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arxiv: 2606.31020 · v1 · pith:3HB6ZO5Cnew · submitted 2026-06-30 · 💰 econ.TH · cs.GT

Robust Aggregation of Calibrated Forecasts

Xinxiang Guo , Yingkai Li , Yifen Mu This is my paper

Pith reviewed 2026-07-01 03:18 UTC · model grok-4.3

classification 💰 econ.TH cs.GT
keywords calibrated forecastsforecast aggregationrobust optimizationmax-min benchmarkonline algorithmsdecision making under uncertaintyinformation aggregation
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The pith

The joint distribution of calibrated forecasts contains decision-relevant information unavailable from any single expert.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

When experts each issue calibrated probabilistic forecasts, the collection of those forecasts together can reveal more about the outcome than any forecast reveals by itself. This means the usual optimal-in-hindsight benchmark, which assumes the outcome is known after the fact, can understate the performance a decision-maker can actually achieve. The paper therefore introduces a robust max-min benchmark that computes the best guaranteed payoff consistent with every possible way the experts' information could be linked while still satisfying individual calibration. The benchmark is computable by linear programming and is reached by online algorithms that observe only the forecasts themselves. It improves on the hindsight benchmark yet remains below what full knowledge of the experts' information structures would allow.

Core claim

The paper establishes 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 benchmark may substantially understate attainable performance. It formalizes this with 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 admits a linear-programming formulation, dominates the OIH benchmark up to calibration error, and can be strictly below the Bayesian benchmark, while online algorithms attain it under forecast-only feedback.

What carries the argument

The robust max-min benchmark over all profile-wise conditional-mean mappings compatible with individual calibration, solved via linear programming.

If this is right

  • The robust benchmark dominates the optimal-in-hindsight benchmark up to calibration error.
  • Online algorithms attain the robust benchmark under forecast-only feedback.
  • Stronger benchmarks are attainable when the decision-maker also observes the realized state.
  • The robust benchmark lies strictly below the Bayesian benchmark that assumes full knowledge of experts' information structures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Soliciting forecasts from several calibrated experts can be valuable even without knowing their information overlap, because the robust benchmark exploits possible joint structure.
  • The same max-min approach could be adapted to approximate calibration by relaxing the conditional-mean constraints with an explicit error term.
  • Empirical tests would require enumerating worst-case conditional means consistent with observed calibration frequencies in historical forecast data.

Load-bearing premise

The decision-maker knows only that each expert's forecast is calibrated and nothing else about how the experts' signals relate to one another or to the outcome.

What would settle it

A concrete strategy that guarantees a strictly higher payoff than the linear program's max-min value, for some profile of calibrated forecasts, would show the benchmark is not the best attainable guarantee.

Figures

Figures reproduced from arXiv: 2606.31020 by Xinxiang Guo, Yifen Mu, Yingkai Li.

Figure 1
Figure 1. Figure 1: An illustrative example. through their correlation with each other and with the state, jointly reveal substantially sharper information about the relevant posterior belief. The central question of this paper is whether, and how, a decision-maker can exploit such information using calibration alone. This question is also about the appropriate benchmark for online decision-making with forecasts. A standard b… view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a framework for aggregating multiple individually calibrated probabilistic forecasts when the decision-maker knows only the calibration property. It introduces a robust max-min benchmark defined as the best guaranteed payoff against all profile-wise conditional-mean mappings consistent with individual calibration. This benchmark admits a linear-programming formulation, dominates the optimal-in-hindsight (OIH) benchmark up to calibration error, lies strictly below the Bayesian benchmark in some cases (via explicit counterexamples), and is attainable by online algorithms under forecast-only feedback.

Significance. If the central claims hold, the work is significant for supplying a tractable, parameter-free benchmark that lies between the conservative OIH benchmark and the informationally demanding Bayesian benchmark. The LP formulation, dominance result, counterexamples separating the benchmarks, and online algorithms from forecast-only feedback are concrete strengths that could influence practical aggregation of expert forecasts. The focus on calibration as the sole maintained assumption and the demonstration that joint calibrated forecasts can carry decision-relevant information unavailable from any single expert add to the contribution.

minor comments (3)
  1. [Abstract] The abstract states that the benchmark 'dominates the OIH benchmark up to calibration error' and 'admits a linear-programming formulation'; a one-sentence clarification of the precise sense of dominance and the dimension of the LP would improve readability for readers who stop at the abstract.
  2. [Online algorithms] The online algorithms section would benefit from an explicit statement of the regret bound relative to the robust benchmark (e.g., whether the bound is O(1) or depends on the number of experts or the calibration error).
  3. A short table or paragraph comparing the three benchmarks (OIH, robust, Bayesian) along the dimensions of maintained assumptions, computational tractability, and attainable value would help readers locate the contribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. No specific major comments or criticisms were raised in the report, so we interpret the recommendation as pertaining to minor editorial or expositional adjustments that we will address in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from external calibration condition

full rationale

The paper defines its max-min robust benchmark directly from the external calibration condition (conditional expectation equals forecast) without reducing it to a fitted parameter or self-referential definition. The LP formulation, dominance over OIH (up to calibration error), separation from Bayesian benchmark via counterexamples, and online algorithms are all derived from this independent premise and standard optimization/regret analysis. No self-citation chains, ansatzes smuggled via prior work, or renamings of known results appear as load-bearing steps. The construction is externally falsifiable and does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claim rests on the domain assumption that forecasts are calibrated and that the decision-maker has no further information about the experts' joint distribution.

axioms (1)
  • domain assumption Each expert's forecast is calibrated: the conditional expectation of the outcome given the forecast equals the forecast itself.
    Stated as the only known property of the experts in the abstract.

pith-pipeline@v0.9.1-grok · 5675 in / 1121 out tokens · 55204 ms · 2026-07-01T03:18:54.187009+00:00 · methodology

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Reference graph

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