I suppose this is a pretty common problem since when we use ranks we are often more interested in how similar they are at the top (or at the bottom, or both extremes) than in how they compare in the middle. Example of two ordered objects according to two different external criteria (I am writing them as adjacent letters for simplicity, which doesn't imply any previous relation like a<b in the 2nd sequence):
The pair of sequences abcdefghijk-abcdefkjihg would be closer than abcdefghijk-defghijkabc because the first 3 objects have more weight than the last 5.
The pair of sequences abcdefghijk-defghijkabc would be closer than abcdefghijk-defghicbajk because despite b and c being closer to the top, cba is the oppposite of abc. This second example might show a goal that is not feasible to conciliate with the first example in an objective way, so each example probably belong to two different measures, which would not be a problem in my case.
Kendall's tau seems adequate for a non-weighted comparison, please correct if I am wrong and would need another dissimilarity or rank correlation measure for the non-weighted case.
Besides that, here I am specifically interested in giving the highest weight to the first position and the lowest for the last position, and respective intermediate weights for the other positions. This answer presents a review with several measures, but I am not sure they provide the kind of measure I need.
ps. I am looking for something I can reference, i.e. some well stablished method if possible. I have found this paper, which might not be a widely accepted approach.
