Abstract:We propose and study a ``classification'' of Borel ideals based on a natural infinite game involving a pair of ideals. The game induces a pre-order $\sqsubseteq$ and the corresponding equivalence relation. The pre-order is well founded and ``almost linear''. We concentrate on $F_{\sigma}$ and $F_{\sigma\delta}$ ideals. In particular, we show that all $F_{\sigma}$-ideals are $\sqsubseteq$-equivalent and form the least equivalence class. There is also a least class of non-$F_{\sigma}$ Borel ideals, and there are at least two distinct classes of $F_{\sigma\delta}$ non-$F_{\sigma}$ ideals.
Keywords: ideals on countable sets, comparison game, Tukey order, games on integers
AMS Subject Classification: 03E15 03E05