  Ranking data arises in a wide variety of application areas but remains
difficult to model, learn from, and predict. Datasets often exhibit
multimodality, intransitivity, or incomplete rankings---particularly when
generated by humans---yet popular probabilistic models are often too rigid to
capture such complexities. In this work we leverage recent progress on similar
challenges in discrete choice modeling to form flexible and tractable
choice-based models for ranking data. We study choice representations, maps
from rankings (complete or top-$k$) to collections of choices, as a way of
forming ranking models from choice models. We focus on the repeated selection
(RS) choice representation, first used to form the Plackett-Luce ranking model
from the conditional multinomial logit choice model. We fully characterize, for
a prime number of alternatives, the choice representations that admit ranking
distributions with unit normalization, a desirably property that greatly
simplifies maximum likelihood estimation. We further show that only specific
minor variations on repeated selection exhibit this property. Our choice-based
ranking models provide higher out-of-sample likelihood when compared to
Plackett-Luce and Mallows models on a broad collection of ranking tasks
including food preferences, ranked-choice elections, car racing, and search
engine relevance tasks.
