A better mathematical engine for sports ratings than softmax

Most ranking systems you've encountered — Elo, softmax-based logit models, the probability lines behind a betting market — share a convenient, load-bearing assumption: that once you know each competitor's rating, their performance on the day is independent of everyone else's. It's a simplification that has stuck around less because it's realistic and more because the honest alternative was computationally out of reach. This paper goes after that computational barrier directly.

The thirty-year-old bottleneck

Multinomial probit choice probabilities — the probability that a given competitor "wins" once you let performances be draws from correlated Gaussian noise around each one's underlying strength — are Gaussian orthant integrals. Computing just one of these has, for three decades, required simulation: one expensive integral per alternative. Going the other direction — inversion, meaning starting from a prescribed vector of win probabilities (the odds a market or a ranking system implies) and solving backward for the underlying strength parameters that would reproduce them — is even harder, and has been considered impractical for correlated contests once the number of competitors gets large.

The paper's answer scales linearly for the correlation structures that actually show up in practice — factor, block, and hierarchical structures, which cover most of the ways real-world correlation arises: teammates sharing a system, competitors sharing a venue or weather, a league-wide shock hitting everyone at once. The method is tested at up to one million alternatives, reproducing prescribed probabilities to high accuracy even in the extreme tail of the distribution.

How much faster is "linear instead of quadratic-ish"

To make any comparison possible at all against the existing standard — a simulation-based approach called the Geweke-Hajivassiliou-Keane simulator — the author has to shrink the problem back down to sizes the old method can still handle, because performance testing the incumbent at a million alternatives simply isn't feasible. Even at a modest 200 alternatives, GHK is already measured as roughly 200 times slower, and its cost grows at close to a rate of n to the power of 2.8, compounding badly as the contest grows. The new method's cost grows linearly. That gap is precisely why correlated-performance models have historically stayed confined to small, special-case problems instead of becoming a practical default.

At only 200 alternatives, the traditional simulator is already two hundred times slower — and its cost keeps compounding from there.

Why correlation, not just speed, is the point

Speed unlocks something conceptually bigger: it makes the Thurstone-Mosteller family of models — a richer, more general class of choice models that has always allowed correlated performances in theory — a practical alternative to logit-style models at modern scale, rather than a textbook curiosity. That matters anywhere competitors don't perform in isolation: teammates who share tactics, drivers who share machinery, horses who share course conditions, or tournament brackets where results share structure. A model honest about those shared factors can, in principle, separate genuine individual strength from the correlated noise everyone in a group experiences together — something an independence-assuming model cannot do even in principle, no matter how much data you feed it.

Honest caveats

This is a computational and methodological contribution, not an empirical sports-forecasting study — the million-alternative test demonstrates that the calibration reproduces prescribed probabilities accurately and quickly, not that any particular real-world ranking system built on top of it would outperform existing sports models on genuine predictive accuracy. Whether the gains from modelling correlation translate into materially better forecasts in any specific sport is a separate empirical question the paper doesn't set out to answer, and would need its own dedicated study against real outcomes.

Why it matters

Sports ranking and betting-market inversion are the most intuitive use case, but the reach is broader: any large choice or recommendation system — where a user picks one item from many, and items plausibly share unobserved factors — faces exactly this same inversion problem. Making million-item multinomial probit calibration computationally practical is the kind of infrastructure change that doesn't show up as a headline result on its own, but quietly reopens a whole class of richer models that had been shelved for being too slow to use.