Distribution Aggregation via Continuous Thiele's Rules

TL;DR

This paper introduces Continuous Thiele's Rules, a new class of distribution aggregation methods based on maximizing agents' satisfaction functions, and analyzes their trade-offs and fairness properties.

Contribution

It generalizes Thiele's rules to distribution aggregation, introduces the concept of Inequality Aversion, and provides bounds on fairness and welfare trade-offs.

Findings

Nash Product Rule satisfies Average Fair Share.

Bounds established for Egalitarian and welfare losses.

Quantifies trade-offs between fairness and efficiency.

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 $\sum_if(\pi^i)$ where $\pi^i$ 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.