Distribution Aggregation via Continuous Thiele's Rules

We introduce the class of Continuous Thiele's Rules that generalize the familiar Thiele's rules {janson2018phragmens} of multi-winner voting to distribution aggregation problems. Each rule in that class maximizes ∑ᵢf(πⁱ) where πⁱ 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 'Inequality Aversion' of f (elsewhere known as "Relative Risk Aversion"), we derive bounds on the Egalitarian loss, welfare loss and the approximation of Average Fair Share, leading to a quantifiable, continuous presentation of their inevitable trade-offs. In particular, we show that the Nash Product Rule satisfies Average Fair Share in our setting.