We consider ordinal online problems, i.e., tasks that only require pairwise comparisons between elements of the input. A classic example is the secretary problem and the game of googol, as well as its multiple combinatorial extensions such as $(J,K)$-secretary, $2$-sided game of googol, ordinal-competitive matroid secretary. A natural approach to these tasks is to use ordinal 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 can improve upon ordinal algorithms. We give first 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 $\varepsilon> 0$. As an implication, previous lower bounds for the aforementioned variants of secretary problems hold not only against ordinal algorithms, but also against any online algorithm. However, the value range of the input elements in our construction is huge: $N=O\left(\frac{n^3\cdot n!\cdot n!}{\varepsilon}\right)\uparrow\uparrow(n-1)$ (tower of exponents) for an input sequence of length $n$. As a second result, we identify a class of natural ordinal problems and find cardinal algorithm with a matching advantage of $1+ \Omega \left(\frac{1}{\log^{(c)}N}\right),$ where $\log^{(c)}N=\log\ldots\log N$ with $c$ iterative logs and $c$ is an arbitrary constant. Further, we introduce the cardinal complexity for any given ordinal online task: the minimum size $N(\varepsilon)$ of different numerical values in the input such the advantage of cardinal over ordinal algorithms is at most $1+\varepsilon$. As a third result, we show that the game of googol has much lower cardinal complexity of $N=O\left(\left(\frac{n}{\varepsilon}\right)^n\right)$.

翻译：我们考虑在线序贯问题，即仅需对输入元素进行两两比较的任务。经典例子包括秘书问题、古戈尔博弈，及其多种组合扩展，如$(J,K)$-秘书问题、双边古戈尔博弈、序贯竞争拟阵秘书问题。处理这些任务的自然方法是使用序贯算法，该算法在每一步仅考虑已到达元素的相对排序，而不关注输入的数值大小。我们正式研究基数算法相较于序贯算法的改进程度。首先，我们针对任意在线序贯问题给出输入分布的通用构造，使得任意基数算法相对于序贯算法的优势至多为$1+\varepsilon$，其中$\varepsilon>0$可任意小。这意味着，前述秘书问题变体的已知下界不仅适用于序贯算法，也适用于任意在线算法。然而，我们构造中的输入元素取值范围极大：对于长度为$n$的输入序列，$N=O\left(\frac{n^3\cdot n!\cdot n!}{\varepsilon}\right)\uparrow\uparrow(n-1)$（指数塔）。第二个结果中，我们识别了一类自然序贯问题，并发现基数算法具有匹配的优势$1+ \Omega \left(\frac{1}{\log^{(c)}N}\right)$，其中$\log^{(c)}N=\log\ldots\log N$含$c$次迭代对数，$c$为任意常数。进一步，我们为给定的在线序贯任务引入基数复杂度：输入中不同数值的最小规模$N(\varepsilon)$，使得基数算法相对于序贯算法的优势至多为$1+\varepsilon$。第三个结果表明，古戈尔博弈的基数复杂度显著更低，为$N=O\left(\left(\frac{n}{\varepsilon}\right)^n\right)$。