World Cup 2026 — Calibrated Monte Carlo Tournament Model

A from-scratch probabilistic model of the FIFA World Cup 2026, built to be calibrated rather than merely plausible. Every parameter is estimated by maximum likelihood from 49,547 international matches, every modelling choice is tested out-of-sample on the six World Cups from 2002 to 2022, and the results are reported with paired standard errors rather than cherry-picked wins.

Pure Python standard library. No numpy, no scipy, no pandas. The Poisson GLM, the Newton–Raphson solver, the Fisher-information standard errors and the likelihood-ratio tests are all implemented directly.

python3 fit_goal_model.py     # estimate the model from match history  (~2 min)
python3 backtest.py           # walk-forward validation, 2002-2022     (~3 min)
python3 main.py 100000        # 100k tournament simulations            (~80 s)
python3 make_figures.py       # regenerate the figures below           (instant)

Every figure in this README is generated by make_figures.py, which emits SVG directly from the standard library — the same no-dependency rule as the rest of the repository, applied to the plotting.

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The day-1 forecast, and what happened

Before a ball was kicked, the model ranked all 48 teams. Its top four were Spain, Argentina, France and England — and those turned out to be the four semifinalists. It priced that outcome, in advance, at 1 in 214.

Here is that forecast. It came from the Elo snapshot of 11 June 2026 and parameters fitted on matches ending 10 June 2026 — the day before the opening match. That cutoff is not a claim to be taken on trust, it is recorded in the artefact itself: data/goal_model.json carries "fit_window_end": "2026-06-10". Nothing below saw a single minute of the tournament. The forecast is preserved verbatim in data/day1_forecast.csv — sealed, seeded (12345), and reproducible with python3 main.py 100000 on a data/results.json with no scores entered.

The same numbers, with the rounds in between:

#	team	champion	final	semifinal	quarter	advance
1	Spain ✅	18.28%	28.35%	41.71%	55.17%	99.2%
2	Argentina ✅	13.38%	22.54%	34.69%	50.17%	96.5%
3	France ✅	9.68%	16.99%	30.33%	45.75%	93.3%
4	England ✅	6.38%	12.17%	22.18%	37.49%	95.6%
5	Brazil	4.99%	9.84%	19.45%	34.40%	93.9%
6	Portugal	4.60%	9.32%	17.29%	32.07%	90.2%
7	Colombia	4.47%	8.77%	16.37%	30.57%	89.4%
8	Netherlands	3.23%	6.85%	14.86%	29.28%	89.4%
9	Ecuador	3.06%	6.62%	14.53%	26.49%	94.3%
10	Germany	2.98%	6.46%	14.20%	26.10%	93.9%
13	Mexico (host)	2.25%	5.91%	14.98%	36.78%	95.3%
20	United States (host)	1.11%	2.87%	6.75%	16.82%	67.6%
24	Canada (host)	0.90%	2.93%	8.53%	27.57%	94.6%

✅ = reached the semifinals.

Forty-eight teams entered and forty-four of them did not reach the semifinals. Spain, the day-1 favourite, went on to beat France 2–0 in the first semifinal and reach the final.

What that forecast was actually worth

Less than it looks, and the model itself says so. Asked how often all four of its top-ranked teams reach the semifinals together, it answers:

outcome	model probability
0 of the top 4 reach the SF	18.10%
1 of 4	43.33%
2 of 4	30.59%
3 of 4	7.51%
4 of 4 — what happened	0.47% (1 in 214)

Two things follow, and both matter more than the headline:

• The four events are negatively dependent. Multiplying the marginals naively gives 0.974%, more than twice the true joint probability of 0.468%. These teams sit in the same bracket and can eliminate each other — France and Spain in fact met in the semifinal — so "all four survive" is strictly harder than independence implies. This is the kind of error a marginals-only model makes and a full bracket simulation does not.
• n = 1. A 1-in-214 event occurring once is not evidence that the model is well calibrated. It is one draw from a distribution, and the honest prior is that it was mostly luck. The evidence for this model is §3 below — 384 matches, proper scoring rules, paired standard errors — and that evidence is equivocal. Both results are in this README, at the same level of prominence, because reporting only the flattering one is how forecasting repositories mislead people.

The host numbers illustrate the model's structure nicely. Mexico ranks 5th on P(reach the quarter-finals) at 36.78% but only 13th on P(champion) at 2.25%: a 108-Elo home bonus is worth a great deal across three group matches and a Round of 32, and very little across seven matches against progressively stronger opposition.

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Why this repository might interest you

Joint estimation	Home advantage and the goal model are estimated together by fixed-point iteration, because the Elo replay that produces the covariate depends on the home-advantage parameter being estimated.
Proper scoring throughout	Ranked Probability Score (the standard for ordered football outcomes), multiclass Brier, log-loss — aggregate metrics only. The single-tournament result above is reported with the model's own price attached and labelled as the anecdote it is.
Strict walk-forward validation	Parameters are re-fitted on the 10 years ending the day before each tournament starts. Ratings are replayed match by match. No information from the future touches any prediction.
Paired inference	Models are compared on identical matches, so differences are reported as paired means ± SE with t statistics — removing the common match-randomness variance that swamps unpaired comparisons.
Honest effect sizes	The fitted model beats both baselines on every aggregate group-stage metric, and the paired difference is 1.1σ — not significant at n = 288. That is reported here as prominently as the win.
A live out-of-sample call	On day 1, before a ball was kicked, the model's four highest-ranked teams were Spain, Argentina, France and England. Those were the four semifinalists — an outcome it priced at 1 in 214. Reported above with that price attached, and with the n = 1 caveat it deserves.
Exact competition rules	The full 495-row FIFA Annex C third-place allocation table, validated on load against each match's permitted source groups; FIFA Article 13 tiebreakers with head-to-head mini-tables computed only among currently tied teams.

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1. The model

1.1 Ratings

Team strength is the eloratings.net Elo rating. The win expectancy for a rating difference $d$ is

$$E(d) = \frac{1}{1 + 10^{-d/400}}$$

Ratings update inside every simulated tournament using the site's own rule,

$$\Delta R = K \cdot G(|\text{GD}|) \cdot (W - E),\qquad K = 60 \text{ (WC finals)}$$

with margin multiplier $G = 1$ for a one-goal win, $1.5$ for two, $(11+N)/8$ for $N \ge 3$. A team on a deep run therefore carries its earned rating into later rounds, and a match decided on penalties counts as a draw for rating purposes — matching the official bookkeeping.

1.2 Scorelines

Elo gives an expectancy, not a scoreline, and group standings need goals. Each team's scoring rate is modelled as a log-linear function of the effective rating difference:

$$\log \lambda_A = \alpha + \beta, d_{\text{eff}}, \qquad \log \lambda_B = \alpha - \beta, d_{\text{eff}}, \qquad d_{\text{eff}} = R_A - R_B + H,v$$

where $v \in {-1, 0, +1}$ encodes who, if anyone, is at home. Scores are drawn from the resulting bivariate distribution with the Dixon–Coles correction $\tau$ applied to the four low-score cells ${0,0},{0,1},{1,0},{1,1}$, which repairs the well-known underdispersion of independent Poisson at 0–0 and 1–1. Draw probabilities then fall out of the model rather than being assumed.

Three alternative goal models remain selectable via --goal-model for ablation: regression (no Dixon–Coles), elo_total (empirical total $T(d) = 2e^{\alpha}\cosh(\beta d)$ split to match $E(d)$ exactly), and elo_fixed (fixed 2.70-goal total — the naive baseline).

1.3 Knockout rounds

90 minutes are sampled from the same distribution. A draw goes to 30 minutes of extra time sampled at one third the rates, then to penalties, modelled as

$$P(\text{A wins the shootout}) = 0.5 + w,(E(d) - 0.5), \qquad w = 0.25$$

Shootouts are mostly noise, and $w = 0.25$ encodes that explicitly instead of pretending either that they are coin flips or that they are won by the better team.

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2. Parameter estimation (fit_goal_model.py)

The covariate $d_{\text{eff}}$ is a pre-match rating difference, which must be reconstructed rather than looked up. The estimation pipeline:

1. Replay the entire Elo history — 49,547 matches, 1872 to today — under eloratings.net rules, with the K-factor keyed off the competition tier (60 World Cup, 50 continental, 40 qualifiers/Nations League, 20 friendlies), recovering every match's pre-match rating difference.
2. Restrict to the last 10 years of competitive matches (friendlies dropped): 6,860 matches, 2016-06-10 to 2026-06-10.
3. Fit the Poisson GLM by maximum likelihood — Newton–Raphson with an analytic Hessian, solved by Gaussian elimination with partial pivoting.
4. Resolve the circularity. The replay in step 1 needs $H$; the fit in step 3 produces $H = \gamma/\beta$. The two are iterated to a fixed point (converges to within 1 Elo point in a handful of iterations).
5. Test whether home advantage depends on the rating gap via a likelihood-ratio test on a $v \cdot |d|$ interaction, plus bucketed estimates with CIs.
6. Estimate the World Cup host effect separately across all host matches.
7. Fit the Dixon–Coles $\rho$ by profile likelihood (ternary search).
8. Rescale to the official Elo scale. The replayed ratings drift from the published ones; regressing replayed on official across the 48 finalists gives a scale factor of $0.874$, applied to $\beta$ and $H$ before they are written out.

Standard errors come from the Fisher information, with the delta method for the ratio $H = \gamma/\beta$.

Fitted parameters (data/goal_model.json)

Parameter	Estimate	Interpretation
$\alpha$	0.1740	log baseline scoring rate — 1.19 goals/team for equals
$\beta$	$1.656 \times 10^{-3}$	rate response per Elo point
$\rho$	−0.0344	Dixon–Coles low-score dependence
$H$	107.6 ± 5.2 Elo	generic home advantage
$H_{\text{host}}$	186 ± 34 Elo	World Cup hosts specifically (1930–2022, era-pooled)
$n$	6,860	competitive matches in the fitting window

Two findings worth stating plainly:

• Home advantage does not vary with the rating gap. The interaction test gives $\text{LR} = 1.69$ against a $\chi^2_1$ critical value of $3.84$ — not significant. A constant $H$ is the right specification, which is not obvious a priori and is usually assumed rather than tested.
• Home advantage is declining. Re-fitting on each pre-tournament window gives $H = 119, 120, 126, 117, 111, 98$ Elo for 2002 → 2022. The classic eloratings.net value of 100 sits comfortably inside today's confidence interval; the 1990s value does not.

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3. Validation (backtest.py)

Six World Cups, 288 group matches and 96 knockout matches, scored under strict walk-forward discipline: for each tournament, parameters are re-estimated on the ten years of competitive football ending the day before the opening match, and ratings are replayed up to each match. Late-tournament predictions see tournament form; nothing sees the future.

Three models are compared on identical matches:

• ours — fitted regression + Dixon–Coles + fitted host advantage
• noH — identical, host advantage forced to zero (ablation)
• std — "standard Elo": raw $E(d)$, no fitted parameters, minimal draw model (baseline)

Group stage — 288 matches, 3-way W/D/L

model	RPS	Brier	log-loss
ours	0.2013	0.5788	0.9837
noH	0.2035	0.5832	0.9907
std	0.2037	0.5838	1.0017

Paired differences (negative favours the first model):

comparison	ΔRPS	SE	t	n
ours vs std	−0.00238	0.00209	−1.14	288
ours vs noH	−0.00211	0.00149	−1.41	288
noH vs std	−0.00028	0.00137	−0.20	288

The honest read: the fitted model wins on every metric, and none of it is statistically significant. At 288 matches the standard error on ΔRPS is the same order as the effect. The correct conclusion is "better, and consistent with noise" — not "beats the baseline". Reporting it the other way would be the easiest possible way to make this table look stronger, and would be wrong.

Where the improvement is most visible:

Draw calibration — independent Poisson systematically underpredicts draws, which is exactly the failure Dixon–Coles addresses:

model	predicted draw rate	observed
ours	24.3%	23.3%
noH	24.4%	23.3%
std	22.3%	23.3%

The full picture across the probability range — each match contributing one point per possible outcome, binned, with Wilson 95% intervals:

Neither model is meaningfully better calibrated. The n-weighted mean absolute calibration error is 0.0485 for the fitted model against 0.0484 for the baseline, and most bin deviations sit inside their intervals — which, at 864 points spread over ten bins, is what one should expect.

The difference between them is confidence, not reliability. The baseline spreads its forecasts markedly wider (sd 0.195 against 0.174) without being any more often right, and log-loss — which punishes confident errors hardest — is where that unearned confidence shows up: 1.0017 against 0.9837 overall, widening to 1.2571 against 1.1218 on the large-gap slice below. The histogram beneath the panel shows where forecasts actually live: only 3.8% of them exceed 0.70, which is why that slice is only 24 matches.

Large rating gaps (|d| ≥ 300, n = 24) — where a raw Elo expectancy is most overconfident and the fitted floor on the underdog's scoring rate matters most:

model	RPS	log-loss
ours	0.2485	1.1218
std	0.2558	1.2571

Knockout stage — 96 matches, P(advance)

model	Brier	log-loss
ours	0.1986	0.5810
noH	0.1987	0.5826
std	0.1976	0.5807

Here the fitted model is not better — the three are statistically indistinguishable (ours vs std: ΔBrier = +0.0011 ± 0.0066, t = +0.16). Two knockout rounds' worth of extra-time and shootout machinery buys nothing over a raw Elo expectancy on this sample. That is a genuine negative result and it is left in.

Per-match predictions for all 384 matches are written to output/backtest_matches.csv for independent scoring.

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4. Tournament simulation (main.py)

The 2026 format is new (48 teams, 12 groups of four, a Round of 32) and its third-place qualification rule is genuinely intricate. It is implemented exactly, not approximated:

• Group ranking follows FIFA Article 13: points → head-to-head points → h2h goal difference → h2h goals → overall GD → overall goals → ranking. The head-to-head mini-table is recomputed over only the currently tied teams, which is the subtle part — a three-way tie that resolves into a two-way tie must be re-evaluated among the remaining two.
• Third-place allocation uses the complete Annex C table: which eight of twelve third-placed teams qualify determines the bracket, and there are $\binom{12}{8} = 495$ cases. All 495 are present in data/annex_c.txt and each is validated on load — that the assignment is a permutation of the qualified groups, and that every pairing respects its match's permitted source groups.
• Host advantage is stage-dependent. Mexico and Canada are at home through the Round of 16; from the quarter-finals every match is on US soil, so only the USA retains the bonus. Toggle the whole mechanism with --no-home-advantage.
• Conditioning on real results. Scores entered in data/results.json are replayed identically in every simulation including their Elo effect, so all downstream probabilities are properly conditional. Once the group stage is complete, the knockout bracket is inferred (standings → Annex C → R32 → …) and the fixture list is written back automatically.

Throughput: ~1,250 tournaments/second single-core (103 matches each, CPython 3.12, Apple silicon) — 100,000 simulations in about 80 seconds of CPU time, with no compiled dependencies. Score distributions are cached at 1-Elo resolution and sampled via a marginal/conditional CDF pair, which reproduces the exact Dixon–Coles joint rather than an independent approximation.

Outputs (output/)

file	contents
team_stats.csv	per team: P(win group), P(advance), P(reach R16/QF/SF/Final), P(champion)
group_match_probs.csv	every group fixture: P(home win / draw / away win)
knockout_match_probs.csv	per bracket slot: P(pairing), 90-minute W/D/L, P(advance)
knockout_stats.csv	per team: expected knockout draws after 90' and shootouts per tournament
backtest_matches.csv	all 384 historical predictions, all three models

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5. Usage

python3 main.py                      # 100,000 simulations (default)
python3 main.py 10000                # fewer, for a quick look
python3 main.py --refetch-elo        # pull current ratings from eloratings.net first
python3 main.py --no-home-advantage  # neutral-ground counterfactual
python3 main.py --goal-model elo_fixed   # ablate to the naive fixed-total model

--refetch-elo scrapes eloratings.net with a mirror fallback, and on any failure keeps the existing data/teams.csv values and warns — a bad fetch never clobbers the data.

To condition on real results, edit data/results.json and rerun:

{ "group": "A", "home": "Mexico", "away": "South Africa", "score": [2, 0] }
{ "teams": ["Mexico", "Croatia"], "score": [1, 1], "penalties_winner": "Mexico" }

Knockout scores are recorded after extra time; set "extra_time": true when a match was decided in ET so the 90-minute outcome is correctly booked as a draw.

Layout

main.py                  tournament simulator, format rules, output
fit_goal_model.py        Elo replay, Poisson GLM, home-advantage estimation
backtest.py              walk-forward validation, 2002-2022
make_figures.py          SVG figure generation (stdlib only, no matplotlib)
data/teams.csv           48 finalists: group, draw position, Elo, host flag
data/annex_c.txt         all 495 FIFA Annex C third-place allocations
data/goal_model.json     fitted parameters + provenance
data/results.json        editable real results (conditioning input)
data/day1_forecast.csv   sealed pre-tournament forecast (seed 12345)
output/                  generated CSVs
docs/                    generated figures

Set RNG_SEED at the top of main.py to an integer for reproducible runs.

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6. Known limitations

Stated explicitly, because a model whose limitations are unstated should not be trusted:

• Elo is the only strength signal. No squad quality, no injuries, no player-level data, no market prices. A single rating per team cannot know that a key striker is out.
• Teams are stationary within a tournament apart from Elo updates. No fatigue, no rotation ahead of a dead rubber, no motivation effects in settled groups.
• The host advantage estimate is era-pooled. $H_{\text{host}} = 186 \pm 34$ Elo spans 1930–2022, an era over which generic home advantage measurably declined. The simulator therefore uses the contemporary generic $H \approx 108$ by default, not the larger historical host figure.
• Backtest power is limited. 288 group and 96 knockout matches. The group-stage improvement is real in sign and insignificant in size; the knockout result is a wash. Six tournaments cannot settle this.
• The 2026 format has never been played. The Annex C logic is validated against the regulations, not against observed outcomes.
• Fair-play tiebreakers are not modelled (disciplinary records are not forecastable); the model falls through to rating, then to a random draw of lots.

Data sources

• Elo ratings — eloratings.net, snapshot 11 June 2026
• Match history — martj42/international_results, 49,547 matches, 1872–2026
• Groups & bracket — FIFA World Cup 2026 regulations; Final Draw of 5 December 2025 plus March 2026 playoff winners