Softmax variant based on rank of vector indices Given a $K$-dimensional real-valued vector $\mathbf{z} = (z_1, z_2, \ldots, z_K)$, I know that the softmax function returns a vector $\sigma(\mathbf{z})$ with positive elements summing to 1 via the following formula:
$$
\sigma(\mathbf{z})_j = \frac{e^{z_j}}{\sum_{k=1}^K e^{z_k}}, j = 1, \ldots, K
$$
Recently a colleague mentioned to me a variant of the softmax function, which I'll call $\sigma'(\cdot)$, that takes the following form:
$$
\sigma'(\mathbf{z})_j = \frac{\alpha^{r(\mathbf{z}, j)}}{\sum_{k=1}^K \alpha^{r(\mathbf{z}, k)}}, j = 1, \ldots, K
$$
Here, $\alpha > 0$ is a constant and $r(\mathbf{z}, j)$ is the rank of $z_j$ within the vector $\mathbf{z}$, so the smallest value takes rank 1, the second smallest rank 2, and so on. The largest index takes rank $K$. 
Does this softmax variant (or a similar one based on the ranks of a vector's values instead of the values themselves) have a name, and is it used in practice?
 A: I could not find this variant after quite a bit of Googling, Binging, Duckduckgoing... whoever uses it probably keeps it to themselves.
I found this paper on Time series forecasting using a weighted cross-validation
evolutionary artificial neural network ensemble by Donate et al. (Neurocomputing, 2013) where as an alternative to softmax weighting they use rank-based weight in the same way as in the $\sigma^\prime$ where they set $\alpha = e$;  (2.7182...) but that's about it.
I suspect you know of the hierarchical softmax function as a variant of the softmax (or normalized exponential) but this is not related to ranks. A distant relation I can see is with the softrank function by Taylor et al., which is the normalized discounted cumulative gain (see here for an application in a GP setting).
A: Yes, it is used in practice, generally in winner-take-all situation or situation where you want to assign 1 (close to) to maximum value and 0 (close to) to all others. And it is still called softmax. 
If you think about it, the value for $\sigma'(\mathbf{z})_K$ will be very close to 1 and value for all others will be close to 0. See this paper titled "Softmax to Softassign: Neural Network Algorithms for Combinatorial Optimization" for more information: 
https://www.researchgate.net/profile/Anand_Rangarajan/publication/2458489_Softmax_to_Softassign_Neural_Network_Algorithms_for_Combinatorial_Optimization/links/0deec5214b1cb9dbde000000.pdf#page=6
