From Proper Scoring Rules to Max-Min Optimal Forecast Aggregation

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 subcontracts 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); and the class of all QA pooling methods is characterized by a natural set of axioms (generalizing classical work by Kolmogorov on quasi-arithmetic means).