On A Mallows-type Model For (Ranked) Choices

TL;DR

This paper introduces a Mallows-type ranking model using a novel distance function called Reverse Major Index, providing a simple closed-form for choice probabilities and effective parameter estimation methods, demonstrated on real data.

Contribution

It proposes a new Mallows-type model with the RMJ distance for ranked choice data, with closed-form choice probabilities and estimation techniques.

Findings

Closed-form choice probability expressions.

Effective parameter estimation methods.

Model generalizes well with limited display set diversity.

Abstract

We consider a preference learning setting where every participant chooses an ordered list of $k$ most preferred items among a displayed set of candidates. (The set can be different for every participant.) We identify a distance-based ranking model for the population's preferences and their (ranked) choice behavior. The ranking model resembles the Mallows model but uses a new distance function called Reverse Major Index (RMJ). We find that despite the need to sum over all permutations, the RMJ-based ranking distribution aggregates into (ranked) choice probabilities with simple closed-form expression. We develop effective methods to estimate the model parameters and showcase their generalization power using real data, especially when there is a limited variety of display sets.