A Generalized Bradley-Terry Model: From Group Competition to Individual Skill

Abstract

\"The Bradley-Terry model for obtaining individual skill from paired comparisons has been\r\npopular in many areas. In machine learning, this model is related to multi-class probability\r\nestimates by coupling all pairwise classification results. Error correcting output codes\r\n(ECOC) are a general framework to decompose a multi-class problem to several binary\r\nproblems. To obtain probability estimates under this framework, this paper introduces a\r\ngeneralized Bradley-Terry model in which paired individual comparisons are extended to\r\npaired team comparisons. We propose a simple algorithm with convergence proofs to solve\r\nthe model and obtain individual skill. Experiments on synthetic and real data demonstrate\r\nthat the algorithm is useful for obtaining multi-class probability estimates. Moreover, we\r\ndiscuss four extensions of the proposed model: 1) weighted individual skill, 2) home-field\r\nadvantage, 3) ties, and 4) comparisons with more than two teams\"