title: How to merge a collection of ordered preferences
body: <p>I have a set of r reviewers who are rating a set of n objects. Each reviewer independently produces an ordered list of the objects he or she chooses to rank. The goal is to produce one list that is the collation of the various ordered lists. We can assume that each reviewer's point of view is equally weighted.</p>
  
  <p>This differs from most merging and ordered list questions in that there is no global ordering. One reviewer can rate A > B while another can rate B > A. As mentioned, each object is not necessarily rated by each reviewer. </p>
  
  <p>My current thought is to decompose each reviewer's list into a set of ordered tuples for each of the m * (m-1) * .5 unique pairs of entries in the list, where m is the number of objects rated. Now take all the tuples from all reviewers. For a given combination (a,b) find all such tuples and take the majority vote (of those voting) as the determinator of whether a &lt; b. </p>
  
  <p>Now I have a set of ordered tuples that represents the wisdom of all. But how do I turn these into one ordered list? I can start with a randomly chosen pair of objects, and order them, then add another one in the right order, but the output will depend on which one I choose to start with. Also there may be loops.</p>
  
  <p>I'd appreciate any ideas.</p>