The proposed question is rather complicated. As analystic already pointed out, I don't think all these measures can be compared straightforwardly, because rank correlation coefficients, Gini coefficient, and AUC (area under ROC curve) are generally defined on different domains.
However, there is a very close relation between Kendall's $\tau$ and Spearman's $\rho$, the two rank correlation coefficients in the list. While the paper cohoz mentioned has demonstrated their relation empirically (Figure 3), this relation can actually be quantified theoretically. Let $\pi$ and $\sigma$ be two rankings, and $\pi(i)$ and $\sigma(i)$ be the ranks of item $i$ in $\pi$ and $\sigma$, respectively. The Kendall distance and Spearman distance between $\pi$ and $\sigma$ are defined as follows:
$$
K(\pi,\sigma) = \# \lbrace \; (i,j) \, \vert \, \pi(i)>\pi(j) \text{ and } \sigma(i)<\sigma(j) \; \rbrace
$$
$$
S(\pi,\sigma) = \sum_i \left( \pi(i) - \sigma(i)\right)^2
$$
We have the following relation between $K$ and $S$ following [Diaconis and Graham 1977]:
$$
\frac{1}{\sqrt{n}}K(\pi,\sigma) \le S(\pi,\sigma) \le 2K(\pi,\sigma)
$$
Because the rank correlation coefficients are just the normalization of the rank distances to the interval $[-1,1]$, similar inequalities can be easily derived between $\tau$ and $\rho$. In the statistical ranking literature, results are mostly represented in terms of distances rather than coefficients.
Two more things:
- The rankings $\pi$ and $\sigma$ must be complete rankings in order to make this inequality hold. That is, they cannot be partial rankings.
- In case one is interested in $\tau$ and $\rho$ defined not only on rankings but on continuous random variables, the situation is more involved. Here is a related paper by Fredicks and Nelsen.
