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Outer Diversity of Structured Domains

Piotr Faliszewski, Krzysztof Sornat, Stanisław Szufa, Tomasz Wąs

TL;DR

This paper introduces outer diversity for ordinal preference domains and analyzes it across key structured domains (single-peaked, single-crossing, group-separable, and Euclidean). It develops sampling-based algorithms to compute outer diversity, characterizes computational aspects for each domain, and provides extensive empirical comparisons. The results identify ${\mathrm{GS/cat}}$ as the most diverse among studied domains and show that outer diversity aligns with intuitions from inner diversity, offering a robust benchmark for evaluating domain richness in elections. The work offers practical guidance for simulation studies and opens avenues to formally relate outer and inner diversity while suggesting further exploration of hard instances and domain design.

Abstract

An ordinal preference domain is a subset of preference orders that the voters are allowed to cast in an election. We introduce and study the notion of outer diversity of a domain and evaluate its value for a number of well-known structured domains, such as the single-peaked, single-crossing, group-separable, and Euclidean ones.

Outer Diversity of Structured Domains

TL;DR

This paper introduces outer diversity for ordinal preference domains and analyzes it across key structured domains (single-peaked, single-crossing, group-separable, and Euclidean). It develops sampling-based algorithms to compute outer diversity, characterizes computational aspects for each domain, and provides extensive empirical comparisons. The results identify as the most diverse among studied domains and show that outer diversity aligns with intuitions from inner diversity, offering a robust benchmark for evaluating domain richness in elections. The work offers practical guidance for simulation studies and opens avenues to formally relate outer and inner diversity while suggesting further exploration of hard instances and domain design.

Abstract

An ordinal preference domain is a subset of preference orders that the voters are allowed to cast in an election. We introduce and study the notion of outer diversity of a domain and evaluate its value for a number of well-known structured domains, such as the single-peaked, single-crossing, group-separable, and Euclidean ones.
Paper Structure (29 sections, 26 theorems, 33 equations, 6 figures, 2 tables, 4 algorithms)

This paper contains 29 sections, 26 theorems, 33 equations, 6 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preference Orders, Domains, and Elections
  4. Structured Domains
  5. Distance Between Elections
  6. $\boldsymbol{k}$-Kemeny and Inner Diversity
  7. Outer Diversity
  8. Computing Outer Diversity
  9. Single-Peaked Domains
  10. Group-Separable Domains
  11. Single-Crossing and Euclidean Domains
  12. Analysis of the Domains
  13. Outer-Diversity for Eight Candidates
  14. Direct Neighborhoods
  15. Popularity
  16. ...and 14 more sections

Key Result

Proposition 3.1

For every domain $D \subseteq \mathcal{L}(C)$, it holds that:

Figures (6)

  • Figure 1: Microscope plots of our domains, where each dot/cross represents a ranking from the domain, colored according to its normalized popularity (see \ref{['rem:microscope']}). Rankings with normalized popularity below $\boldsymbol 1$ are marked with crosses, and the remaining ones with dots. Dots marking rankings with normalized popularity equal to exactly $\boldsymbol 1$ have a black border.
  • Figure 2: Outer diversity of several structured domains as a function of the number of candidates (on the left), or as a function of their size (on the right; including approximations of most diverse domains). For $\boldsymbol{\mathrm{SPOC}}$ and $\boldsymbol{\mathrm{3D\hbox{-}Cube}}$, we omit outer diversity for $\boldsymbol{20}$ candidates, due to computation time.
  • Figure 3: Outer diversity of several structured domains as a function of the number of candidates, compared to the outer diversity of (an approximation of) the most diverse domain of the same size.
  • Figure 4: An illustration of the construction from the proof of \ref{['thm:spog:dist:hardness']}.
  • Figure 5: Comparison of the optimal diversity (red line) and the one achieved by simulated annealing (black line) for $6$ candidates.
  • ...and 1 more figures

Theorems & Definitions (47)

  • Definition 2.1
  • Definition 3.1
  • Proposition 3.1
  • proof
  • Proposition 4.0
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • ...and 37 more