Computing Parametric Ranking Models via Rank-Breaking

Rank breaking is a methodology introduced by Azari Soufiani et al. (2013a) for applying a Generalized Method of Moments (GMM) algorithm to the estimation of parametric ranking models. Breaking takes full rankings and breaks, or splits them up, into counts for pairs of alternatives that occur in particular positions (e.g., first place and second place, second place and third place). GMMs are of interest because they can achieve significant speed-up relative to maximum likelihood approaches and comparable statistical efficiency. We characterize the breakings for which the estimator is consistent for random utility models (RUMs) including Plackett-Luce and Normal-RUM, develop a general sufficient condition for a full breaking to be the only consistent breaking, and provide a trichotomy theorem in regard to single-edge breakings. Experimental results are presented to show the computational efficiency along with statistical performance of the proposed method.

[1]  F. Mosteller.  Remarks on the method of paired comparisons: I. The least squares solution assuming equal standard deviations and equal correlations , 1951 .

[2]  R. Luce,et al.  Individual Choice Behavior: A Theoretical Analysis. , 1960 .

[3]  Vincent Conitzer,et al.  Common Voting Rules as Maximum Likelihood Estimators , 2005, UAI.

[4]  Steven T. Berry,et al.  Automobile Prices in Market Equilibrium , 1995 .

[5]  David C. Parkes,et al.  Random Utility Theory for Social Choice , 2012, NIPS.

[6]  D. Hunter.  MM algorithms for generalized Bradley-Terry models , 2003 .

[7]  R. A. Bradley,et al.  RANK ANALYSIS OF INCOMPLETE BLOCK DESIGNS , 1952 .

[8]  R. Duncan Luce,et al.  Individual Choice Behavior: A Theoretical Analysis , 1979 .

[9]  Ariel D. Procaccia,et al.  When do noisy votes reveal the truth? , 2013, EC '13.

[10]  L. Hansen.  Large Sample Properties of Generalized Method of Moments Estimators , 1982 .

[11]  L. Bottou,et al.  Generalized Method-of-Moments for Rank Aggregation , 2013 .

[12]  R. A. Bradley,et al.  RANK ANALYSIS OF INCOMPLETE BLOCK DESIGNS THE METHOD OF PAIRED COMPARISONS , 1952 .

[13]  David C. Parkes,et al.  Preference Elicitation For General Random Utility Models , 2013, UAI.

[14]  Devavrat Shah,et al.  Ranking: Compare, don't score , 2011, 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[15]  Tie-Yan Liu,et al.  Learning to rank for information retrieval , 2009, SIGIR.

[16]  Tie-Yan Liu,et al.  Learning to Rank for Information Retrieval , 2011 .

[17]  Nicolas de Condorcet.  Essai Sur L'Application de L'Analyse a la Probabilite Des Decisions Rendues a la Pluralite Des Voix , 2009 .

[18]  Edith Hemaspaandra,et al.  The complexity of Kemeny elections , 2005, Theor. Comput. Sci..

[19]  Craig Boutilier,et al.  Learning Mallows Models with Pairwise Preferences , 2011, ICML.

[20]  R. Plackett.  The Analysis of Permutations , 1975 .

[21]  Devavrat Shah,et al.  Iterative ranking from pair-wise comparisons , 2012, NIPS.

[22]  R. A. Bradley,et al.  Rank Analysis of Incomplete Block Designs: I. The Method of Paired Comparisons , 1952 .

[23]  David C. Parkes,et al.  Generalized Random Utility Models with Multiple Types , 2013, NIPS.