Peer-reviewed studies, preprints and classic methods papers on pricing, efficiency, risk and research practice. Each entry is our summary, with the original link and a plain statement of what the study does not show.
Summaries are original paraphrases of published abstracts and records; no paper text is reproduced. Bibliographic details were checked against publisher, index and author listings on 25 October 2026. Where a paper was later corrected, the correction is shown. Preprints are labelled as not peer reviewed.
24 sources
Maximising the expected logarithm of wealth gives the fastest long-run exponential growth; with fair odds that maximum growth rate equals the channel's information rate. The staking rule derived here became known as the Kelly criterion.
Does not show: That full Kelly stakes are sensible when probabilities are estimated, or that any real betting edge exists.
The model describes score outcomes well, and the authors report a positive return when its probabilities were used to select bets against bookmaker odds in their historical evaluation.
Does not show: That a Dixon–Coles model beats today's football markets after margins, limits and competition.
To a first approximation, wagering prices are efficient forecasts of outcomes and absorb information as lines move, while documented exceptions such as the favourite–longshot bias remain important.
Does not show: That every market or price is efficient, or that the anomalies it lists can be exploited today.
Probabilities from Shin's model were more accurate than basic normalisation or regression in the samples studied; bookmakers differed as sources of probability, and exchange prices were not always the best source, especially in smaller markets.
Does not show: That any de-vig method recovers the true probability, or that de-vigged prices reveal profitable bets.
When probabilities are estimated with error, Kelly stakes should be shrunk; shrunken strategies outperformed raw Kelly out of sample in their simulations and tennis illustration.
Does not show: A universal fraction of Kelly that is safe for everyone, or that the underlying tennis model had a real edge.
In simulations, risk-constrained Kelly bets achieved better trade-offs than fractional Kelly: more growth at the same drawdown risk, or less drawdown risk at the same growth.
Does not show: That drawdown constraints protect a bettor whose probability estimates are wrong.
The authors report that the strategy was profitable in a historical simulation, in a simulation on more granular odds data and in a short real-money trial, and that bookmakers restricted their accounts during the trial.
Does not show: That cross-bookmaker discrepancies remain available or executable at meaningful stakes today.
Bettor disagreement combined with bookmaker risk aversion can generate a favourite–longshot pattern (worse returns on long shots) without assuming that bettors love risk.
Does not show: That backing favourites is profitable, or how large the bias is in any particular market today.
Apparent inefficiencies appear in individual seasons but do not persist systematically across seasons or leagues, and simulations show that significant single-season results are likely even in a fully efficient market.
Does not show: That no bettor can ever find an edge; it shows that single-season evidence is weak.
Much of the original out-of-sample profit depended on a single outlier bet recorded at erroneously long odds; after correction only one strategy remained significantly profitable in the original period, returns came mainly from individual bookmakers' mispricing rather than the buzz signal, and there were no further profits in data after 2020.
Does not show: That attention-based signals never contain information; it shows this published profit was not robust.
Sportsbook forecasts were mostly reliable, yet line changes were negatively autocorrelated (overreaction), forecast quality did not always improve toward game time, and simple strategies would have earned significant hypothetical profits.
Does not show: That football markets behave the same way, or that the hypothetical strategies were executable at scale.
Selecting models by calibration produced better simulated betting outcomes than selecting by accuracy. After the 2025 corrigendum, average returns were negative for both selection rules (about −9.8% for calibration versus −26.8% for accuracy), while the best single models reached about +23.1% and +10.9%.
Does not show: That calibrated models are profitable; the corrected average returns were negative.
Correction: Corrigendum, Machine Learning with Applications 19 (2025), 100627 (doi:10.1016/j.mlwa.2025.100627). The original 2024 return figures were withdrawn; use the corrected values.
Bookmakers facing less competition set prices with a favourite–longshot bias, while the competitive market shows little of it; the model's prediction about returns on draws is also supported. Working-paper versions describe a sample of tens of thousands of matches (later drafts cite over 150,000).
Does not show: That a favourite strategy beats competitive markets; the bias is concentrated where competition is weaker.
Tests based on normalised probabilities behave well; tests based on raw inverse odds do not, and are biased against finding a favourite–longshot bias.
Does not show: Any profitable strategy; it shows how test design can change conclusions.
A bookmaker protecting against insiders sets odds that produce a favourite–longshot bias; the model underlies the Shin de-vig method.
Does not show: How much insider trading exists in a given market.
Provides the estimation approach later used to convert odds into probabilities under Shin's model.
Does not show: That Shin probabilities are always the most accurate.
The power method outperformed normalisation on all three datasets and was comparable to or better than Shin, while always producing valid probabilities.
Does not show: That one method is best for every football market.
Team-strength Poisson models fit football scores reasonably well; allowing modest dependence between the two scores improves the fit.
Does not show: Anything about betting profitability.
Introduces what is now called the Brier score, which rewards forecasts that are both well calibrated and sharp.
Does not show: Whether a forecast can make money against a market.
Explains why strictly proper scores such as the Brier and logarithmic scores reward honest probability forecasts.
Does not show: That a better-scored forecast will be profitable against a market.
Explains Kelly's growth properties and why many practitioners bet a fraction of Kelly to reduce volatility.
Does not show: That any particular sports-betting edge exists.
Kelly maximises long-run growth but can be very risky in the short run; fractional Kelly trades some growth for substantially lower risk.
Does not show: Which fraction suits an individual bettor.
With enough trials, an impressive in-sample backtest is likely even when no strategy has skill, and overfit strategies tend to perform worse out of sample.
Does not show: That every strong backtest is overfit; it shows why the number of trials must be reported.
Reported that attention-related signals were associated with mispricing and profitable strategies; later corrected by Clegg & Cartlidge.
Does not show: A durable profitable strategy, once the outlier bet is corrected.
