# From Proper Scoring Rules to Max-Min Optimal Forecast Aggregation

arXiv: https://arxiv.org/abs/2102.07081 · HTML: https://mlsanity.com/paper/2102.07081/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Feb 2021.

## Built on by (most cited first, 5 of 5)

- [No-Regret Learning with Unbounded Losses: The Case of Logarithmic Pooling](https://mlsanity.com/paper/2202.11219/) (2022)
- [Deep interpretable ensembles](https://mlsanity.com/paper/2205.12729/) (2022)
- [An Axiomatic Approach to Loss Aggregation and an Adapted Aggregating Algorithm](https://mlsanity.com/paper/2406.02292/) (2024)
- [Jailbreaking LLMs via Calibration](https://mlsanity.com/paper/2602.00619/) (2026)
- [Tree-Based Formalization of Multi-Agent Complementarity in Human-AI Interactions](https://mlsanity.com/paper/2606.04779/) (2026)

## Abstract (arXiv)

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).