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Candidate Monotonicity and Proportionality for Lotteries and Non-Resolute Rules

Jannik Peters

TL;DR

Addresses designing multiwinner rules that are candidate monotone and proportional (PSC). Shows PSC-committees are candidate monotone in the non-resolute setting and constructs a candidate-monotone PSC lottery by converting Phragmén's Ordered Rule into a fractional rule (PFR) with dependent rounding. Extends to party-list apportionment and provides experiments illustrating the distribution of PSC-eligible committees, highlighting trade-offs between determinism and randomness. The work offers a pathway to monotone, proportionally representative outcomes while outlining open questions for resolute rules and stronger proportionality axioms.

Abstract

We study the problem of designing multiwinner voting rules that are candidate monotone and proportional. We show that the set of committees satisfying the proportionality axiom of proportionality for solid coalitions is candidate monotone. We further show that Phragmén's Ordered Rule can be turned into a candidate monotone probabilistic rule which randomizes over committees satisfying proportionality for solid coalitions.

Candidate Monotonicity and Proportionality for Lotteries and Non-Resolute Rules

TL;DR

Addresses designing multiwinner rules that are candidate monotone and proportional (PSC). Shows PSC-committees are candidate monotone in the non-resolute setting and constructs a candidate-monotone PSC lottery by converting Phragmén's Ordered Rule into a fractional rule (PFR) with dependent rounding. Extends to party-list apportionment and provides experiments illustrating the distribution of PSC-eligible committees, highlighting trade-offs between determinism and randomness. The work offers a pathway to monotone, proportionally representative outcomes while outlining open questions for resolute rules and stronger proportionality axioms.

Abstract

We study the problem of designing multiwinner voting rules that are candidate monotone and proportional. We show that the set of committees satisfying the proportionality axiom of proportionality for solid coalitions is candidate monotone. We further show that Phragmén's Ordered Rule can be turned into a candidate monotone probabilistic rule which randomizes over committees satisfying proportionality for solid coalitions.

Paper Structure

This paper contains 9 sections, 5 theorems, 2 equations, 3 figures.

Table of Contents

  1. Proportionality and Monotonicity
  2. Preliminaries
  3. Non-Resoluteness and Monotonicity
  4. A Probabilistic Version of Phragmén's Ordered Rule
  5. Consequences for Party List Elections
  6. Experiments
  7. Conclusion
  8. Acknowledgments
  9. Compliance with Ethical Standards

Key Result

Theorem 2

A committee $W$ satisfies PSC if and only if it can be obtained by a minimal demand rule.

Figures (3)

  • Figure 1: Number of instances with a given number of candidates for $k = 3$ and $k = 4$.
  • Figure 2: Both plots depict the fraction of candidates who are assigned are assigned a probability in a certain range as a stackplot (for $k = 3$ on the left and $k = 4$ on the right).
  • Figure 3: Average number of committees satisfying PSC, having a positive probability of being sampled after 50 thousand samples, and how many committees are needed to get a probability mass of at least $75\%$, $90\%$, $95\%$, and $99\%$, for $k = 3$ on the left and $k = 4$ on the right. The black dashed line indicates the total number of committees.

Theorems & Definitions (19)

  • Definition 1: Solid Coalition
  • Definition 2: Proportionality for Solid Coalitions Dumm84a
  • Example 1
  • Definition 3: Candidate Monotonicity EFSS17a
  • Definition 4: Candidate Monotonicity for Probabilistic Rules
  • Theorem 2: Characterization of PSC AzLe22a
  • Theorem 3
  • proof
  • Example 2
  • Example 3
  • ...and 9 more