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.