Stability for Take-Away Games

Abstract

In this paper, we study a family of take-away games called $\alpha$-tag, parametrized by a real number $\alpha \ge 1$. We show that for any given $\alpha$, there is a half-open interval $I_{\alpha }$ containing $\alpha$ such that the set of losing positions for $\alpha$-tag is the same as the set of losing positions for $\beta$-tag if and only if $\beta \in I_{\alpha }$. We then end with some results and conjectures on the nature of these intervals.

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