Probabilistic Prediction Markets with Intermittent Contributions

Michael Vitali; Pierre Pinson

arxiv: 2510.13385 · v3 · pith:GGYXMVBVnew · submitted 2025-10-15 · 💻 cs.LG

Probabilistic Prediction Markets with Intermittent Contributions

Michael Vitali , Pierre Pinson This is my paper

Pith reviewed 2026-05-18 07:23 UTC · model grok-4.3

classification 💻 cs.LG

keywords prediction marketsforecast combinationrobust regressionintermittent submissionsprobabilistic forecastspayoff allocationtime-varying conditions

0 comments

The pith

A prediction market uses robust regression to combine agent forecasts while handling missing submissions and adapting to time-varying conditions.

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

The paper introduces a prediction market where independent agents trade probabilistic forecasts for rewards without sharing raw data. It employs robust regression to learn how to optimally combine these forecasts, even when agents submit intermittently or miss rounds entirely. The design also includes a payoff allocation that factors in both historical accuracy and current performance to meet economic fairness standards. This allows the market to function in settings where participants have competitive interests and varying availability. If the approach works as claimed, it offers a flexible way to aggregate forecasts dynamically without requiring constant participation from all agents.

Core claim

The paper claims that robust regression models can learn the optimal combination of probabilistic forecasts from agents who enter and exit the market at will, while handling missing submissions and time-varying conditions, and that a corresponding payoff mechanism can allocate rewards based on both in-sample and out-of-sample performance while satisfying several desirable economic properties.

What carries the argument

Robust regression models for learning optimal forecast combinations that handle missing submissions, paired with a payoff allocation mechanism that incorporates in-sample and out-of-sample performance.

If this is right

Agents gain flexibility to join or leave without disrupting the overall forecast quality.
The market automatically adjusts weights as agent performance changes.
Payoffs reflect both past reliability and recent accuracy to encourage sustained good contributions.
Collaboration occurs through forecasts alone, preserving data ownership.

Where Pith is reading between the lines

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

This structure might apply to domains like energy trading or financial forecasting where contributors have irregular availability.
Testing against real-time data streams could reveal whether the regression updates keep pace with rapid shifts.
The economic properties of the payoffs could be checked in lab experiments with human participants to confirm incentive alignment.

Load-bearing premise

That robust regression can reliably learn the best way to weight and combine forecasts when some agents submit only intermittently and when accuracy patterns shift over time.

What would settle it

A controlled test showing that the market's aggregated forecasts fail to beat simple averaging or individual agent accuracy under realistic patterns of intermittent submissions and non-stationary conditions.

Figures

Figures reproduced from arXiv: 2510.13385 by Michael Vitali, Pierre Pinson.

**Figure 1.** Figure 1: Market design overview and not sum to one. For a comprehensive overview of the state of the art with forecast combination, we refer the reader [16]. E. Pay-off allocation A pay-off function is central to the design of a market mechanism as it distributes the generated utility among the market players (sellers) according to their performances. For this reason, it is critical to design a pay-off function tha… view at source ↗

**Figure 3.** Figure 3: pay-off allocation for QR (top) and RQR (bottom) with three quantile [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 2.** Figure 2: Convergence of estimated weights for QR (top) and RQR (bottom) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** Figure 4: Convergence of estimated weights for QR (top) and RQR (bottom) [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 6.** Figure 6: Wind energy power generation reported by [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

read the original abstract

Although both data availability and the demand for accurate forecasts are increasing, collaboration between stakeholders is often constrained by data ownership and competitive interests. In contrast to recent proposals within cooperative game-theoretical frameworks, we place ourselves in a more general framework, based on prediction markets. There, independent agents trade forecasts of uncertain future events in exchange for rewards. We introduce and analyse a prediction market that (i) accounts for the historical performance of the agents, (ii) adapts to time-varying conditions, while (iii) permitting agents to enter and exit the market at will. The proposed design employs robust regression models to learn the optimal forecasts' combination whilst handling missing submissions. Moreover, we introduce a pay-off allocation mechanism that considers both in-sample and out-of-sample performance while satisfying several desirable economic properties. Case-studies using simulated and real-world data allow demonstrating the effectiveness and adaptability of the proposed market design.

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.

Desk Editor's Note private letter to a colleague

read the letter

The paper sets up a prediction market that uses robust regression to combine forecasts when agents contribute only intermittently and includes a payoff rule based on both in-sample and out-of-sample performance. This directly targets the practical problem of data ownership that blocks full collaboration on forecasts across groups or firms. The market framing lets agents enter and exit without breaking the system, which is more flexible than the cooperative game theory approaches referenced in the abstract. Time adaptation and performance weighting are built in from the start, which fits real settings where conditions shift and not everyone submits every round. The simulated and real-world case studies are presented as evidence that the design adapts and remains effective, which adds some applied grounding. The main limitation is that the abstract shows no equations, no specific quantitative results, and no error bars or implementation details. This leaves the claims about optimality and the economic properties of the payoff rule difficult to verify without the full derivations and numbers. The assumption that robust regression will reliably handle the missing submissions under time variation needs checking against the actual model and experiments. Readers working on forecast combination, mechanism design, or collaborative prediction in applied domains like energy or finance would find the setup worth examining. It is an incremental but targeted step within existing prediction market and regression work rather than a broad shift. I would bring this to a reading group to discuss the regression choice and payoff construction. It deserves a serious referee to go through the math and results in detail. I recommend sending it for peer review.

Referee Report

2 major / 1 minor

Summary. The paper claims to introduce and analyse a prediction market that accounts for the historical performance of the agents, adapts to time-varying conditions, while permitting agents to enter and exit the market at will. The proposed design employs robust regression models to learn the optimal forecasts' combination whilst handling missing submissions. Moreover, it introduces a pay-off allocation mechanism that considers both in-sample and out-of-sample performance while satisfying several desirable economic properties. Case-studies using simulated and real-world data demonstrate the effectiveness and adaptability of the proposed market design.

Significance. If the result holds, this work is significant as it provides a flexible framework for prediction markets that can handle real-world challenges like intermittent agent participation and changing conditions. By using robust regression and a performance-based payoff, it could enable better collaboration in data-sensitive environments, offering an alternative to cooperative game theory approaches mentioned in the abstract. The handling of missing submissions via regression is a practical strength.

major comments (2)

Abstract: The abstract asserts effectiveness via simulated and real-world case-studies but provides no equations, error bars, exclusion rules, or quantitative results; central claims about optimality and economic properties cannot be verified from available text.
Market Design and Payoff Allocation: The claim that the payoff allocation satisfies desirable economic properties under intermittent submissions and time-varying conditions is load-bearing for the central contribution, yet the description does not include explicit derivations showing how these properties are preserved when agents enter/exit or submissions are missing.

minor comments (1)

The notation and definitions for the robust regression models in the methods could be expanded for clarity, particularly regarding how missing submissions are imputed or weighted.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive summary and for identifying areas where the presentation can be strengthened. We address the two major comments below and commit to revisions that improve clarity without altering the core contributions.

read point-by-point responses

Referee: Abstract: The abstract asserts effectiveness via simulated and real-world case-studies but provides no equations, error bars, exclusion rules, or quantitative results; central claims about optimality and economic properties cannot be verified from available text.

Authors: We agree that the abstract, as a high-level summary, does not contain the detailed equations or quantitative results that appear in the body of the paper. The full manuscript presents the robust regression formulation, the explicit handling of missing submissions, the payoff rule combining in-sample and out-of-sample performance, and the numerical outcomes from both simulated and real-world experiments. To address the concern, we will revise the abstract to include a concise statement of the key quantitative improvements and the specific economic properties that are satisfied, while respecting length limits. revision: yes
Referee: Market Design and Payoff Allocation: The claim that the payoff allocation satisfies desirable economic properties under intermittent submissions and time-varying conditions is load-bearing for the central contribution, yet the description does not include explicit derivations showing how these properties are preserved when agents enter/exit or submissions are missing.

Authors: The payoff mechanism is constructed from the weights obtained via robust regression on available forecasts, with an additional out-of-sample adjustment term. Because the regression operates only on observed submissions, missing data are handled automatically. We acknowledge that the current text states the resulting properties (individual rationality, budget balance, and incentive compatibility) but does not supply step-by-step derivations for the dynamic case. We will add a dedicated subsection containing these derivations, showing how the properties continue to hold when agents enter or exit and when submissions are intermittent. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces a prediction market design that uses robust regression to combine agent forecasts under intermittent submissions and time-varying conditions, along with a payoff rule incorporating in- and out-of-sample performance. No equations or derivations are presented that reduce the claimed market properties, predictions, or payoffs to fitted parameters by construction, nor do self-citations form a load-bearing chain for the central mechanism. The design is advanced as an original framework with case-study validation, remaining self-contained and independent of tautological re-expression of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proposal rests on standard prediction-market assumptions and the effectiveness of robust regression for missing data; no explicit free parameters, new entities, or ad-hoc axioms are stated in the abstract.

axioms (2)

domain assumption Agents participate rationally and respond to payoff incentives in the market
Implicit in any prediction-market framework; invoked when claiming the design permits entry/exit at will and allocates payoffs fairly.
domain assumption Robust regression can produce stable combined forecasts despite missing submissions and time variation
Central to the claim that the market learns optimal combinations; location: abstract description of the design.

pith-pipeline@v0.9.0 · 5671 in / 1381 out tokens · 32308 ms · 2026-05-18T07:23:20.792398+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The proposed design employs robust regression models to learn the optimal forecasts' combination whilst handling missing submissions... pay-off allocation mechanism that considers both in-sample and out-of-sample performance

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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J. Dowell and P. Pinson, “Very-short-term probabilistic wind power forecasts by sparse vector autoregression,”IEEE Transactions on Smart Grid, vol. 7, no. 2, pp. 763–770, 2016

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J. W. Messner and P. Pinson, “Online adaptive lasso estimation in vector autoregressive models for high dimensional wind power forecasting,”Int. Journal of F orecasting, vol. 35, no. 4, pp. 1485–1498, 2019

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L. Cavalcante, R. J. Bessa, M. Reis, and J. Browell, “Lasso vector autoregression structures for very short-term wind power forecasting,” Wind Energy, vol. 20, no. 4, pp. 657–675, 2017

work page 2017

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Online distributed learning in wind power forecasting,

B. Sommer, P. Pinson, J. W. Messner, and D. Obst, “Online distributed learning in wind power forecasting,”International Journal of F orecast- ing, vol. 37, no. 1, pp. 205–223, 2021

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C. Gonc ¸alves, R. J. Bessa, and P. Pinson, “Privacy-preserving distributed learning for renewable energy forecasting,”IEEE Transactions on Sus- tainable Energy, vol. 12, no. 3, pp. 1777–1787, 2021

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work page 2014

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J. Witkowski, R. Freeman, J. W. Vaughan, D. M. Pennock, and A. Krause, “Incentive-compatible forecasting competitions,”Manage- ment Science, vol. 69, no. 3, p. 1354–1374, 2022

work page 2022

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A. A. Raja, P. Pinson, J. Kazempour, and S. Grammatico, “A market for trading forecasts: A wagering mechanism,”International Journal of F orecasting, vol. 40, no. 1, pp. 142–159, 2024

work page 2024

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Adaptive optimization for prediction with missing data,

D. Bertsimas, A. Delarue, and J. Pauphilet, “Adaptive optimization for prediction with missing data,”Machine Learning, vol. 114, 2025, art. no. 124

work page 2025

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Learning data-driven uncertainty set partitions for robust and adaptive energy forecasting with missing data,

A. Stratigakos and P. Andrianesis, “Learning data-driven uncertainty set partitions for robust and adaptive energy forecasting with missing data,” 2025. [Online]. Available: https://arxiv.org/abs/2503.20410

work page arXiv 2025

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work page 2024

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A value for n-person games,

L. S. Shapleyet al., “A value for n-person games,”Contributions Theory Games, vol. 2, no. 28, p. 307–317, 1953

work page 1953

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Online coalitional games for real- time payoff distribution with applications to energy markets,

A. A. Raja and S. Grammatico, “Online coalitional games for real- time payoff distribution with applications to energy markets,”IEEE Transactions on Energy Markets, Policy and Regulation, vol. 1, no. 2, pp. 97–106, 2023

work page 2023

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X. Wang, R. J. Hyndman, F. Li, and Y . Kang, “Forecast combinations: An over 50-year review,”International Journal of F orecasting, vol. 39, no. 4, pp. 1518–1547, 2023

work page 2023

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Online quantile regression,

Y . Shen, D. Xia, and W.-X. Zhou, “Online quantile regression,” 2025. [Online]. Available: https://arxiv.org/abs/2402.04602

work page arXiv 2025

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Wind power production estimation and forecast on belgian grid

Elia, “Wind power production estimation and forecast on belgian grid.” [Online]. Available: https://opendata.elia.be/explore/dataset/ ods031/table/?sort=datetime&refine.offshoreonshore=Offshore

work page

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Ecmwf ifs high-resolution operational forecasts,

European Centre for Medium-Range Weather Forecasts, “Ecmwf ifs high-resolution operational forecasts,” 2016. [Online]. Available: https://rda.ucar.edu/datasets/d113001/

work page 2016

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Ncep gfs 0.25 degree global forecast grids historical archive,

National Centers for Environmental Prediction, National Weather Service, NOAA, U.S. Department of Commerce, “Ncep gfs 0.25 degree global forecast grids historical archive,” 2015. [Online]. Available: https://rda.ucar.edu/datasets/d084001/

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Bayesian regression markets,

T. Falconer, J. Kazempour, and P. Pinson, “Bayesian regression markets,” Journal of Machine Learning Research, vol. 25, no. 180, p. 1–38, 2024. 24th Power Systems Computation Conference PSCC 2026 Limassol, Cyprus — June 8 – June 12, 2026 8 APPENDIXA ANALYSIS OFMARKETPROPERTIES We gather here the proofs for the various market properties mentioned and discu...

work page 2024