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