On the Likelihood of Single-Peaked Preferences

TL;DR

This paper analyzes the probability of single-peaked preferences in elections, providing bounds and exact results under various models, revealing that such preferences are generally unlikely under uniform assumptions but more probable in certain models.

Contribution

It offers a comprehensive combinatorial analysis of single-peaked preferences, deriving bounds and exact probabilities under different probabilistic models and configurations.

Findings

Single-peaked preferences are unlikely in random elections under Impartial Culture.

Refined bounds show higher likelihood of single-peakedness in certain models.

Exact probabilities are provided for elections with few voters or candidates.

Abstract

This paper contains an extensive combinatorial analysis of the single-peaked domain restriction and investigates the likelihood that an election is single-peaked. We provide a very general upper bound result for domain restrictions that can be defined by certain forbidden configurations. This upper bound implies that many domain restrictions (including the single-peaked restriction) are very unlikely to appear in a random election chosen according to the Impartial Culture assumption. For single-peaked elections, this upper bound can be refined and complemented by a lower bound that is asymptotically tight. In addition, we provide exact results for elections with few voters or candidates. Moreover, we consider the P\'{o}lya urn model and the Mallows model and obtain lower bounds showing that single-peakedness is considerably more likely to appear for certain parameterizations.