Explaining Preferences by Multiple Patterns in Voters' Behavior

TL;DR

This paper investigates the complexity of explaining voters' preferences using multiple axes or decision trees, providing polynomial algorithms for the case of two such patterns and proving hardness for three or more.

Contribution

It introduces a polynomial-time algorithm for recognizing preferences explainable by two axes or trees and establishes hardness results for three or more patterns.

Findings

Polynomial-time algorithms for k=2 patterns in certain preference domains

Hardness results for k≥3 patterns in value-restricted and group-separable preferences

Forbidden minor characterizations for caterpillar group-separable preferences

Abstract

In some preference aggregation scenarios, voters' preferences are highly structured: e.g., the set of candidates may have one-dimensional structure (so that voters' preferences are single-peaked) or be described by a binary decision tree (so that voters' preferences are group-separable). However, sometimes a single axis or a decision tree is insufficient to capture the voters' preferences; rather, there is a small number $k$ of axes or decision trees such that each vote in the profile is consistent with one of these axes (resp., trees). In this work, we study the complexity of deciding whether voters' preferences can be explained in this manner. For $k=2$, we use the technique developed by Yang~[2020] in the context of single-peaked preferences to obtain a polynomial-time algorithm for several domains: value-restricted preferences, group-separable preferences, and a natural subdomain of group-separable preferences, namely, caterpillar group-separable preferences. For $k\ge 3$, the problem is known to be hard for single-peaked preferences; we show that this is also the case for value-restricted and group-separable preferences. Our positive results for $k=2$ make use of forbidden minor characterizations of the respective domains; in particular, we establish that the domain of caterpillar group-separable preferences admits a forbidden minor characterization.