Robust Aggregation of Calibrated Forecasts

Decision-makers often rely on multiple probabilistic forecasts that are individually calibrated but need not be fully informative.

We develop a framework for aggregating such forecasts when the decision-maker knows only that experts satisfy calibration.

We show that the joint distribution of calibrated forecasts can contain decision-relevant information that is unavailable from any single expert, so the standard optimal-in-hindsight (OIH) benchmark may substantially understate attainable performance.

To formalize this idea, we introduce a robust max-min benchmark: the best payoff a decision-maker can guarantee against all profile-wise conditional-mean mappings compatible with calibration.

This benchmark is tractable, admits a linear-programming formulation, and dominates the OIH benchmark up to calibration error.

It can nevertheless be strictly below the Bayesian benchmark, clarifying the value of knowing experts' information structures.

Finally, we provide online algorithms that attain the robust benchmark under forecast-only feedback and stronger contextual benchmarks under state feedback.