The direct model

Plackett-Luce. Every driver gets a strength, a linear function of their features: grid position and season pace history. The probability of winning is the softmax over those strengths. Remove the winner, renormalise over who is left, and draw again for second, and so on down to twentieth.

That structure is the reason the output is coherent without any patching. A race is a ranking, so the win probabilities sum to one, and exactly one driver occupies each position. The weights are fit by maximum likelihood on the finishing orders actually observed, in plain numpy and scipy. Ordinal regression was the fallback. A boosted tree was rejected: it would overfit two hundred and forty outcome rows and nobody could read the result.

The simulator

Ten thousand runs of the race, one lap at a time. Each run draws a form deviation per driver, then plays the race out: tyre degradation, fuel burn, a pit strategy chosen by search, safety cars, virtual safety cars, red flags and retirements. Counting how often each driver finishes in each position gives the same matrix the direct model emits, from a completely different direction.

The two are not variations on one model. They fail differently, which is the point of running both.

Pace, per driver, never per team

Each driver carries their own pace term and their own degradation slope per dry compound; intermediates and wets are pooled because there are too few laps to do otherwise. There is no team term. Once every driver has a parameter a team term is collinear with them and buys nothing, and worse, it erases the thing the strategy search consumes: teammates diverge most in degradation, because setup and driving style are theirs alone.

Those parameters are fit on lap rows, roughly eleven thousand of them, not on race outcomes. Per-driver parameters are cheap at that scale.

The cleaning matters more than the model

Raw lap times are contaminated. In-laps, out-laps, the first two racing laps and every lap under a non-green track status come out before a single parameter is fit, and what is left is corrected for fuel load, compound and tyre age. A pace model fit on uncleaned laps is measuring traffic and safety cars, not cars.

Strategy is a search, not a rule

Every stint plan is chosen by enumerating one, two and three-stop plans over candidate pit laps and taking the minimum total race time. A threshold rule like "pit when the tyre is this old" is what produces a hundred stops in a race, because it never compares the roughly twenty-three second cost of a green-flag stop against what the stop is supposed to gain.

The plan is re-solved whenever race control changes the sums. Under a safety car the pit loss falls to about twelve seconds, and that flips the answer, which is why real races reshuffle behind one. Each car solves separately, or all twenty stop on the same lap and the Monte Carlo loses the variance that is the entire output.

Events are sampled, not assumed

Safety car rates come from each circuit's own 2018 to 2026 history, shrunk toward a street or permanent base rate so that one dramatic year does not dominate. Occurrence, lap and duration are all sampled, and virtual safety cars are modelled separately because they cost a different amount of time. Red flags get one pooled rate across all circuits: sixteen in ninety-eight races is not enough to estimate per circuit, and pretending otherwise would be fitting noise. A red flag gives a free tyre change, closes the gaps to zero and re-runs the first-lap incident draw at the standing restart.

Retirements are split by who owns them. A per-team per-lap hazard covers mechanical and mid-race collision, since 2026 timing data reports both as "Retired". First-lap incidents are pooled by circuit only: the rate turned out flat across grid quartiles, between 2.4 and 3.0 percent, so a grid-position term would have been invented rather than measured.

The one honest compromise

Cars do not interact. There is no dirty air, no DRS, no blocking. In place of an overtaking model there is a single calibration term: each grid slot costs a car a fixed amount of lap time over the race, re-issued by running order whenever a safety car bunches the field. Its job is narrow, to cancel the strategy search's optimism about undercuts, and it is marked in the code with its own ceiling. A rank blend was tried first and thrown out: it capped a fast car at its starting rank, which is the opposite of what a race does.

The backtest is nine races. Two constants in the simulator, the track-position term and the pooled form scatter, were tuned on those same nine, so the backtest number flatters itself. This is why the live season is the real evidence and the backtest is only a smoke test.

Monaco is the race the simulator loses, which is exactly what a model with no overtaking should do at the circuit where overtaking does not happen.

Setup is unobservable. No wing level, differential, brake bias or ride height data exists publicly, and the effect is real. That is irreducible noise, so it widens the intervals rather than adding a parameter that would fit it spuriously.

And there is no qualifying model yet, no bookmaker baseline yet, and no pairwise overtaking model. Each is a known gap with a known cost, not something the page is quiet about.