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Equivalence of Connected and Peak-Pit Maximal Condorcet Domains

Guanhao Li

TL;DR

The paper proves that, within the class of maximal Condorcet domains, the peak-pit, connected, and directly connected notions coincide. It develops a combinatorial framework based on geodesics of alike linear orders on the permutahedron and analyzes sequences of switching pairs to show that restricting geodesics preserves the peak-pit property. A key step is showing that maximal peak-pit Condorcet domains are directly connected, which implies the equivalence of the three properties and resolves ambiguities between maximal peak-pit and peak-pit maximal domains. The results unify prior scattered findings, clarify the structure of maximal Condorcet domains, and have implications for strategyproofness and domain characterization in social choice theory.

Abstract

This paper provides a combinatorial proof to show that, in the study of maximal Condorcet domains, the class of peak-pit Condorcet domains, the class of connected Condorcet domains, and the class of directly connected Condorcet domains are all equivalent.

Equivalence of Connected and Peak-Pit Maximal Condorcet Domains

TL;DR

The paper proves that, within the class of maximal Condorcet domains, the peak-pit, connected, and directly connected notions coincide. It develops a combinatorial framework based on geodesics of alike linear orders on the permutahedron and analyzes sequences of switching pairs to show that restricting geodesics preserves the peak-pit property. A key step is showing that maximal peak-pit Condorcet domains are directly connected, which implies the equivalence of the three properties and resolves ambiguities between maximal peak-pit and peak-pit maximal domains. The results unify prior scattered findings, clarify the structure of maximal Condorcet domains, and have implications for strategyproofness and domain characterization in social choice theory.

Abstract

This paper provides a combinatorial proof to show that, in the study of maximal Condorcet domains, the class of peak-pit Condorcet domains, the class of connected Condorcet domains, and the class of directly connected Condorcet domains are all equivalent.

Paper Structure

This paper contains 3 sections, 1 theorem, 2 equations.

Table of Contents

  1. Introduction
  2. Condorcet domains
  3. Paths of alike linear orders

Key Result

lemma thmcounterlemma

Let $a, b, c$ be three distinct alternatives. Define Let ${\mathcal{D}}\subseteq{\mathcal{L}}(\{a,b,c\})$ be a Condorcet domain. Then the following are equivalent:

Theorems & Definitions (5)

  • definition thmcounterdefinition
  • remark thmcounterremark
  • lemma thmcounterlemma
  • proof
  • definition thmcounterdefinition