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.

翻译：本文引入了\textit{连续蒂勒规则}这一类别，将多赢家投票中熟悉的\textbf{蒂勒规则} \cite{janson2018phragmens}推广到分布聚合问题。该类中的每条规则旨在最大化$\sum_if(\pi^i)$，其中$\pi^i$表示个体$i$的满意度，$f$可以是任意二阶可微、递增且凹的实函数。基于一个我们称之为$f$的\textit{“不平等厌恶度”}的量（在其他文献中亦称为“相对风险厌恶度”），我们推导了关于平等主义损失、福利损失以及\textit{平均公平份额}近似度的界，从而对这些不可避免的权衡关系给出了可量化的连续表述。特别地，我们证明了纳什乘积规则在我们的设定下满足\textit{平均公平份额}。