A Taxonomy of Non-dictatorial Unidimensional Domains

TL;DR

This paper classifies all non-dictatorial, unidimensional preference domains that admit strategy-proof social choice rules, introducing semi-hybrid domains and analyzing their properties and canonical rules.

Contribution

It provides an exhaustive classification of non-dictatorial, unidimensional domains and introduces semi-hybrid domains, expanding the understanding of strategy-proof rules in social choice theory.

Findings

Semi-hybrid domains allow strategy-proof rules to depend on non-peak preferences.

Projection and hybrid rules are canonical for semi-single-peaked and semi-hybrid domains.

Single-peaked and hybrid domains uniquely determine rules by voters' peaks.

Abstract

A preference domain is called a non-dictatorial domain if it allows the design of unanimous social choice functions (henceforth, rules) that are non-dictatorial and strategy-proof. We study a class of preference domains called unidimensional domains and establish that the unique seconds property (introduced by Aswal, Chatterji, and Sen (2003)) characterizes all non-dictatorial domains. The principal contribution is the subsequent exhaustive classification of all non-dictatorial, unidimensional domains and canonical strategy-proof rules on these domains, based on a simple property of two-voter rules called invariance. The preference domains that constitute the classification are semi-single-peaked domains (introduced by Chatterji, Sanver, and Sen (2013)) and semi-hybrid domains (introduced here) which are two appropriate weakenings of single-peaked domains and are shown to allow strategy-proof rules to depend on non-peak information of voters' preferences; the canonical rules for these domains are the projection rule and the hybrid rule respectively. As a refinement of the classification, single-peaked domains and hybrid domains emerge as the only unidimensional domains that force strategy-proof rules to be determined completely by voters' preference peaks.