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Generalized Veto Core and a Practical Voting Rule with Optimal Metric Distortion

Fatih Erdem Kizilkaya, David Kempe

TL;DR

The approach to ties reveals a novel characterization of the (general) veto core, showing it to be identical to the set of candidates who can emerge as winners under a natural class of matching algorithms reminiscent of SerialDictatorship.

Abstract

We revisit the recent breakthrough result of Gkatzelis et al. on (single-winner) metric voting, which showed that the optimal distortion of 3 can be achieved by a mechanism called Plurality Matching. The rule picks an arbitrary candidate for whom a certain candidate-specific bipartite graph contains a perfect matching, and thus, it is not neutral (i.e, symmetric with respect to candidates). Subsequently, a much simpler rule called Plurality Veto was shown to achieve distortion 3 as well. This rule only constructs such a matching implicitly but the winner depends on the order that voters are processed, and thus, it is not anonymous (i.e., symmetric with respect to voters). We provide an intuitive interpretation of this matching by generalizing the classical notion of the (proportional) veto core in social choice theory. This interpretation opens up a number of immediate consequences. Previous methods for electing a candidate from the veto core can be interpreted simply as matching algorithms. Different election methods realize different matchings, in turn leading to different sets of candidates as winners. For a broad generalization of the veto core, we show that the generalized veto core is equal to the set of candidates who can emerge as winners under a natural class of matching algorithms reminiscent of Serial Dictatorship. Extending these matching algorithms into continuous time, we obtain a highly practical voting rule with optimal distortion 3, which is also intuitive and easy to explain: Each candidate starts off with public support equal to his plurality score. From time 0 to 1, every voter continuously brings down, at rate 1, the support of her bottom choice among not-yet-eliminated candidates. A candidate is eliminated if he is opposed by a voter after his support reaches 0. On top of being anonymous and neutral, this rule satisfies many other axioms desirable in practice.

Generalized Veto Core and a Practical Voting Rule with Optimal Metric Distortion

TL;DR

The approach to ties reveals a novel characterization of the (general) veto core, showing it to be identical to the set of candidates who can emerge as winners under a natural class of matching algorithms reminiscent of SerialDictatorship.

Abstract

We revisit the recent breakthrough result of Gkatzelis et al. on (single-winner) metric voting, which showed that the optimal distortion of 3 can be achieved by a mechanism called Plurality Matching. The rule picks an arbitrary candidate for whom a certain candidate-specific bipartite graph contains a perfect matching, and thus, it is not neutral (i.e, symmetric with respect to candidates). Subsequently, a much simpler rule called Plurality Veto was shown to achieve distortion 3 as well. This rule only constructs such a matching implicitly but the winner depends on the order that voters are processed, and thus, it is not anonymous (i.e., symmetric with respect to voters). We provide an intuitive interpretation of this matching by generalizing the classical notion of the (proportional) veto core in social choice theory. This interpretation opens up a number of immediate consequences. Previous methods for electing a candidate from the veto core can be interpreted simply as matching algorithms. Different election methods realize different matchings, in turn leading to different sets of candidates as winners. For a broad generalization of the veto core, we show that the generalized veto core is equal to the set of candidates who can emerge as winners under a natural class of matching algorithms reminiscent of Serial Dictatorship. Extending these matching algorithms into continuous time, we obtain a highly practical voting rule with optimal distortion 3, which is also intuitive and easy to explain: Each candidate starts off with public support equal to his plurality score. From time 0 to 1, every voter continuously brings down, at rate 1, the support of her bottom choice among not-yet-eliminated candidates. A candidate is eliminated if he is opposed by a voter after his support reaches 0. On top of being anonymous and neutral, this rule satisfies many other axioms desirable in practice.
Paper Structure (17 sections, 20 theorems, 3 equations, 2 figures, 2 algorithms)

This paper contains 17 sections, 20 theorems, 3 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Metric Distortion
  4. Domination Graphs
  5. The Veto Core and its Equivalence to Dominant Candidates
  6. The Proportional Veto Core and its Generalization
  7. Relation to Dominant Candidates
  8. A Class of Voting Rules with Ties Characterizing The Veto Core
  9. Ties are Convex
  10. SerialVeto
  11. Characterization of the Veto Core via SerialVeto
  12. A Practical Voting Rule Electing from the Veto Core
  13. SimultaneousVeto
  14. Axiomatic Properties of SimultaneousPluralityVeto
  15. Axioms Violated by SimultaneousPluralityVeto
  16. ...and 2 more sections

Key Result

Lemma 1

For any weight vectors $\mathbf{p}\xspace \in \Delta(V)$ and $\mathbf{q}\xspace \in \Delta(C)$, candidate $a$ is ${(\mathbf{p}\xspace, \mathbf{q}\xspace)}\xspace$-dominant if and only if $p(T) \leq q(\mathcal{N}_a(T))$ for all $T \subseteq V$.

Figures (2)

  • Figure 1: A consistent embedding of the election in \ref{['ex:convexity']} into a 2-dimensional $\ell_1$-normed space such that the plurality veto core is non-convex.
  • Figure 2: A bottom trading cycle $\phi = (v_1, c_1, \ldots, v_4, c_4)$ of a ${(\mathbf{p}\xspace, \mathbf{q}\xspace)}\xspace$-matching $\mathbf{M}\xspace$ is illustrated above. Notice that $\mathbf{M}\xspace \circ \phi$ is the same matching as $\mathbf{M}\xspace$ except that the weight on each bold edge is decremented by 1 and the weight on each dotted edge is incremented by 1.

Theorems & Definitions (31)

  • Example 1
  • Example 2
  • Definition 1: Domination Graphs gkatzelis:halpern:shah:resolving
  • Lemma 1: gkatzelis:halpern:shah:resolving, Lemma 1
  • Lemma 2: gkatzelis:halpern:shah:resolving, Theorem 1
  • Definition 2: Proportional Veto Core moulinProportionalVetoPrinciple1981
  • Definition 3: ${(\mathbf{p}\xspace, \mathbf{q}\xspace)}\xspace$-Veto Core
  • Proposition 1
  • Theorem 1
  • Corollary 1
  • ...and 21 more