A Characterization of Scoring Rules for Linear Properties

We consider the design of proper scoring rules, equivalently proper losses, when the goal is to elicit some function, known as a property, of the underlying distribution. We provide a full characterization of the class of proper scoring rules when the property is linear as a function of the input distribution. A key conclusion is that any such scoring rule can be written in the form of a Bregman divergence for some convex function. We also apply our results to the design of prediction market mechanisms, showing a strong equivalence between scoring rules for linear properties and automated prediction market makers.