Given a collection of sample rankings, what is the best way to compile them into an aggregate ranking? (Don't worry, I'm working towards a well-defined question.) There are at least two obvious applications: voters' ballots in an election, and game rankings.
The Kemeny-Young method is intended for the former situation. It works on the assumption that each sample ranking includes all of the candidates to be included in the final ranking, as is normal for electoral ballots. It uses the model that there is an objective "correct" (discrete) ranking such that each individual ballot is a noisy version of the ranking; the algorithm is equivalent to a form of maximum likelihood estimation.
This model is of questionable relevance to political elections, but seems appropriate for game rankings. Consequently, I am interested in a modified version, where
most "ballots" do not rank all or even a majority of the candidates, and
every candidate has an independent real-valued "score" which follows a normal distribution with variance 1 and an unknown mean "base score"; each ballot takes sample scores for a subset of the candidates and then ranks them.
The question is, how to find a maximum likelihood estimate - or decent approximation thereof - of the candidates' base scores (up to an additive constant)? I tried to find a solution for the simplified case where each ballot is a single pairwise comparison, but got bogged down in equations. This isn't really a strong area of mine. Any suggestions?
