∞ Magnifying

In Winning Ways Volume 2 (pg. 334) they say

In many ways the games $\infty$ and $\pm\infty$ behave like enormously magnified versions of $\uparrow$ and $*$

& give us an infinitely magnifying operation defined by

$$\int^\mathbb{Z}G=\{\int^\mathbb{Z}G^L+n|\int^\mathbb{Z}G^R-n\}_{n=0,1,2,...}$$

As an example they also give:

$$\int^\mathbb{Z}*=\{\mathbb{Z}|\mathbb{Z}\}=\pm\infty$$

Which reminds me of the function $*+n=\{n|n\}$..


After reading the affine games paper & then seeing the formula in my notes again, I got to thinking…

They essentially define $0=\{-\infty|\infty\}$. No problems there. But they also define $\rightmoon=\{\infty|-\infty\}=\pm\infty$. I don’t have a problem with that. I just think $\rightmoon$ is, well.. bigger 🤷‍♀️

Aside interval arithmetic & fuzzy numbers

It made me wonder if something like

$$\int^{\mathbb{O}n}*=\{\mathbb{O}n|\mathbb{O}n\}=\pm\text{on}$$

would be valid.

I also wondered about this equation, because I find it rather aesthetic:

$$\int^{\rightmoon} *=\{\rightmoon|\rightmoon\}=\pm\rightmoon=\rightmoon$$

I might call this a moon magnifying operation & speculate that

$$\int^{\rightmoon} n=\rightmoon:\forall n$$


I was also trying to think about game analysis for projective games. I think chess endgames could be a good fit since loopy situations are possible. Also, affine theory defines check games. So it seems a rather natural setting.

Was playing around with some custom cases as well just for fun ( $1$ x $n$ boards w/ 2 kings).

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