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A General Theory of Proportionality with Additive Utilities

Piotr Skowron

TL;DR

The paper tackles proportional representation in a highly general setting where a feasible subset of candidates must be selected under arbitrary downward-closed constraints and voters express additive utilities. It develops two core families of rules: PropRank, a dynamic purchasing-with-virtual-money method extending Phragmén to additive utilities, and an Equal Shares–based approach generalized to general constraints, both accompanied by foresight mechanisms. The authors prove strong proportionality guarantees, including a pro rata proportionality degree $d(\alpha)=(\alpha-u_{\max})/2$ for PB constraints and a parameterized $d(\alpha)=(\alpha-u_{\max})/(\kappa+1)$ for PropRank and MES under broader constraints, with PJR properties formalized accordingly. Experimental results on real PB data show substantial improvements in EJR+ and overall utility, especially when using κ=1 and the BOS heuristics, suggesting practical effectiveness alongside theoretical guarantees.

Abstract

We consider a model where a subset of candidates must be selected based on voter preferences, subject to general constraints that specify which subsets are feasible. This model generalizes committee elections with diversity constraints, participatory budgeting (including constraints specifying how funds must be allocated to projects from different pools), and public decision-making. Axioms of proportionality have recently been defined for this general model, but the proposed rules apply only to approval ballots, where each voter submits a subset of candidates she finds acceptable. We propose proportional rules for cardinal ballots, where each voter assigns a numerical value to each candidate corresponding to her utility if that candidate is selected. In developing these rules, we also introduce methods that produce proportional rankings, ensuring that every prefix of the ranking satisfies proportionality.

A General Theory of Proportionality with Additive Utilities

TL;DR

The paper tackles proportional representation in a highly general setting where a feasible subset of candidates must be selected under arbitrary downward-closed constraints and voters express additive utilities. It develops two core families of rules: PropRank, a dynamic purchasing-with-virtual-money method extending Phragmén to additive utilities, and an Equal Shares–based approach generalized to general constraints, both accompanied by foresight mechanisms. The authors prove strong proportionality guarantees, including a pro rata proportionality degree for PB constraints and a parameterized for PropRank and MES under broader constraints, with PJR properties formalized accordingly. Experimental results on real PB data show substantial improvements in EJR+ and overall utility, especially when using κ=1 and the BOS heuristics, suggesting practical effectiveness alongside theoretical guarantees.

Abstract

We consider a model where a subset of candidates must be selected based on voter preferences, subject to general constraints that specify which subsets are feasible. This model generalizes committee elections with diversity constraints, participatory budgeting (including constraints specifying how funds must be allocated to projects from different pools), and public decision-making. Axioms of proportionality have recently been defined for this general model, but the proposed rules apply only to approval ballots, where each voter submits a subset of candidates she finds acceptable. We propose proportional rules for cardinal ballots, where each voter assigns a numerical value to each candidate corresponding to her utility if that candidate is selected. In developing these rules, we also introduce methods that produce proportional rankings, ensuring that every prefix of the ranking satisfies proportionality.
Paper Structure (19 sections, 18 theorems, 92 equations, 1 figure, 1 table, 3 algorithms)

This paper contains 19 sections, 18 theorems, 92 equations, 1 figure, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. The Model
  3. Feasibility Constraints
  4. Proportionality Axioms
  5. Ranking Rules
  6. Proportionality Degree with Additive Utilities
  7. The PropRank Rule
  8. General Scheme
  9. Scaling Factors
  10. Proportionality Degree of PropRank
  11. Proportional Justified Representation of PropRank
  12. Remark on Global Scaling Factors
  13. Method of Equal Shares for General Constraints
  14. The Main Loop of the Algorithm
  15. The Equal Shares with a Foresight Subroutine
  16. ...and 4 more sections

Key Result

Theorem 1

Consider participatory budgeting constraints, and assume that the number of voters is divisible by the budget. If a committee $W$ satisfies EJR, then it has the proportionality degree of $d(\alpha) = (\alpha - u_{\max})/2$, where $u_{\max}$ is the highest utility a voter assigns to a candidate.

Figures (1)

  • Figure 1: The plot shows the proportionality degree implied by the EJR axiom as a function of the size of the cohesive group of voters $S$, with $\gamma = n / |S|$. It illustrates the value of the proportionality coefficient (the coefficient next to $\alpha$ in the proportionality degree) which, in the case of participatory budgeting constraints and approval-based elections with matroid constraints, equals to $1/2$. In general case the coefficient converges to $1/2$ in the limit. The coefficient is approximately $0.39$ and $0.43$ for $\gamma = 2$ and $\gamma = 3$, respectively.

Theorems & Definitions (41)

  • Definition 1: Matroid constraints
  • Definition 2: Participatory budgeting (PB) constraints
  • Definition 3: Cohesiveness for general feasibility constraints
  • Definition 4: Cohesiveness for participatory budgeting constraints with the budget $b$
  • Definition 5
  • Definition 6: Extended Justified Representation
  • Definition 7: Extended Justified Representation for Ranking Rules
  • Definition 8: Proportionality Degree
  • Theorem 1
  • proof
  • ...and 31 more