From Proper Scoring Rules to Max-Min Optimal Forecast Aggregation

Abstract

This paper forges a strong connection between two seemingly unrelated forecasting problems: incentive-compatible forecast elicitation and forecast aggregation. Proper scoring rules are the well-known solution to the former problem. To each such rule s we associate a corresponding method of aggregation, mapping expert forecasts and expert weights to a consensus forecast, which we call quasi-arithmetic (QA) pooling with respect to s. We justify this correspondence in several ways: QA pooling with respect to the two most well-studied scoring rules (quadratic and logarithmic) corresponds to the two most well-studied forecast aggregation methods (linear and logarithmic). Given a scoring rule s used for payment, a forecaster agent who sub-contracts several experts, paying them in proportion to their weights, is best off aggregating the experts reports using QA pooling with respect to s, meaning this strategy maximizes its worst-case profit (over the possible outcomes). The score of an aggregator who uses QA pooling is concave in the experts weights. As a consequence, online gradient descent can be used to learn appropriate expert weights from repeated experiments with low regret. The class of all QA pooling methods is characterized by a natural set of axioms (generalizing classical work by Kolmogorov on quasi-arithmetic means).