  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
$N=O\left(\left(\frac{n}{\varepsilon}\right)^n\right)$.
