# Inferring values based on rankings

I'm looking to infer items' values from only ranked lists of these items. I'm assuming that each item has some value, for which a higher value makes it more likely to appear first. This value can be on any scale (meaning, I don't need this value to be on any particular scale) as long as it can be compared to other items' values.

I have N items, and M ranked lists of these N items, where each ranked list contains a subset of N (anywhere from 2 to N items).

How can I infer these values--or at least, get some idea about a value's distribution?

(P.S.: I'm not sure if "order-statistics" is the right tag for this post.)

• @Nick I believe the situation may be subtler than that. Suppose I have $M=4$ lists in which subsets of a fixed collection of $N=3$ items are ranked AB, AC, CB, and ABC, from first to last in each case. Could we posit some reasonable probability model to handle the discrepancies among these lists and from that develop, say, estimates of the probabilities of the correct ranks of each of A, B, and C? This takes a very general approach to the concept of "value" as meaning only "relative value within the $N$ items"--rather than of any absolute value, as you have suggested. – whuber Dec 2 '15 at 18:08
• I'm not positive what you mean, @whuber, but "develop[ing] [...] estimates of the probabilities of the correct ranks of each of A, B, and C" is precisely what I want to do. Is there a way I could revise my question to better reflect that? – uberpro Dec 2 '15 at 19:14
• One possible direction into which you could look are Bradley-Terry type models which infer "worth parameters" on a latent scale based on paired comparisons. And it is possible to convert the rankings into paired comparisons. In R, the packages prefmod, BradleyTerry2, eba, or psychotools would be useful for this. But there are also models that directly work on rankings but I'm not sure which of these also infer positions on a latent continuous scale. – Achim Zeileis Dec 2 '15 at 19:43