Distribution Aggregation via Continuous Thiele's Rules

We introduce the class of $\mathit{\text{Continuous}}\mathit{\text{Thiele’s}}\mathit{\text{Rules}}$ that generalize the familiar $\mathbf{\text{Thiele’s}}\mathbf{\text{rules}}$ \cite{janson2018phragmens} of multi-winner voting to distribution aggregation problems. Each rule in that class maximizes ${\sum }_{i}f({\pi }^i)$ 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 $\mathit{\text{’Inequality}}\mathit{\text{Aversion’}}$ of $f$ (elsewhere known as “Relative Risk Aversion”), we derive bounds on the Egalitarian loss, welfare loss and the approximation of $\mathit{\text{Average}}\mathit{\text{Fair}}\mathit{\text{Share}}$, leading to a quantifiable, continuous presentation of their inevitable trade-offs. In particular, we show that the Nash Product Rule satisfies$\mathit{\text{Average}}\mathit{\text{Fair}}\mathit{\text{Share}}$ in our setting.