Monte Carlo
The Justice Table
Keep every match of 2025/26 exactly as it was played — same chances, same expected goals — and let only the finishing fall differently, 10,000 times per league. Bottom-placed Pisa go down in only 38% of those alternate histories.
Five leagues, resimulated
Each row below is a team's position-probability strip: how often they finish in each place across 10,000 xG resimulations of the season, ranked by expected points. The outlined cell is where they actually finished — the further it sits from the dark cells, the stranger the universe we live in. Sort by luck to find the season's frauds and victims.
The justice table · Premier League 2025/26
P(finishing position) across 10,000 xG resimulations · outlined cell = actual finish · luck = actual points − xPts · Title/CL/EL/Rel = share of universes finishing 1st / top 4 / 5th–7th / bottom 3 · click a column head to sort, click a team for their universes
| # | Team | P(finishing position) · 1st → 20th | Act | Δ | Pts | xPts | Luck | Title | CL | EL | Rel |
|---|
A first honest surprise: in all five leagues, the actual champion also tops the justice table — Arsenal win 63.9% of Premier League resimulations, Barcelona 60.0% of La Liga's, Bayern 87.5%, Inter 68.9%, PSG 71.8%. Justice was served at the top. The injustice lives further down.
One caution reading the Luck column at the very top: it is not all luck. Across twelve seasons of all five leagues, top-three clubs beat their xPts by about +8.6 points a year — and roughly five of those points survive when you sort clubs by the previous season instead, so they cannot be chance. Elite sides carry a real, repeatable edge in finishing and goalkeeping that pure expected goals cannot see. So a strong club sitting a few points above its xPts is over-performing partly on skill, not only on fortune; the further down the table you go, the more the gap is genuine variance.
Sunderland finished 7th on 54 points — but earned just 43.1 xPts, which ranks them 18th in the justice table and sends them down in 29.4% of universes1. Levante suffered the mirror-image fate: relegated in 18th, they survive in 89.5% of resimulations. And Osasuna climb seven places, from 16th to 9th, the moment you rank the season by chances rather than outcomes2.
1 Relegation = bottom three in every league; for the Bundesliga and Ligue 1 this counts the relegation-playoff spot as down. 2 Tiebreakers are Pts → GD → GF in all leagues; Serie A and La Liga head-to-head rules are not applied, and residual ties are broken at random. 3 Ligue 1 has 305 of its 306 fixtures in the source data.
Every universe for one team
Pick a team (or click one in the table above) to see all 10,000 of their alternate seasons at once. Pisa's real total of 18 points sits at the 0.01st percentile of their own distribution — in the median universe they collect 40 points and finish around 16th. Not one alternate season in a hundred treats them this badly.
10,000 alternate seasons: —
share of simulated 2025/26 seasons ending on each final points total
- simulated seasons
- actual points
- xPts (expected points)
The title race, as a movie
The justice table is clairvoyant — it knows every chance the season produced. This chart is honest about time: before each Premier League matchweek, a strength model is refit using only matches played before that week, then the rest of the season is simulated 3,000 times. Press play to watch what the model believed, week by week, as 2025/26 unfolded.
Chance of winning the league, week by week
P(outcome), forecast before each matchweek · Premier League 2025/26 · 3,000 sims per matchweek
The movie's plot: Liverpool entered the season as favourites at 43%, peaked at 67% before week 6 — and never lifted the trophy. The model had cut them below 2% by week 19, and to zero from week 30; they finished 5th. Arsenal crossed 50% for good before week 9. Manchester City clung on to 37% at their week-17 peak and still held 19.6% entering week 37 — the season's single biggest one-week swing was their −20 points-percentage collapse into the final week.
Switch tabs for the quieter dramas: fourth place went to the wire (Aston Villa 62% vs Liverpool 38% on the final morning — Villa took it), and Tottenham reached a 54% relegation probability before week 35, then stayed up in 17th.
How the universes are built
The core idea comes from Maher (1982) and Dixon & Coles (1997): football goals behave like Poisson counts. The modern twist is to centre each team's Poisson rate on the expected goals they actually created in that specific match, rather than on season-long averages — so every alternate universe replays the same chances, and only the finishing changes. Independent Poissons underrate low-scoring draws, so a Dixon–Coles correction τ reshuffles probability among the 0-0, 1-0, 0-1 and 1-1 scorelines, with ρ = −0.083 fitted by maximum likelihood on 19,837 matches, 2014–24.
One trap avoided: per-match win/draw/loss probabilities and xPts are computed exactly on a score grid, not sampled — simulating a single match 10,000 times gives unstable two-decimal probabilities (the KU Leuven "Monte Carlo trap"). Monte Carlo is reserved for the season level, where the joint distribution of 380 matches plus ranking tiebreakers is genuinely intractable.
What the model got right — and wrong
A forecast is only as honest as its scoring rule. On the ordered win/draw/loss outcome, the weekly-refit model scores a Ranked Probability Score of 0.2073 against 0.2277 for a baseline that predicts the same three probabilities for every match — the league's long-run home-win / draw / away-win shares, ignoring who is playing — a 9.0% improvement. That gap is real, and it is also humblingly small: most of a football season is noise no model is entitled to.
Right: it called the collapse early — Liverpool were under 2% for the title by week 20, months before the table made it obvious. It had West Ham at 91% to go down by week 22; they went down. And its retrospective verdict matched reality at the top of all five leagues: every actual champion also tops their justice table.
Wrong — or at least uncomfortable: Tottenham hit a 55% relegation probability at week 35 and survived. The promoted-team prior gave Sunderland a 54% chance of going straight back down; they finished 7th. But was that the model being wrong, or Sunderland being lucky? The justice engine votes lucky: their 54 points sit at the 94th percentile of their own 10,000 alternate seasons, and they are relegated in 29.4% of them. A single probability can't be wrong; a pattern of them can. The honest summary is the RPS gap above — 9.0% better than base rates, no more, no less.