  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
maximum likelihood problem is non-concave, making inference difficult. In this
work we generalize this trend, showing that there cannot exist any parametric
model with a concave log-likelihood function that can exhibit intransitive
preferences. Given this result, we motivate a new 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.
