Extended Formulations for Order Polytopes through Network Flows

TL;DR

This paper introduces extended formulations using network flow models to efficiently describe the complex polytopes of probabilistic choice models in mathematical psychology, enabling scalable analysis for many alternatives.

Contribution

It develops network flow-based extended formulations for choice polytopes, overcoming combinatorial complexity limits of previous methods.

Findings

Extended formulations simplify polytope descriptions for many choice alternatives.

Network flow models provide more parsimonious representations than standard methods.

Facilitates computational testing of probabilistic choice models on larger datasets.

Abstract

Mathematical psychology has a long tradition of modeling probabilistic choice via distribution-free random utility models and associated random preference models. For such models, the predicted choice probabilities often form a bounded and convex polyhedral set, or polytope. Polyhedral combinatorics have thus played a key role in studying the mathematical structure of these models. However, standard methods for characterizing the polytopes of such models are subject to a combinatorial explosion in complexity as the number of choice alternatives increases. Specifically, this is the case for random preference models based on linear, weak, semi- and interval orders. For these, a complete, linear description of the polytope is currently known only for, at most, 5--8 choice alternatives. We leverage the method of extended formulations to break through those boundaries. For each of the four types of preferences, we build an appropriate network, and show that the associated network flow polytope provides an extended formulation of the polytope of the choice model. This extended formulation has a simple linear description that is more parsimonious than descriptions obtained by standard methods for large numbers of choice alternatives. The result is a computationally less demanding way of testing the probabilistic choice model on data. We sketch how the latter interfaces with recent developments in contemporary statistics.