Research Publications
Independent mathematical research, closed-form derivations, and empirical probability studies for sports wagering efficiency.
No-Vig Fair Odds & Bookmaker Overround Decomposition
Mathematical methods for stripping bookmaker margins from odds: Multiplicative, Additive, Power and Shin. Each produces probability estimates under its own margin-allocation assumptions.
P_fair = (1 / O_i) / Σ(1 / O_k) - • Exchange Commission vs Bookmaker Margin: Compare the Final Odds 8 min →
- • How Bookmakers Set Odds: The 5-Stage Compilation Pipeline & Market Microstructure 16 min →
- • The Favourite-Longshot Bias: Empirical Evidence Across 2,490 Matches & Mathematical Mechanics 19 min →
- • Multiplicative vs. Shin Devigging Methods: Mathematical Derivation & 2,490-Match EPL Benchmark 22 min →
- • One-Way Market Devig: Extracting Fair Odds from Props and Moneyline-Only Markets 22 min →
Expected Value, Overround & Model Assumptions
Formal derivation of EV = (P_win × (Odds - 1)) - (P_loss × 1). Distinguishes quoted overround from model expected loss and explains why the probability estimate determines selection EV.
EV = (P_win × (Odds - 1)) - (P_loss × 1) - • Closing Line Value (CLV) Explained: Why Beating the Close Predicts Long-Term Sports Betting ROI 18 min →
- • Value Bet Identification: The 6-Step Quantitative Workflow 15 min →
- • Parlay Margin Compounding: The Mathematics of Multi-Leg Accumulators 18 min →
- • EV, Variance, and Sample Size: How Many Bets to Prove Real Skill? 22 min →
- • Cash Out vs. Manual Hedging: Model Prices and Settlement States 22 min →
Kelly Criterion Staking: Full, Half & Quarter Kelly with Risk of Ruin Analysis
Optimal geometric growth bet sizing derived from Kelly's 1956 paper. Covers fractional Kelly variants, tilt protection, drawdown limits, and risk of ruin simulation.
f* = (b × p - q) / b - • Fractional Kelly Criterion: Full vs Half vs Quarter Kelly Staking Compared 17 min →
- • Simultaneous Kelly Portfolio Optimization: Optimal Allocation for Concurrent Bets 22 min →
- • Risk of Ruin in Sports Betting: Monte Carlo Simulation and Bankroll Survival 18 min →
- • Kelly Criterion vs. Flat Staking: Which Bankroll Strategy Wins? 15 min →
Poisson Goal Expectancy Engine: 1X2, Over/Under & Correct Score Probabilities
Poisson distribution modeling for football and hockey outcomes. Calculates independent goal probabilities for 1X2, Over/Under 2.5, correct score matrices, and BTTS markets.
P(X=k) = (λ^k · e^-λ) / k! - • Deriving Over/Under Probabilities from Poisson Distribution: A Mathematical Guide 14 min →
- • Dixon-Coles Model Explained: Low-Score Correlation and Time-Decay Weighting in Football 22 min →
- • Attack & Defence Strength Ratings: Parameterizing Football Poisson Models 18 min →
- • Correct Score Matrix: Building the Full Probability Grid from Poisson Parameters 18 min →
All Mathematical Publications
Browse and search our full curriculum of open-source derivations, staking proofs, and match models.
Exchange Commission vs Bookmaker Margin: Compare the Final Odds
Convert exchange odds after commission, compare them with bookmaker quotes, and avoid confusing market-level charges with overround.
No-Vig Fair Odds Explained: The Mathematics of Bookmaker Margins
Discover the mathematical foundations of bookmaker margins (overround) and learn how to compare model-based fair odds using Multiplicative, Additive, Power, and Shin models.
Expected Value (+EV) Betting: The Mathematical Path to Profit
Master the core equation of profitable sports betting. Learn how to calculate Expected Value (EV), understand market efficiency, and quantify Closing Line Value (CLV).
Kelly Criterion Staking Guide: The Mathematics of Bankroll Growth
Mathematical framework for optimal bankroll compounding, risk management, and the growth-variance trade-off across Full, Half, and Quarter Kelly.
Poisson Goal Expectancy Model: Predictive Football Analytics
Mathematical formulation of bivariate Poisson goal models, Dixon-Coles corrections, xG integration, and derivative betting market pricing.
How Bookmakers Set Odds: The 5-Stage Compilation Pipeline & Market Microstructure
An investigative breakdown of how quantitative sportsbooks calculate probabilities, inject overround, manage dynamic risk, and exploit public betting bias.
The Favourite-Longshot Bias: Empirical Evidence Across 2,490 Matches & Mathematical Mechanics
Comprehensive empirical analysis of the favourite-longshot bias across 2,490 Premier League matches (source: football-data.co.uk), exploring behavioral drivers, the Shin devigging model, and quantitative exploitation strategies.
Multiplicative vs. Shin Devigging Methods: Mathematical Derivation & 2,490-Match EPL Benchmark
A rigorous mathematical and econometric comparison of Multiplicative, Additive, Power, and Shin devigging methods across 2,490 Premier League matches (football-data.co.uk), analyzing calibration and Kelly staking impact.
Closing Line Value (CLV) Explained: Why Beating the Close Predicts Long-Term Sports Betting ROI
A definitive empirical and econometric guide to Closing Line Value (CLV), the Efficient Market Hypothesis in sports betting, statistical significance tests, and empirical evaluation across 2,490 Premier League matches.
Fractional Kelly Criterion: Full vs Half vs Quarter Kelly Staking Compared
A definitive mathematical and stochastic comparison of Full, Half, and Quarter Kelly staking. Explores volatility drag, drawdown distributions, parameter estimation errors, and 1,000-trial Monte Carlo evidence.
Deriving Over/Under Probabilities from Poisson Distribution: A Mathematical Guide
A formal mathematical derivation of Over/Under goal probabilities using the Poisson convolution theorem, cumulative distribution functions, 2D score matrices, and Asian totals.
Value Bet Identification: The 6-Step Quantitative Workflow
An institutional step-by-step operating procedure for discovering, validating, and executing +EV wagers across global sportsbooks using sharp benchmarks and fractional Kelly sizing.
Parlay Margin Compounding: The Mathematics of Multi-Leg Accumulators
A formal mathematical proof of exponential margin compounding in sports betting parlays and accumulators, Same Game Parlay (SGP) copula pricing, and rare +EV exceptions.
Dixon-Coles Model Explained: Low-Score Correlation and Time-Decay Weighting in Football
A comprehensive mathematical breakdown of the Dixon-Coles bivariate Poisson adjustment, low-score tau correction matrix, exponential time-decay parameter estimation, and football betting market efficiencies.
Simultaneous Kelly Portfolio Optimization: Optimal Allocation for Concurrent Bets
A mathematical treatise on multivariate Kelly criterion staking for concurrent sports bets, covariance matrix estimation, Karush-Kuhn-Tucker optimization, and risk management heuristics.
One-Way Market Devig: Extracting Fair Odds from Props and Moneyline-Only Markets
One-way prices do not identify probability and margin. Compare conditional proxy and synthetic models, event definitions, valid domains and numerical checks.
Risk of Ruin in Sports Betting: Monte Carlo Simulation and Bankroll Survival
A quantitative study on Risk of Ruin (RoR), gambler ruin analytical derivations, continuous diffusion approximations, and 10,000-path Monte Carlo bankroll survival models.
Kelly Criterion vs. Flat Staking: Which Bankroll Strategy Wins?
A definitive mathematical and empirical comparison between proportional Kelly staking and fixed flat betting, with a 10,000-path simulation, odds-band returns from 2,490 Premier League matches (football-data.co.uk), drawdown profiles, and fractional Kelly rules.
EV, Variance, and Sample Size: How Many Bets to Prove Real Skill?
A rigorous mathematical guide to sports betting sample size requirements, single-bet variance formulas, Central Limit Theorem confidence intervals, and Brier calibration.
Attack & Defence Strength Ratings: Parameterizing Football Poisson Models
A step-by-step mathematical guide to calculating attack and defence ratings from empirical league standings, home advantage factors, and time-decay weighting.
Correct Score Matrix: Building the Full Probability Grid from Poisson Parameters
A mathematical guide to constructing 7x7 correct score probability surfaces, Dixon-Coles correlation adjustments, and deriving 1X2, Over/Under, and BTTS fair odds.
Cash Out vs. Manual Hedging: Model Prices and Settlement States
An illustrative two-layer cashout pricing model, hedge calculations across settlement states, and comparison of linear expectation with wealth utility. Model inputs are not measured operator fees.
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