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I am currently reading the book Combinatorial Game Theory by Aaron Siegel. I tried to solve Problem 3.10 in Chapter II, which is a stronger form of the Number Avoidance Theorem:

Suppose $x$ is equal to a number and $G$ is not, and let $H$ be an arbitrary game. If left has a winning move of the form $G + H + x^L$, then she has a winning move of the form $G^L + H + x$.

I tried to solve this on my own, but failed so far. If $x$ is not just equal to a number but actually is a number, then the statement is clear by comparing incentives of $x$ and $G$. I tried to solve the general case by reducing $x$ to canonical form step-by-step. This worked fine as long as the option $x^L$ from the statement is not reversible. If it is reversible, I do not see any way to proceed.

Does anyone know of a better method to solve this problem?

Tzimmo
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1 Answers1

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The statement from Siegel's book is false. Here is a counterexample:

Let $G \equiv {\uparrow}$, $H \equiv *2$, and $x \equiv * + * = 0$. Then $G + H + x^L \equiv {\uparrow} + *2 + * = {\uparrow}{*}3 > 0$ is a win for Left. If the statement from the problem were true, there would be a left option $G^L$ such that $G^L + H + x \ge 0$. However, $G^L + H + x \equiv 0 + *2 + (* + *) = *2 \not\gtrless 0$.

Tzimmo
  • 1,299