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Approval-Based Voting with Mixed Goods

Xinhang Lu, Jannik Peters, Haris Aziz, Xiaohui Bei, Warut Suksompong

TL;DR

This work considers a voting scenario in which the resource to be voted upon may consist of both indivisible and divisible goods, and proposes two variants of the extended justified representation (EJR) notion from multiwinner voting.

Abstract

We consider a voting scenario in which the resource to be voted upon may consist of both indivisible and divisible goods. This setting generalizes both the well-studied model of multiwinner voting and the recently introduced model of cake sharing. Under approval votes, we propose two variants of the extended justified representation (EJR) notion from multiwinner voting, a stronger one called EJR for mixed goods (EJR-M) and a weaker one called EJR up to 1 (EJR-1). We extend three multiwinner voting rules to our setting -- GreedyEJR, the method of equal shares (MES), and proportional approval voting (PAV) -- and show that while all three generalizations satisfy EJR-1, only the first one provides EJR-M. In addition, we derive tight bounds on the proportionality degree implied by EJR-M and EJR-1, and investigate the proportionality degree of our proposed rules.

Approval-Based Voting with Mixed Goods

TL;DR

This work considers a voting scenario in which the resource to be voted upon may consist of both indivisible and divisible goods, and proposes two variants of the extended justified representation (EJR) notion from multiwinner voting.

Abstract

We consider a voting scenario in which the resource to be voted upon may consist of both indivisible and divisible goods. This setting generalizes both the well-studied model of multiwinner voting and the recently introduced model of cake sharing. Under approval votes, we propose two variants of the extended justified representation (EJR) notion from multiwinner voting, a stronger one called EJR for mixed goods (EJR-M) and a weaker one called EJR up to 1 (EJR-1). We extend three multiwinner voting rules to our setting -- GreedyEJR, the method of equal shares (MES), and proportional approval voting (PAV) -- and show that while all three generalizations satisfy EJR-1, only the first one provides EJR-M. In addition, we derive tight bounds on the proportionality degree implied by EJR-M and EJR-1, and investigate the proportionality degree of our proposed rules.
Paper Structure (15 sections, 22 theorems, 51 equations, 1 figure, 1 table)

This paper contains 15 sections, 22 theorems, 51 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Preliminaries
  4. EJR Notions and Rules
  5. GreedyEJR-M
  6. Generalized Method of Equal Shares
  7. Generalized PAV
  8. Proportionality Degree
  9. Proportionality Degree Implied by EJR-M and EJR-1
  10. Proportionality Degree of Specific Rules
  11. Case 1: $t \geq 2$.
  12. Case 2: $1 \leq t < 2$.
  13. Conclusion
  14. Proof of \ref{['prop:EJR-M-strong-EJR-1-no-logic-relation']}
  15. (Almost) Tightness of \ref{['thm:prop-degree-GPAV']}

Key Result

Proposition 3.2

For each constant $\beta \in [0,1)$, there exists an indivisible-goods instance in which no allocation satisfies EJR-$\beta$. This remains true even if we relax the inequality $u_j(A) > t-\beta$ in def:EJR-beta to $u_j(A) \ge t-\beta$.

Figures (1)

  • Figure 1: A mixed-goods instance with two agents $N = \{1,2\}$, two indivisible goods $G = \{g_1,g_2\}$, a cake $C$ of length $0.9$, and $\alpha = 2$. Agent $1$ approves $R_1 = \{g_1\}\cup C$, while agent $2$ approves $R_2 = \{g_2\}\cup C$. If the allocation $A = \{g_1,g_2\}$ is chosen, both agents receive a utility of $1$.

Theorems & Definitions (47)

  • Definition 3.1: EJR-$\beta$
  • Proposition 3.2
  • proof
  • Definition 3.3: EJR-M
  • Proposition 3.4
  • proof
  • Corollary 3.5
  • Proposition 3.6
  • proof
  • Example 3.7
  • ...and 37 more