# How to assign a score to items based on a set of partial rankings

I have the following setup:

There is a collection of items I and a collection of partial rankings V. That is, an element of V is a total ordering on a subset of I. There is no expectation of consistency among the elements of V: it may be that x < y for one element and y < x for another.

I would like to assign a score $s : I \to \mathbb{R}$ which in some sense captures these rankings. That is, I would like s(x) < s(y) to mean "x tends to be less than y for elements of V which have both in their domain". I'm not sure of what a good way to do this is.

Arrow's impossibility theorem puts some constraints on what can be achieved here, because given a set of votes and a scoring function like this we could use the scoring function to define a total order on the items, which is then constrained by the theorem.

I suppose I'm really looking for references rather than an answer to this question (although both would be appreciated): I'm sure there's a body of theory around this, but I have no idea what it is like or what it's called, so I'm at a bit of a loss as to where to start looking for a solution.

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If your partial rankings were all the same size, then the ranking score at least seems to be special --- that is, assign the ranking loser a score of 1, the next-up 2, etc.; and then sum over partial rankings. This score seems to be in a special sub-variety of the scoring simplex, which behaves well w.r.t. permutations --- details escape me, but for a Reference, I'm recalling this much as it was in a recent AMS Notices (in the past one or two years, I think). – some guy on the street May 8 '10 at 15:37
Unfortunately the rankings are not likely to be of the same size - they're probably all going to be of similar size (and small compared to the size of I) though. – David R. MacIver May 8 '10 at 15:41