Sport is a maths problem.
Every match, race and rally reduced to the equations that decide it. Forty models fitted on fifty thousand real results, written up as specifications you can read, taught as courses you can take, set out in five books, and rendered as films you can watch. Licence the models, learn to build them, or just enjoy the mathematics of the games you love.
The mathematics, three ways in
A book to read, a course to take, a model to study. Each one drawn from the same fitted models that run the rest of the site, so nothing here is a toy version of the real thing.
Where do you want to go?
Learn the models in the Academy, read them in the Library, study their specifications, watch the films, turn the dials in the Lab — and then, if you like, price a match and see how the market disagrees.
Academy
Six courses that teach the maths on this site, with figures drawn from the live models, worked examples, quizzes that explain themselves, and a certificate that verifies.
Start learning →Library
Five books, the validation report, model cards and league data sheets as PDFs, the certifications, and the model licences. The first chapter of everything is free.
Browse →Quant Library
Nineteen models written up properly: what goes in, what it does, what comes out, what it was measured at, and where it stops being trustworthy. Free to read, licensed to use.
Read the specs →Films
Short films of the equations, six of them rendered in 3D from the physics, three studio pieces of boot and ball in slow motion, and flight explorers with wind and altitude on the sliders.
Watch →The equations
Forty equations, each with a picture that moves. Fitted, physics, literature or solved: the evidence grade is on the card.
See them move →Lab
Sixteen live models with sliders. Move a parameter and watch the optimal strategy move with it.
Open the lab →Predict
Pick two teams. Every market, the model's fair prices, the confidence band and the words that explain it.
Price a match →Live
This week's fixtures, the tables, the tennis draws and the NBA slate, each with the model's number beside the market's.
See the week →Play
Six games where the model is the opponent. Manage a real match, take ten penalties against a keeper who learns, pace an 800 m.
Enter the arena →Factors
Weather at the ground, altitude, the crowd, the journey, gear and fuelling. Each one graded by its evidence, each one a toggle on the model.
Stack them → GearWhat the kit is worth
Shoes, spikes, rackets, strings, suits and balls, brand by brand, measured by the papers, modelled by the physics, expressed as a change in the chance of winning.
Compare the kit →Research
One event modelled every morning as a short paper, and the news desk that scores the week's stories against what the models say.
Read today's →Paper desk
The model's price beside the market's on every fixture, a paper bankroll to test any rule you like, and a walk-forward that says plainly that not betting beats all of them.
Test a rule →The equations that decide it
Nothing here is a hunch. Each model is written down, fitted on results, and scored against the closing line. These are the six that do the most work.
Goals are Poisson arrivals
Fitted on every result since 2014, blended with shots on target because shots are a less noisy signal than goals. Taught in Match Pricing →
The chase is real
Second-half scoring by half-time score, after controlling for how good both sides are. A side one goal down scores faster; a side ahead barely slows. The Manager game runs on this.
When to press, when to park
Chase variance when you are behind, kill it when ahead, time the switch by the clock. Try to beat it in the Manager game.
A 2% edge becomes 25%
Morris importance says which points matter. Spend your big serves there. Break Point tests whether you do.
Pace is a store, not a feeling
Split Second lets you set your own splits against the optimal ghost and watch the store drain. Taught in The Athlete's Equations →
The rim is bigger from above
The same integrator bends the free kick round the wall and dips the kick serve inside the line. See it in 3D on Films. Taught in Physics of the Ball →
Measured, and reported either way
A model that is never scored is a story. Every one here is tested on results it did not see, and the report says where it wins and where it does not — including where the market forecasts better than the model.
One method, five arenas
Football
Dixon–Coles with time decay and shots, contextual effects, referee tendencies, state-space form, style clusters, game-state effects, the strategy solver, pitch control, penalties as a game, free-kick flight.
Tennis
Nested Markov chain, Morris point importance, live ratings from this season's tour results, serve flight with topspin.
Basketball
Expected points per shot location, possession value, endgame decision trees, Pythagorean expectation, the forgiving arc.
Athletics & Olympics
Keller optimal pacing, Banister fitness–fatigue and the taper, medal-table allocation by marginal return.