Neutrality and geometry of mean voting

Mean proximity rules provide a simple geometric framework to achieve consensus among a collection of rankings (votes) over a set of alternatives. They embed all rankings into a Euclidean space, take the mean of the embeddings of the input votes, and return the ranking whose embedding is closest to the mean. Previous work on mean proximity rules has not integrated an important axiom---neutrality---into the framework. By drawing on ideas from the representation theory of finite groups, we show that integrating neutrality actually helps achieve a succinct representation for every mean proximity rule. Various connections are drawn between mean proximity rules and other prominent approaches to social choice.