# Defining “average rank” when not every ranking covers the whole set

Here's a mathematical modeling problem I came across while working on a hobby project.

I have a website that presents each visitor with a list of movie titles. The user has to rank them from most to least favorite. After each visit, I want to create a cumulative ranking that takes into account each visitor's individual ranking. Normally I would just take the mean ordinal rank: e.g., if Person A rated "Avatar" 10th and Person B rated it 20th, its cumulative rank would be 15th. However, new movies will be added to the list as the website grows, so each person will have ranked only a subset of the full movie list.

Any thoughts on how I can define "average rank" when some rankings do not cover the whole set? My best idea so far is to model this as a directed graph, where nodes are movies and weighted edges are preferences (e.g. "10 people ranked 'Avatar' right above 'District 9'"), and then finding sinks and sources. How else could one go about this?

(Sorry if this question is too applied.)

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It's not a duplicate, but there's some relation to this question: mathoverflow.net/questions/6019/… – Tom Leinster Feb 21 '10 at 3:48
What is the final goal of this; i.e. what do you want your cumulative ranking to reflect? Depending on this, there are various possibilities. E.g. if you just want to get something that generally reflects the popularity of each movie, you can go with a trivial extension of what you do now. For each movie, just compute the fraction of people who ranked it 1st, 2nd, etc. -- the fraction is relative to the total number of people who ranked this movie. Then do something with this (take the average w.r.t. the fractions, or the mode, etc.) – user3035 Feb 21 '10 at 4:04
this is question related to the netflix prize? – john mangual Feb 21 '10 at 4:14
@John: not related to the Netflix Prize, although maybe there is something I can learn from that. – RexE Feb 21 '10 at 4:20