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Distribution Aggregation via Continuous Thiele's Rules

Jonathan Wagner, Reshef Meir

Abstract

We introduce the class of \textit{Continuous Thiele's Rules} that generalize the familiar \textbf{Thiele's rules} \cite{janson2018phragmens} of multi-winner voting to distribution aggregation problems. Each rule in that class maximizes $\sum_if(π^i)$ where $π^i$ is an agent $i$'s satisfaction and $f$ could be any twice differentiable, increasing and concave real function. Based on a single quantity we call the \textit{'Inequality Aversion'} of $f$ (elsewhere known as "Relative Risk Aversion"), we derive bounds on the Egalitarian loss, welfare loss and the approximation of \textit{Average Fair Share}, leading to a quantifiable, continuous presentation of their inevitable trade-offs. In particular, we show that the Nash Product Rule satisfies\textit{ Average Fair Share} in our setting.

Distribution Aggregation via Continuous Thiele's Rules

Abstract

We introduce the class of \textit{Continuous Thiele's Rules} that generalize the familiar \textbf{Thiele's rules} \cite{janson2018phragmens} of multi-winner voting to distribution aggregation problems. Each rule in that class maximizes where is an agent 's satisfaction and could be any twice differentiable, increasing and concave real function. Based on a single quantity we call the \textit{'Inequality Aversion'} of (elsewhere known as "Relative Risk Aversion"), we derive bounds on the Egalitarian loss, welfare loss and the approximation of \textit{Average Fair Share}, leading to a quantifiable, continuous presentation of their inevitable trade-offs. In particular, we show that the Nash Product Rule satisfies\textit{ Average Fair Share} in our setting.
Paper Structure (15 sections, 10 theorems, 43 equations, 2 figures, 1 table)

This paper contains 15 sections, 10 theorems, 43 equations, 2 figures, 1 table.

Table of Contents

  1. Preference Modeling
  2. Axiomatic Demands And Their Incompatibilities
  3. Contribution
  4. Related Work
  5. Preliminaries
  6. Aggregation Rules and their Properties
  7. Continuous Theile's Rules
  8. Optimal allocations
  9. Axiomatic Properties of CTRs
  10. Utilitarian vs. Egalitarian Welfare
  11. Welfare Loss
  12. Approximate AFS
  13. Egalitarian Loss and Individual Shares
  14. Discussion
  15. Randomized approval voting

Key Result

Proposition 1

$x \in \arg \max_{y \in \Delta^m}\sum_i f(\pi^{i}_{y})$ if and only if

Figures (2)

  • Figure 1: Green lines show welfare loss upper bounds, scarlet lines show Egalitarian loss. Thick lines are for $m=15$ and $n=100$, dashed lines for $m=3$ and $n=20$.
  • Figure :

Theorems & Definitions (30)

  • Definition 1
  • Definition 2: Continuous Thiele's Aggregation Rules
  • Definition 3
  • Definition 5: marginal contributions
  • Proposition 1: MRS condition
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Example 4
  • ...and 20 more