Converting a list of partial rankings into a global ranking I'm working on something like the following problem. I have a bunch of users and N books. Each user creates an ordered ranking of all the books he's read (which is likely a subset of the N books), e.g., Book 1 > Book 40 > Book 25.
Now I want to turn these individual user rankings into a single ordered ranking of all the books.
Are there any good or standard approaches to try? So far, I'm thinking of Bradley-Terry models applied to pairwise comparisons, but I'm wondering if there's anything else.
 A: Plackett-Luce ranking models deal with this problem and are a likelihood based technique where the likelihood is maximized using a majorization-maximization routine, which is similar to Expectation Maximization, in the sense that they use an auxiliary objective function over the likelihood function which is optimized to guarantee iterative monotonic maximization of the likelihood function. (see MM algorithms for Plackett-Luce ranking models by David Hunter). He provides code as well. 
From a ranking perspective, they are an extension of Bradley-Terry models which you mention in your post. Bradley-Terry models estimate a global ranking from a sample of pairwise rankings. Plackett-Luce models extend this to rankings of length $>=$ 2. They also allow for each sample being a ranking of a different length.
This fits your dataset perfectly:
Book 1 > Book 40 > Book 25
Book 40 > Book 30
Book 25 > Book 17 > Book 11 > Book 3 etc.
A: If you're interested in use (more than in development), you should give a try to rankade, our ranking system.
Rankade is free and easy to use, and it's different from Bradley-Terry model and Elo ranking system (here's a comparison) because it can manage matches with 2+ factions (i.e. books, in your scenario). Inserting user's ordered rankings (as matches between two or more books, with detailed final standings, including ties) you'll obtain the single ordered ranking of all the books you're looking for.
In addiction, rankade give you the opportunity to check time evolution for books ranking, and stats for books match-ups, and more.
