On the likelihood of single-peaked preferences

Area of Science:

• Social Choice Theory
• Combinatorial Mathematics
• Computational Social Choice

Background:

• The single-peaked domain restriction is a fundamental concept in social choice theory.
• Understanding the prevalence of such restrictions is crucial for analyzing voting systems.
• The Impartial Culture (IC) assumption is a standard benchmark for random elections.

Purpose of the Study:

• To conduct a combinatorial analysis of the single-peaked domain restriction.
• To investigate the likelihood of elections being single-peaked under various models.
• To establish general bounds for the probability of domain restrictions.

Main Methods:

• Derivation of a general upper bound for domain restrictions based on forbidden configurations.
• Refinement of the upper bound with a lower bound for single-peaked elections.
• Exact calculations for small numbers of voters and candidates.
• Analysis using the Pólya urn and Mallows models.

Main Results:

• The single-peaked restriction is shown to be very unlikely under the Impartial Culture assumption.
• An asymptotically tight lower bound for single-peaked elections is established.
• Exact probabilities are computed for small-scale elections.
• The Pólya urn and Mallows models demonstrate increased likelihood of single-peakedness for certain parameters.

Conclusions:

• The study quantifies the rarity of single-peaked elections in a general random setting.
• Specific probabilistic models offer more favorable conditions for single-peakedness.
• The findings contribute to a deeper understanding of preference aggregation and domain restrictions in voting theory.