Stability for Take-Away Games

TL;DR

This paper investigates a family of take-away games called alpha-tag, revealing that the set of losing positions remains constant within specific intervals of the parameter alpha, and explores the properties of these intervals.

Contribution

It introduces the concept of intervals of stability for the set of losing positions in alpha-tag games, providing new insights into their structure and conjecturing about their nature.

Findings

Existence of half-open intervals I_alpha where losing positions are invariant

Characterization of stability intervals for different alpha values

Open questions and conjectures about the structure of these intervals

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.