  This paper introduces an objective for optimizing proper scoring rules. The
objective is to maximize the increase in payoff of a forecaster who exerts a
binary level of effort to refine a posterior belief from a prior belief. In
this framework we characterize optimal scoring rules in simple settings, give
efficient algorithms for computing optimal scoring rules in complex settings,
and identify simple scoring rules that are approximately optimal. In
comparison, standard scoring rules in theory and practice -- for example the
quadratic rule, scoring rules for the expectation, and scoring rules for
multiple tasks that are averages of single-task scoring rules -- can be very
far from optimal.
