Optimization of Scoring Rules

We characterize the optimal reward functions (scoring rules) that incentivize an agent to acquire information and report it truthfully to the principal. The optimal scoring rules let the agent make a simple binary bet in single-dimensional problems, and choose the dimension with the most surprising signal to be scored on in symmetric multi-dimensional problems. This scoring rule format remains approximately optimal for asymmetric distributions. These results demonstrate the importance of linking incentives to obtain high-quality information in multi-dimensional problems. In contrast, standard scoring rules like the quadratic scoring rule, or averages of single-dimensional scoring rules can be far from optimal.