We consider ordinal online problems, i.e., tasks that only depend on the pairwise comparisons between elements of the input. E.g., the secretary problem and the game of googol. The natural approach to these tasks is to use ordinal online algorithms that at each step only consider relative ranking among the arrived elements, without looking at the numerical values of the input. We formally study the question of how cardinal algorithms (that can use numerical values of the input) can improve upon ordinal algorithms. We give a universal construction of the input distribution for any ordinal online problem, such that the advantage of any cardinal algorithm over the ordinal algorithms is at most $1+\varepsilon$ for arbitrary small $\varepsilon> 0$. However, the value range of the input elements in this construction is huge: $O\left(\frac{n^3\cdot n!}{\varepsilon}\right)\uparrow\uparrow (n-1)$ for an input sequence of length $n$. Second, we identify a natural family of core problems and find a cardinal algorithm with a matching advantage of $1+ \Omega \left(\frac{1}{\log^{(c)}N}\right),$ where $\log^{(c)}N=\log\log\ldots\log N$ with $c$ iterative logs and $c$ is an arbitrary constant $c\le n-2$. Third, we construct an input distribution of only exponential size $N=O((n/\varepsilon)^n)$ for the game of googol such that any cardinal algorithm has advantage of at most $1+\varepsilon$ over ordinal algorithms for arbitrary small $\varepsilon> 0$. Finally, we study the dependency on $n$ of the core problem. We provide an efficient construction of size $O(n)$, if we allow cardinal algorithms to have constant factor advantage against ordinal algorithms.

翻译：我们考虑的在线问题, 也就是说, 任务只能取决于输入元素之间的对等比较 。 例如, 秘书问题 和googol 游戏 。 这些任务的自然方法是使用 ordinal 在线算法, 每一步只考虑对到达元素的相对排序, 而不看输入的数值 。 我们正式研究基本算法( 能够使用输入的数值) 如何改进 ordal 算法 。 我们为任何 ordin 在线问题提供一个通用的输入分布 。 例如, 任何基本算法相对于 ordinal 算法的优势, 最多是 1 $ varepsl > 。 然而, 此构造中输入元素的价值范围是巨大的 : $left( orc{n\\\\\ cdrentral rial) $( n_ ral_ ral_ ral_ ral_ ral_ ral_ ral_ ral_ ral_ ral_ $_ ral_ ral_ ral_ ral_ ral_ ral_ ral_ ral_ ral_ ral_ $_ ral_ ral_ ral_ ral_ $_ ral_ ral_ ral_ $_ $____ ral______________________ ral________________ ral___ r_r_r_r_r_r______________________r_r_r_r_r_r_r__________________________________ral_ral_ral_ral_ral_ral_ral_ral_r_r_r_r_r_r_r_r_r_r_r_r_r_r_______r_r_r_