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Deepest voting on rankings

Jean-Baptiste Aubin, Antoine Rolland, Ioana Gavra, Irène Gannaz, Jacques Anderson Kouassi

Abstract

This article aims to present a unified framework for ranking-based voting rules based on the use of depth functions on permutations, as a counterpart of deepest voting rules on evaluation introduced in Aubin et al. [2022]. It introduces the notion of depth functions, in continuous sets and in permutation sets, the later using the notion of Fr{é}chet means. Deepest voting procedures are then formally defined, and some classical voting rules are expressed as deepest voting procedures, using a large variety of distances on the set of permutations. Links are done between the depth functions mathematical properties and some behaviours of the voting rule, such as Neutrality, Anonymity, Universality, Condorcet winner/loser property and so on.

Deepest voting on rankings

Abstract

This article aims to present a unified framework for ranking-based voting rules based on the use of depth functions on permutations, as a counterpart of deepest voting rules on evaluation introduced in Aubin et al. [2022]. It introduces the notion of depth functions, in continuous sets and in permutation sets, the later using the notion of Fr{é}chet means. Deepest voting procedures are then formally defined, and some classical voting rules are expressed as deepest voting procedures, using a large variety of distances on the set of permutations. Links are done between the depth functions mathematical properties and some behaviours of the voting rule, such as Neutrality, Anonymity, Universality, Condorcet winner/loser property and so on.
Paper Structure (19 sections, 11 theorems, 16 equations, 4 tables)

This paper contains 19 sections, 11 theorems, 16 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Definition of depth functions
  3. Depth functions in continuous case
  4. Depth functions on permutations
  5. Distances on permutations
  6. Weighted distances on permutations
  7. Depth functions associated to a distance
  8. Deepest voting
  9. Continuous deepest voting
  10. Deepest voting on permutations
  11. Classical rules as Fréchet voting
  12. Unweighted distances
  13. Weighted distances
  14. Voting rules properties
  15. Conclusion
  16. ...and 4 more sections

Key Result

Proposition 1

Consider $n$ voters and $m$ candidates. Let $e_{cv}$ be the rank in $\{1,\dots,m\}$ given by voter $v$ to candidate $c$, and denote $\Phi=(e_{cv})_{v,c}\in\mathfrak{S}_m$ the obtained ranks. Then,

Theorems & Definitions (21)

  • Definition 1
  • Definition 2: Goibert
  • Definition 3
  • Definition 4: Goibert
  • Definition 5: Goibert
  • Remark 1
  • Remark 2
  • Definition 6: Deepest Voting
  • Proposition 1
  • Definition 7
  • ...and 11 more