On the Likelihood of Single-Peaked Preferences

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 Pólya urn model and the Mallows model and obtain lower bounds showing that single-peakedness is considerably more likely to appear for certain parameterizations.

References in corpus (3)

• On the Likelihood of Single-Peaked Preferences
• Computational Aspects of Nearly Single-Peaked Electorates
• Central binomial coefficients also count (2431,4231,1432,4132)-avoiders

Cited by in corpus (6)

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• Computational Aspects of Nearly Single-Peaked Electorates
• An Optimal Procedure to Check Pareto-Optimality in House Markets with Single-Peaked Preferences
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• A heuristic search algorithm for discovering large Condorcet domains
• Flexible Level-1 Consensus Ensuring Stable Social Choice: Analysis and Algorithms