Computing similarity between two lists

Question

EDIT: as everyone is getting confused, I want to simplify my question. I have two ordered lists. Now, I just want to compute how similar one list is to the other.

Eg

Answer 1

Is the list of documents exhaustive? That is, is every document rank ordered by system 1 also rank ordered by system 2? If so a Spearman's rho may serve your purposes. When they don't share the same documents, the big question is how to interpret that result. I don't think there is a measurement that answers that question, although there may be some that implement an implicit answer to it.