Parametric Models for Intransitivity in Pairwise Rankings

There is a growing need for discrete choice models that account for the complex nature of human choices, escaping traditional behavioral assumptions such as the transitivity of pairwise preferences. Recently, several parametric models of intransitive comparisons have been proposed, but in all cases the model log-likelihood is non-concave, making inference difficult. In this work we generalize this trend, showing that there cannot exist an parametric model that both (i) has a log-likelihood function that is concave in item-level parameters and (ii) can exhibit intransitive preferences. Given this observation, we also contribute a new simple model for analyzing intransitivity in pairwise comparisons, taking inspiration from the Condorcet method (majority vote) in social choice theory. The majority vote model we analyze is defined as a voting process over independent Random Utility Models (RUMs). We infer a multidimensional embedding of each object or player, in contrast to the traditional one-dimensional embedding used by models such as the Thurstone or Bradley-Terry-Luce (BTL) models. We show that a three-dimensional majority vote model is capable of modeling arbitrarily strong and long intransitive cycles, and can also represent arbitrary pairwise comparison probabilities on any triplet. We provide experimental results that substantiate our claims regarding the effectiveness of our model in capturing intransitivity for various pairwise choice tasks such as predicting choices in recommendation systems, winners in online video games, and elections.