Rank Centrality: Ranking from Pair-wise Comparisons Sahand Negahban Statistics Department, Yale University, 24 Hillhouse Ave, New Haven, CT 06510 ,
[email protected] Sewoong Oh Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, 104 S. Mathews Ave., Urbana, IL 61801,
[email protected] Devavrat Shah* Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Massachusetts Ave., Cambridge, MA 02139,
[email protected] The question of aggregating pair-wise comparisons to obtain a global ranking over a collection of objects has been of interest for a very long time: be it ranking of online gamers (e.g. MSR's TrueSkill system) and chess players, aggregating social opinions, or deciding which product to sell based on transactions. In most settings, in addition to obtaining a ranking, finding `scores' for each object (e.g. player's rating) is of interest for understanding the intensity of the preferences. In this paper, we propose Rank Centrality, an iterative rank aggregation algorithmy for discovering scores for objects (or items) from pair-wise comparisons. The algorithm has a natural random walk interpretation over the graph of objects with an edge present between a pair of objects if they are compared; the score, which we call Rank Centrality, of an object turns out to be its stationary probability under this random walk. To study the efficacy of the algorithm, we consider the popular Bradley-Terry-Luce (BTL) model (equiv- alent to the Multinomial Logit (MNL) for pair-wise comparisons) in which each object has an associated score which determines the probabilistic outcomes of pair-wise comparisons between objects.