  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).
