§ IV · Reference

6 min readlast revised 2026-04-30snapshot 2026-06-15T03:47Z

Bibliography

The academic works and mathematical methods cited across this project, grouped by the role each one plays in the methodology rather than by year.

By The 45% Problem project

Contents

Every formal claim in the Vault that rests on prior academic work is listed here. Entries are grouped by the role the work plays in the project: probabilistic football modeling, de-vigging and market microstructure, forecast scoring rules, pairwise comparison and small-sample inference, bootstrap procedures, bet sizing under uncertainty, and the macroeconomic-determinants literature. Within each section, citations are alphabetical by first author. The "used in" line after each entry points to the Vault page or pages that lean on the work.

A canonical citation for this project itself lives at Citation. A canonical symbol table lives at Notation.

Probabilistic football modeling

Dixon, M. J., and Coles, S. G. (1997). Modelling association football scores and inefficiencies in the football betting market. Journal of the Royal Statistical Society: Series C (Applied Statistics), 46(2), 265 to 280.

Used in: Simulation (the $\tau(x, y)$ low-score correction with $\rho = -0.05$ ), Models, Notation.

Dixon, M. J., and Robinson, M. E. (1998). A birth process model for association football matches. Journal of the Royal Statistical Society: Series D (The Statistician), 47(3), 523 to 538.

Used in: Simulation (the empirical baseline for extra-time scoring rates that informed the

\lambda^{\mathrm{ET}} = 0.6 \cdot \lambda^{90}

calibration).

Karlis, D., and Ntzoufras, I. (2003). Analysis of sports data by using bivariate Poisson models. Journal of the Royal Statistical Society: Series D (The Statistician), 52(3), 381 to 393.

Used in: Simulation (the $X = W_1 + W_3, \; Y = W_2 + W_3$ common-shock decomposition), Models. The authors' empirical range of $\lambda_3 \in [0.05, 0.20]$ on European football data is the basis for the project's locked $\lambda_3 = 0.10$ .

Lee, J., Kim, J., Kim, H., and Lee, J.-S. (2022). A Bayesian approach to predict football matches with changed home advantage in spectator-free matches after the COVID-19 break. Entropy, 24(3), 366.

Used in: contextual reading on home-advantage modeling. The project sets $H = 0$ for the FIFA 2026 simulation (single-host-complex final), but the modeling of home advantage as a hierarchical Bayesian shift parameter follows the paper's framing.

Maher, M. J. (1982). Modelling association football scores. Statistica Neerlandica, 36(3), 109 to 118.

Used in: Simulation (the independent Poisson case recovered when $\lambda_3 = 0$ ), Models. Maher's paper is the original separable-Poisson formulation that Karlis and Ntzoufras (2003) generalize.

Ogundeji, R., Aleem, A., and Obute, D. (2024). A Bayesian approach for predicting match outcomes: FIFA World Cup 2026. Journal of Innovative Applied Mathematics and Computational Sciences, 4(2), 153 to 174. DOI: 10.58205/jiamcs.v4i2.1848.

Used in: contextual reading on Bayesian World Cup forecasting. This project's frequentist multinomial logit and Bayesian shrinkage prior in M3 share the spirit of the paper's hierarchical regression framework without adopting the same likelihood.

De-vigging and market microstructure

Nyberg, H. (2014). A multinomial logit-based statistical test of association football betting market efficiency. HECER Discussion Paper No. 380, University of Helsinki.

Used in: Evaluation (the multinomial logit specification with Draw as reference and the $\chi^{2}_{2}$ likelihood ratio test on $H_0: \beta_{2H} = \beta_{2A} = 0$ ). The pre-registered critical values (5.991 for M★ at $\alpha = 0.05$ , 8.668 for the Bonferroni-corrected shadow tests) trace directly to this paper's framework.

Shin, H. S. (1993). Measuring the incidence of insider trading in a market for state-contingent claims. The Economic Journal, 103(420), 1141 to 1153.

Used in: Market Layer (the original Power method de-vigging derivation that motivates $\sum_i r_i^{z} = 1$ ). Shin's paper introduces the favorite-longshot bias correction implicit in the Power method's $z$ exponent.

Štrumbelj, E. (2014). On determining probability forecasts from betting odds. International Journal of Forecasting, 30(4), 934 to 943.

Used in: Market Layer. Štrumbelj's empirical comparison of the Power method against Shin's quadratic-surplus and proportional de-vigging on more than 100,000 two-way and three-way markets is the basis for the project's choice of the Power method as the canonical de-vigging procedure.

Forecast scoring rules

Brier, G. W. (1950). Verification of forecasts expressed in terms of probability. Monthly Weather Review, 78(1), 1 to 3.

Used in: Evaluation, Glossary, Notation. The original definition of what is now called the Brier score; the project uses the multiclass form summed over $K = 3$ outcomes.

Epstein, E. S. (1969). A scoring system for probability forecasts of ranked categories. Journal of Applied Meteorology, 8(6), 985 to 987.

Used in: Evaluation (the cumulative-distribution form of the Ranked Probability Score with normalization $1/(K-1)$ ).

Gneiting, T., and Raftery, A. E. (2007). Strictly proper scoring rules, prediction, and estimation. Journal of the American Statistical Association, 102(477), 359 to 378.

Used in: Why probabilities, Evaluation. The two-axis frame of calibration and sharpness used to argue that reporting all three scoring rules (Brier, log-loss, RPS) is the honest commitment.

Pairwise comparison and small-sample inference

Diebold, F. X., and Mariano, R. S. (1995). Comparing predictive accuracy. Journal of Business and Economic Statistics, 13(3), 253 to 263.

Used in: Evaluation, Kill criteria. The DM test is the inferential procedure under the kill criterion's 2-SE log-loss gap rule.

Harvey, D., Leybourne, S., and Newbold, P. (1997). Testing the equality of prediction mean squared errors. International Journal of Forecasting, 13(2), 281 to 291.

Used in: Evaluation. The HLN small-sample correction

\mathrm{DM}^{*} = \mathrm{DM} \cdot \sqrt{(N + 1 - 2h + h(h-1)/N) / N}

is applied because $N$ in the dozens of matches plus a small hold-out is well inside the regime where the asymptotic normal approximation is unreliable.

Newey, W. K., and West, K. D. (1987). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55(3), 703 to 708.

Used in: Evaluation. The Bartlett kernel HAC variance with bandwidth $h = \max(1, \lfloor N^{1/3}\rfloor)$ used inside the DM statistic.

Bootstrap and uncertainty quantification

Hall, P., and Mueller, U. K. (1997). Bias-corrected bootstrap methods for scale-invariant statistics. (Bias-correction procedure adapted in this project as the Hall-Mueller bootstrap Sharpe.)

Used in: Evaluation (the

\mathrm{SR}_{\mathrm{HM},b} = 2 \cdot \widehat{\mathrm{SR}} - \mathrm{SR}_{\mathrm{boot},b}

bias correction applied to the stationary block bootstrap of the CLV Sharpe ratio). The exact 1997 reference is loosely identified; the correction structure as implemented matches the standard bias-correction form for ratio statistics in the bootstrap-Sharpe literature.

Politis, D. N., and Romano, J. P. (1994). The stationary bootstrap. Journal of the American Statistical Association, 89(428), 1303 to 1313.

Used in: Evaluation (the stationary block bootstrap with block-length parameter

\bar{L} = \mathrm{clamp}(\lfloor 1.75 \cdot N^{1/3} \rfloor, [4, 20])

applied to the CLV series).

Bet sizing under uncertainty

Kelly, J. L., Jr. (1956). A new interpretation of information rate. Bell System Technical Journal, 35(4), 917 to 926.

Used in: Volatility Gate. The full Kelly fraction $f_{\mathrm{full}} = (p \cdot o - 1)/(o - 1)$ is the optimal log-growth solution under known $p$ .

MacLean, L. C., Thorp, E. O., and Ziemba, W. T. (Eds.) (2010). The Kelly Capital Growth Investment Criterion: Theory and Practice. World Scientific Handbook in Financial Economics Series. The collected volume includes "Good and Bad Properties of the Kelly Criterion" (2010, Risk and Decision Analysis, 1(1)).

Used in: Volatility Gate. The mean-variance argument that fractional Kelly with $\phi = 1/4$ dominates full Kelly under realistic estimation error on $p$ is the basis for the project's locked Kelly fractions.

Thorp, E. O. (1997). The Kelly criterion in blackjack, sports betting, and the stock market. Originally presented at the 10th International Conference on Gambling and Risk Taking, Montreal.

Used in: Volatility Gate. The standard quarter-Kelly recommendation ( $\phi = 1/4$ on mainline markets) traces to the practical analysis here.

Macroeconomic and structural determinants

Hoffmann, R., Ging, L. C., and Ramasamy, B. (2002). The socio-economic determinants of international soccer performance. Journal of Applied Economics, 5(2), 253 to 272.

Used in: The 45% Problem, Models (the M3 Bayesian macro prior on GDP, population, and climate variables), Methodology. The "approximately half of tournament variance is left to chance" framing the project's name builds on traces back to this paper's explanatory power decomposition.

Open scientific infrastructure

Center for Open Science. The Open Science Framework (OSF). DOI infrastructure for pre-registrations and time-stamped scholarly artifacts. https://osf.io.

Used in: Pre-registration. The project's sealed pre-registration carries DOI 10.17605/OSF.IO/8B5HD.

Mintz, J., and Hung, T. (2015). KaTeX: a fast, easy-to-use JavaScript math typesetting library. https://katex.org.

Used in: every Vault page that renders mathematical notation.

Citation conventions

In essay prose this project cites as "Author and Coauthor (Year)" for two-author works and "Author (Year)" for single-author works. Three authors are written out in full at first mention and abbreviated to "Author et al. (Year)" at subsequent mentions. References that share a canonical short name with a named procedure (Diebold-Mariano test, the Hall-Mueller bootstrap, the Newey-West kernel) use the hyphenated form when referring to the procedure and the author-year form when referring to the source paper.

For citations of this project as a whole, see Citation.

Where to go next

Citation: how to cite this project itself.
Notation: the symbol table that names every quantity referenced above.
Glossary: plain-language definitions of every term used in the essays.
Methodology: the long-form methodology that puts the citations above into a single argument.