpler such games, with birthday equal to $3$. They satisfy $$ \downarrow*\text{}<G<\text{}\uparrow*\qquad\text{and}\qquad\downarrow\text{}< G+*<\quad\uparrow, $$ giving an idea of how "unfuzzy" they are; but $G$ is fuzzy with $\uparrow$ and $\downarrow$. (Note that $*$ behaves exactly the same as $G$ in all these particulars $-$ it is fuzzy with $\uparrow$ and $\downarrow$, strictly between $\uparrow*$ and $\downarrow*$, and becomes between $\uparrow$ and $\downarrow$ after adding $*$ to it.)

Answer 2

You can check that $G$ has no dominated or reversible options (this is easy once you've checked that $\uparrow*$, $G$, and $G + *$ are all fuzzy), so the form you've given is already the canonical form. Being in canonical form doesn't mean it couldn't be a sum of simpler objects, though.

However, in this case, $G$ doesn't simplify to any sum of ups, downs, or nimbers. Two (or more) ups plus a nimber is positive, whereas $G$ is not. A single up is also positive. The game $\uparrow *n$, if $n>1$, is also positive. The game $\uparrow *$ is fuzzy, but 2 of them added together give $\uparrow \uparrow$ which is positive whereas $G+G=0$.