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arxiv: 2604.17194 · v1 · submitted 2026-04-19 · 📊 stat.ML · cs.LG

Forecast Sports Outcomes under Efficient Market Hypothesis: Theoretical and Experimental Analysis of Odds-Only and Generalised Linear Models

Kaito Goto , Naoya Takeishi , Takehisa Yairi This is my paper

Pith reviewed 2026-05-10 06:26 UTC · model grok-4.3

classification 📊 stat.ML cs.LG
keywords betting odds conversionsports forecastingefficient market hypothesisgeneralized linear modelsfootballfavorite longshot biasprobability estimation
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The pith

New odds-only and generalized linear models convert betting odds to outcome probabilities more accurately than existing methods.

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

The paper introduces two approaches to convert betting odds into probabilities under the efficient market hypothesis. The first is an odds-only method called OO-EPC that aligns with bookmakers' equal profitability confidence without needing historical data. The second is FL-GLM, a generalized linear model that adjusts for favorite-longshot bias using historical data. Empirical tests on 90,014 football matches show these methods outperform prior odds-only and GLM approaches for most or all bookmakers. This matters because accurate probabilities from odds can benchmark forecasting models and test market efficiency.

Core claim

The authors propose the Odds-Only-Equal-Profitability-Confidence (OO-EPC) method for converting odds without historical fitting by enforcing equal confidence in profitability across outcomes, and the Favourite-Longshot-Bias-Adjusted GLM (FL-GLM) that fits a single parameter for bias. On a dataset of 90,014 football matches from five bookmakers, OO-EPC beats existing odds-only methods for the majority of bookmakers, and FL-GLM beats multinomial and logistic GLMs for all bookmakers. They also test OO-EPC in real basketball forecasting competitions.

What carries the argument

The OO-EPC method, which sets probabilities so that the implied profitability confidence is equal for each outcome, and the FL-GLM, which incorporates a single parameter to capture the favorite-longshot bias in odds.

If this is right

  • Improved benchmarks for sports outcome forecasting models using betting odds.
  • More reliable analysis of whether betting markets are efficient.
  • Practical use in annual forecasting competitions without historical fitting for odds-only cases.
  • Simpler interpretable models for probability conversion from odds.

Where Pith is reading between the lines

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

  • These methods could extend to other sports if the observed biases are similar.
  • Future work might test whether the equal profitability confidence assumption holds across different betting markets.
  • Combining both methods or using them in ensemble could further improve forecasts.
  • The real-world application in basketball competitions suggests robustness beyond the football training data.

Load-bearing premise

The patterns of bias and relationships found in the 90,014 football matches will hold for future matches and other sports, and that matching equal profitability confidence is the right goal for an odds-only converter.

What would settle it

Applying the OO-EPC and FL-GLM to a fresh dataset of matches from a different sport or later football seasons and checking whether they still outperform the baselines on accuracy metrics.

Figures

Figures reproduced from arXiv: 2604.17194 by Kaito Goto, Naoya Takeishi, Takehisa Yairi.

Figure 1
Figure 1. Figure 1: Comparison of expected Number of Draws implied by each Odds-Only Method and the Actual Draws under each bookmaker. Each given group of six bars represent an evaluation under the given bookmaker; each given colour represents the amount of expected draws implied by the given odds-only method or the actual number of draws. Odds-Only Method Correlation between Log-Loss and Booksum p-value Multiplicative 0.1085… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of expected Number of Draws implied by each Generalised Linear Model and the Actual Draws under each bookmaker. Each given group of four bars represent an evaluation under the given bookmaker; each given colour represents the amount of expected draws implied by the given generalised linear model or the actual number of draws. make assumptions with no supportive evidence, whereas our proposed OO-… view at source ↗
read the original abstract

Converting betting odds into accurate outcome probabilities is a fundamental challenge in order to use betting odds as a benchmark for sports forecasting and market efficiency analysis. In this study, we propose two methods to overcome the limitations of existing conversion methods. Firstly, we propose an odds-only method to convert betting odds to probabilities without using historical data for model fitting. While existing odds-only methods, such as Multiplicative, Shin, and Power exist, they do not adjust for biases or relationships we found in our betting odds dataset, which consists of 90014 football matches across five different bookmakers. To overcome these limitations, our proposed Odds-Only-Equal-Profitability-Confidence (OO-EPC) method aligns with the bookmakers' pricing objectives of having equal confidence in profitability for each outcome. We provide empirical evidence from our betting odds dataset that, for the majority of bookmakers, our proposed OO-EPC method outperforms the existing odds-only methods. Beyond controlled experiments, we applied the OO-EPC method under real-world uncertainty by using it for six iterations of an annual basketball outcome forecasting competition. Secondly, we propose a generalised linear model that utilises historical data for model fitting and then converts betting odds to probabilities. Existing generalised linear models attempt to capture relationships that the Efficient Market Hypothesis already captures. To overcome this shortcoming, our proposed Favourite-Longshot-Bias-Adjusted Generalised Linear Model (FL-GLM) fits just one parameter to capture the favourite-longshot bias, providing a more interpretable alternative. We provide empirical evidence from historical football matches where, for all bookmakers, our proposed FL-GLM outperforms the existing multinomial and logistic generalised linear models.

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

4 major / 2 minor

Summary. The manuscript proposes two methods for converting betting odds to outcome probabilities under the Efficient Market Hypothesis. The first is an odds-only method (OO-EPC) that adjusts existing conversions to enforce equal profitability confidence across outcomes, derived from biases observed in a dataset of 90,014 football matches across five bookmakers. The second is a generalised linear model (FL-GLM) that fits a single parameter to capture the favourite-longshot bias. The paper claims OO-EPC outperforms existing odds-only methods (Multiplicative, Shin, Power) for the majority of bookmakers on this dataset, while FL-GLM outperforms existing multinomial and logistic GLMs for all bookmakers; OO-EPC is additionally evaluated in six iterations of a real basketball forecasting competition.

Significance. If the outperformance claims hold under proper out-of-sample validation, the work could strengthen the use of betting odds as benchmarks for market efficiency tests and improve practical sports forecasting pipelines. The large-scale dataset, the real-world competition application for OO-EPC, and the emphasis on interpretability via a single-parameter GLM are positive features. The odds-only nature of OO-EPC, if it generalises without historical fitting, would be particularly useful for settings where training data are unavailable.

major comments (4)
  1. [Experimental results / abstract] Experimental results section (and abstract): Both OO-EPC and FL-GLM performance claims are evaluated on the identical 90,014-match football dataset used to identify the biases/relationships and to fit the single FL-GLM parameter. No hold-out set, temporal split, or cross-validation is described, so the reported superiority over baselines may reflect in-sample fitting rather than genuine generalisation.
  2. [Methods (OO-EPC derivation)] Methods description of OO-EPC: Although presented as an odds-only converter with no historical model fitting, the functional form is explicitly derived from biases and relationships observed across the full 90,014-match dataset. This makes the outperformance claim on the same data circular and weakens the assertion that OO-EPC is independent of historical information.
  3. [FL-GLM definition and empirical comparison] FL-GLM specification and results: The model fits one parameter directly to the favourite-longshot bias present in the evaluation data. This reduces the procedure to an in-sample correction term; the reported outperformance over multinomial and logistic GLMs on the same data is therefore expected by construction and does not demonstrate superior modelling of the underlying probabilities.
  4. [Real-world application / results] Basketball competition evaluation: The six-iteration real-world test is reported only for OO-EPC and is narrow in scope; no equivalent external validation or statistical comparison is provided for FL-GLM, leaving the generalisation claim for the GLM unsupported.
minor comments (2)
  1. [Abstract / results] The abstract and results sections omit any mention of statistical significance tests, confidence intervals, or error bars on the performance differences, making it difficult to assess whether the reported gains are reliable.
  2. [Methods] Notation for the equal-profitability-confidence objective and the exact functional form of OO-EPC should be stated explicitly with equations to allow reproducibility.

Simulated Author's Rebuttal

4 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which correctly identify the need for stronger validation to support the generalisation claims. We agree that the current presentation relies on in-sample evaluations and will revise the manuscript accordingly by adding temporal splits, cross-validation, parameter stability checks, and statistical comparisons. These changes will clarify the methods' applicability while preserving their core contributions.

read point-by-point responses

  1. Referee: Experimental results section (and abstract): Both OO-EPC and FL-GLM performance claims are evaluated on the identical 90,014-match football dataset used to identify the biases/relationships and to fit the single FL-GLM parameter. No hold-out set, temporal split, or cross-validation is described, so the reported superiority over baselines may reflect in-sample fitting rather than genuine generalisation.

    Authors: We acknowledge the validity of this concern. The full dataset was used to illustrate how the methods capture the observed biases and relationships. In the revision, we will add a dedicated out-of-sample evaluation section using chronological temporal splits (e.g., train on the first 70% of matches by date and test on the remainder) and 5-fold cross-validation. Both OO-EPC and FL-GLM will be assessed under these protocols, with performance metrics reported against the same baselines to confirm whether superiority holds out-of-sample. revision: yes

  2. Referee: Methods description of OO-EPC: Although presented as an odds-only converter with no historical model fitting, the functional form is explicitly derived from biases and relationships observed across the full 90,014-match dataset. This makes the outperformance claim on the same data circular and weakens the assertion that OO-EPC is independent of historical information.

    Authors: We agree that the functional form was identified through analysis of the full dataset to enforce equal profitability confidence. However, once derived, OO-EPC becomes a fixed, parameterised converter that requires only the current betting odds as input and no historical matches or refitting for new applications. This property distinguishes it from data-dependent GLMs. To address circularity, we will move the full derivation to an appendix, add a stability analysis of the derived parameters across random subsets and time periods of the data, and emphasise that the method can be applied without access to the original dataset. revision: partial

  3. Referee: FL-GLM specification and results: The model fits one parameter directly to the favourite-longshot bias present in the evaluation data. This reduces the procedure to an in-sample correction term; the reported outperformance over multinomial and logistic GLMs on the same data is therefore expected by construction and does not demonstrate superior modelling of the underlying probabilities.

    Authors: We concur that fitting the single favourite-longshot parameter on the evaluation data renders the reported results in-sample. In the revised manuscript, we will refit the FL-GLM parameter exclusively on a training subset (chronological first 70%) and evaluate on the held-out test subset. The multinomial and logistic baselines will be trained and evaluated identically. This provides a fair out-of-sample comparison and demonstrates whether the single-parameter adjustment yields superior probability estimates beyond in-sample correction. revision: yes

  4. Referee: Basketball competition evaluation: The six-iteration real-world test is reported only for OO-EPC and is narrow in scope; no equivalent external validation or statistical comparison is provided for FL-GLM, leaving the generalisation claim for the GLM unsupported.

    Authors: The basketball competition provides genuine external, real-world validation for OO-EPC under live conditions and without football-specific historical data. For FL-GLM, which requires historical fitting, we will incorporate the temporal-split evaluation described above as its primary out-of-sample test on the football data. We will also add paired statistical significance tests (e.g., Wilcoxon signed-rank) across bookmakers and iterations to strengthen the comparisons for both methods. revision: partial

Circularity Check

2 steps flagged

OO-EPC and FL-GLM outperformance claims reduce to in-sample adjustments on the 90014-match evaluation dataset

specific steps
  1. fitted input called prediction [Abstract]
    "While existing odds-only methods, such as Multiplicative, Shin, and Power exist, they do not adjust for biases or relationships we found in our betting odds dataset, which consists of 90014 football matches across five different bookmakers. To overcome these limitations, our proposed Odds-Only-Equal-Profitability-Confidence (OO-EPC) method aligns with the bookmakers' pricing objectives of having equal confidence in profitability for each outcome. We provide empirical evidence from our betting odds dataset that, for the majority of bookmakers, our proposed OO-EPC method outperforms the existing"

    The method form is explicitly constructed to correct biases and relationships observed across the entire evaluation dataset; its claimed superiority is then shown on that same dataset, so the performance gain is a direct consequence of the data-driven design rather than an independent odds-only derivation.

  2. fitted input called prediction [Abstract]
    "our proposed Favourite-Longshot-Bias-Adjusted Generalised Linear Model (FL-GLM) fits just one parameter to capture the favourite-longshot bias, providing a more interpretable alternative. We provide empirical evidence from historical football matches where, for all bookmakers, our proposed FL-GLM outperforms the existing multinomial and logistic generalised linear models."

    The single parameter is fitted directly to the favourite-longshot bias present in the historical football matches; the model's outperformance is then reported on those same matches, reducing the superiority claim to an in-sample fit rather than an independent GLM prediction.

full rationale

The paper derives OO-EPC by identifying and correcting biases/relationships in the full 90014-match football dataset (despite claiming no historical fitting), then demonstrates superiority on that same data. FL-GLM explicitly fits its single parameter to the favourite-longshot bias in the historical matches and evaluates outperformance on the identical data. Both central claims therefore rest on data-driven construction rather than independent, out-of-sample derivation. No self-citations or uniqueness theorems are load-bearing; the circularity is limited to the fitted-input pattern and lack of hold-out separation.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the assumption that the favourite-longshot bias is the dominant systematic error and that equal-profitability confidence is the right target for odds conversion; one free parameter is introduced in FL-GLM.

free parameters (1)
  • favourite-longshot bias parameter
    Single scalar fitted to historical data to adjust the GLM for the observed bias.
axioms (1)
  • domain assumption Efficient Market Hypothesis provides the baseline relationships that the GLM should not re-learn
    Paper states that existing GLMs attempt to capture relationships already captured by EMH.

pith-pipeline@v0.9.0 · 5619 in / 1260 out tokens · 24977 ms · 2026-05-10T06:26:09.317075+00:00 · methodology

discussion (0)

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