Why the Poisson model underrates draws in football
Published
The obvious way to model football scores is also slightly wrong, and it is wrong in one specific place. The Poisson model underrates draws, because it assumes the two teams score independently of each other — and in the low-scoring corner of the score grid, where football spends more than a third of its time, that assumption is hardest to defend. The fix has been known since 1997, and understanding why it was needed is the fastest route to understanding what a goal model can and cannot do.
Start with what plain Poisson does
The Poisson approach is elegant. Give every team an attack rating and a defence rating, add a home-advantage term, and you get an expected goal count for each side in a given fixture. Feed each expected count into a Poisson distribution, multiply the two together, and you have a probability for every scoreline: 0-0, 1-0, 3-2, all of them. Every market you could want — match result, over/under 2.5, both teams to score, correct score — is then just a matter of adding up the right cells of that grid.
The elegance comes from one assumption: that the two teams’ goal counts are independent. Whatever the home side does has no bearing on the away side. That assumption is what lets you multiply.
It is also false, and it fails in a particular place.
Why the Poisson model underrates draws
The established result, from Dixon and Coles’ 1997 paper onward, is that an independent Poisson misprices the four lowest scorelines — leaning against 0-0 and 1-1 relative to the 1-0 and 0-1 results beside them. The reasons are the ones any supporter would give without a model: teams that concede first change how they play, teams protecting a narrow lead drop deeper, and a goalless hour makes both sides more cautious rather than less. Goals are not independent events; they are events in a match where the score is known to the participants.
Dixon and Coles’ answer was deliberately modest. Rather than rebuild the model, they added a correction term — usually written τ — that applies to exactly four scorelines: 0-0, 1-0, 0-1 and 1-1. It nudges probability towards the two draws and away from the two narrow wins, by an amount fitted from the data rather than assumed. The mechanics of that correction, with a worked example, are set out in the Dixon-Coles piece; what follows here is how much territory it covers, and how large the bias inside that territory actually looks.
The region is not a rounding detail
It is easy to read “a small correction to four scorelines” as a technicality. The size of the territory says otherwise. Across 17,495 finished matches in the ten competitions we model, 36.2% ended in one of those four results:
- 1-1 — 12.1% of all matches
- 1-0 — 10.1%
- 0-1 — 7.5%
- 0-0 — 6.5%
More than one match in three lands in the exact region where the independence assumption is under most strain. And the size of that region varies enormously by competition, from 28.2% in the Bundesliga — the highest-scoring league we cover — to 43.0% in Brazil’s Série A, where more than two matches in five finish inside it.
That spread is the practical argument for fitting the model per league rather than globally. A single correction estimated across all ten would be too weak for Brazil and too strong for Germany, and the error would land squarely on the draw probability, which is the hardest outcome in football to price.
How big is the bias, actually?
Here is where we have to be careful, because it is easy to overstate this and most articles do.
Measured with each league’s own average scoring rates, an independent Poisson reproduces those four scorelines to within 0.4 percentage points: 1-1 comes out at 11.87% against an actual 12.07%, 1-0 at 9.91% against 10.08%, 0-1 at 7.88% against 7.49%, and 0-0 at 6.62% against 6.53%. Two of the four sit slightly above the prediction and two slightly below.
That is not the crisp, uniform bias the folklore describes — but nor does it refute the correction, and the reason is worth understanding. A test run on league averages asks whether the Poisson shape fits a whole competition. Dixon and Coles’ correction operates on a single fixture, where the expected goals are specific to those two teams. Pooling thousands of very different fixtures blends them together, and the blending can mask an effect that is present in every individual match.
So the honest position is the one we take in our Poisson explainer: the aggregate shape is a good fit, the per-match assumption is still the weak point, and the only way to settle whether a correction earns its place is to score it out of sample rather than to argue about it.
The second flaw: stale ratings
The 1997 paper carried a second contribution that gets less attention and matters just as much. A team’s rating should reflect what it is now, not the average of what it has been over several seasons. Dixon and Coles weighted each historical match by an exponential decay function of its age, so a result from last month carries more influence on the fit than one from two years ago, with the decay rate itself estimated rather than guessed.
Without it, a club that was excellent two seasons ago and mediocre since keeps a rating it no longer deserves. With it, ratings track form on a timescale the data chooses. Our fit uses this same time decay, which is why our ratings move through a season rather than sitting still.
What the correction doesn’t fix
A corrected Poisson model is still a model of goals given team strength. It is not a model of football, and the honest limits are worth stating plainly:
- It knows nothing about lineups. Injuries, suspensions and rotation are invisible to it — the model sees a team name and a rating, and a first-choice XI and a heavily rotated one look identical.
- It has no memory of context. Cup runs, derbies, fixture congestion, a manager sacked on Tuesday: none of it enters.
- Promoted teams are its weakest point. A newly promoted side has no top-flight history to fit against, and what the model does instead is closer to an assumption than an estimate.
- The correction is calibrated, not magic. It repairs a known bias in low-scoring results. It does not make the model right about any individual match.
Why not replace the whole thing with “AI”?
The obvious question is why anyone would still patch a goal model rather than train a gradient-boosted ensemble that could learn the draw bias on its own. The longer answer is here; the short one is that a goal model gives you coherence for free. Because every market is read off one scoreline grid, our 1X2, over/under and both-teams-to-score probabilities cannot contradict each other. A classifier trained separately per market has no such guarantee, and sites that predict each market independently give themselves away constantly.
The genuine tradeoff is flexibility: a booster will absorb any feature you can put in a column, where a goal model has to be re-derived to accept each one. That is a real advantage, and it is where our model will eventually go. It is not, however, what the word “AI” is doing on most prediction sites.
Where to check our work
Our model version, its data sources, its backtest against the closing-odds market and its known failures are all set out on the methodology page, and every prediction it makes is frozen before kick-off and graded — hits and misses alike — on the public track record. A model this old is not a secret, which is rather the point: you can read the original paper, and then read whether ours actually works. As always, treat any prediction as a probability rather than a plan, and stake only what you can afford to lose.