  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.
