haywire

What the World Cup Did to the Robot

Sat, 08 Aug 2026 09:00:00 GMT

Updated 2026-08-08

The Reveal

Spain won the World Cup. 1-0 against Argentina, which happens to be the exact scoreline the robot had settled on the evening before the final: Spain 1-0, an 18% shot at exact and 48/29/23 on the 90-minute outcome. The final is also the one knockout round where the joker requires no thought whatsoever, there’s only a single fixture to put it on. Priced by Part 1’s expected-points formula, that pick was worth $2 \cdot 0.18 + 0.48 = 0.84$ points, doubled to 1.68 by the joker. The robot, in other words, ended its tournament with a 6-point play on the table.

I collected 3 of them.

Here the cruel poetic irony that closed Part 1¹ came to collect its interest. My champion pick had to be locked in by the knockout stages, and at lock time the robot’s favourite was Argentina, so Argentina it was. The robot then spent the following fortnight quietly defecting to Spain, and by the 8th of July it had the pair of them in the final with Spain on top. The scoreline I could still change right up to kickoff, so Spain 1-0 went in and banked its 3 points for exact. The joker I could not: it was locked to Argentina, because that’s who I’d predicted, and no amount of the robot’s revised opinion could move it. Three points banked, three forfeited to a lock I’d have undone a fortnight earlier if the league had let me. The robot was right, which is somehow worse.

Two honest sentences before this reads like a highlight reel. On the eve of the final the model gave Spain 64% and Argentina 36% across 100,000 simulations, so even at its most certain it reserved a one-in-three chance of being wrong. A model like this should be judged on the whole season of numbers it produced, not on whether one 64/36 coin landed the right way up.

Part 1 also left an aside dangling: the model was carrying two different finals, the most frequent pairing across all simulations (Spain v England) and the match-by-match modal path (Spain v Argentina). By the 8th of July the two had re-converged on Spain v Argentina, and reality played the path final. Most common destination and most likely route are different questions, but this time they agreed on the answer.

Below is the bracket exactly as the model predicted it on the 2nd of July, before the round of 16 had been played:

And the same bracket after the final, every tie now a result:

The diff between those two pictures is the story of the knockout rounds. The final pairing was called seventeen days early, as were three of the four quarter-final winners. The hole in the script is Brazil-shaped: predicted to reach the semi-finals, removed in the round of 16 by Norway, 1-2. And the champion call was a 54/46 in Argentina’s favour that went the other way. Reality kept nearly the whole script and rewrote the last line.

That’s the result dealt with. The more interesting story is how the odds got there.

The Race

The chart below is the tournament as the model lived it, one run per day. Every kink is real results being re-fitted and re-simulated; the flat stretches are rest days, because a deterministic model asked the same question twice gives the same answer.

Follow Argentina’s line first. Around 14% on the 13th of June, then three group wins (3-0 Algeria, 2-0 Austria, 3-1 Jordan) while the other favourites wobbled, peaking at 27.4% on the 27th of June. That surge is precisely what talked me into the champion lock, and you can watch it fade as knockout reality priced itself back in.

Germany’s line is the cruellest thing the model produced all summer. They won Group E, at one point 7-1 against Curaçao, and hovered around 5% the whole group stage. Then 0.0%, overnight, on the 30th of June: 1-1 with Paraguay in the round of 32, out on penalties. The line doesn’t dip, it stops. Part 1 promised that the 50/50 shootout coin would matter, and here it is mattering.

The Netherlands are the same story on the neighbouring line. Won Group F, drew 1-1 with Morocco, lost the coin. Two group winners, zero knockout wins, both erased by the one number in the entire model that refuses to hold an opinion. Morocco’s reward is the quiet climb to 7.7%, the best outsider odds on the board, which is what knocking out the Netherlands buys you.

France are the opposite shape. A perfect group (3-1 Senegal, 3-0 Iraq, 4-1 Norway) earned them almost nothing, stuck near 7% because their half of the draw was jammed with rivals. Then the bracket opened, and the line climbed to 11.5% on the 30th of June and 17.7% by the 5th of July without France doing anything differently. The draw is a rating too.

And Spain were never the loudest line on the chart, 15.7% at the parameter lock, around 20% in early July, 32% by the 8th, 64% on the eve of the final, 100% at the end. The eventual champions led the race outright for barely a week of it.

So the race shows what each surprise did to the odds. The next chart asks how surprising each one actually was.

One bar per scored group fixture, and the height is the model’s surprise at the outcome, measured in bits:

$\text{surprise} = -\log_2 P(\text{the outcome that actually happened})$

An outcome the model had at 50% costs exactly 1 bit, one it had at 25% costs 2 bits, and each extra bit doubles the shock. Spain 0-0 Cape Verde was priced at 8%, and $-\log_2(0.08) \approx 3.6$ bits, the tallest bar of the summer.

Why the logarithm rather than just plotting the probability? Because surprise should add: two 25% outcomes back to back (2 + 2 bits) are exactly as shocking as one 6.25% outcome (4 bits), and the log is the only scale with that property. That additivity is what makes the average bar height one honest number for the whole month, and that number is precisely the log-loss the backtest knobs were tuned on in Part 1, wearing friendlier units.

The average over the group stage was 1.24 bits per match. A clairvoyant scores 0 and a know-nothing shrugging 1/3-1/3-1/3 at every fixture scores $\log_2 3 \approx 1.58$ on every single bar, so that gap of a third of a bit is the model’s entire edge over guessing, and it is also why the humans in the league stayed within touching distance.

The labelled spikes hold a lovely counter to the instinct that upset means giant-killing: four of the five biggest shocks were draws the model had priced under 20%. The eventual world champions sit at the top of their own upset-o-meter for failing to score against the side ranked 36th of 48 on the ratings table below. The only non-draw in the top five is Ivory Coast 1-0 Ecuador, which the pundits barely blinked at.

All of these shocks were being fed straight back into the two numbers each team carries. Time to look at what they did there.

The Hidden Numbers

Throughout the tournament, the predictions were published and the ratings behind them never were. This chart is the machinery’s internal state: every team plotted by its two fitted numbers, attack (right = scores more) against defence (up = concedes less). The hollow circle is where a team stood in the fit of the 14th of June, two matchdays in; the solid dot is the final fit; the trail between them is the tournament changing the model’s mind. Remember from Part 1 that these are log-scale, so equal steps mean equal percentage changes: a +0.10 move on either axis is $\exp(0.10) \approx 1.105$ , roughly +10.5% expected goals in every future fixture, and a trail of that length is a big deal.

The full list: all 48 teams' ratings and drift, 2026-06-14 to 2026-07-20 (click to expand)

Sorted by overall strength (attack + defence). att/dfn are the latest fit (2026-07-20); each drift column is the change since the 2026-06-14 fit - the trail in the figure, as a number. Ratings are log-scale (+0.10 of attack = ~+10.5% expected goals in every fixture) and centred on the average of ALL ~200 national teams, which is why every side here is positive. Regenerates with every re-run.

#	team	attack	drift	defence	drift
1	Spain	+1.58	-0.06	+1.86	+0.50
2	Argentina	+1.55	+0.13	+1.34	-0.34
3	Portugal	+1.47	-0.02	+1.33	+0.12
4	France	+1.58	+0.15	+1.13	-0.13
5	Brazil	+1.44	-0.04	+1.25	-0.00
6	England	+1.58	+0.19	+1.07	-0.43
7	Colombia	+1.18	-0.20	+1.46	+0.27
8	Netherlands	+1.51	+0.07	+1.00	-0.06
9	Belgium	+1.44	+0.08	+1.03	-0.05
10	Morocco	+1.11	+0.04	+1.34	-0.15
11	Germany	+1.59	+0.08	+0.86	-0.16
12	Switzerland	+1.25	+0.04	+1.03	+0.05
13	Japan	+1.20	-0.01	+1.06	-0.11
14	Norway	+1.47	+0.10	+0.78	-0.18
15	Mexico	+1.05	+0.09	+1.12	+0.03
16	Croatia	+1.22	+0.02	+0.90	-0.16
17	Uruguay	+0.98	-0.05	+1.12	-0.23
18	Senegal	+1.19	+0.16	+0.79	-0.28
19	Ecuador	+0.63	-0.27	+1.31	-0.15
20	Austria	+1.14	+0.03	+0.77	-0.23
21	Australia	+0.80	-0.24	+1.11	-0.09
22	Ivory Coast	+0.80	-0.00	+1.05	+0.05
23	Turkey	+1.15	+0.01	+0.68	-0.04
24	Sweden	+1.23	+0.08	+0.55	-0.12
25	Canada	+0.93	+0.05	+0.81	-0.07
26	Egypt	+0.87	+0.13	+0.87	-0.09
27	DR Congo	+0.72	+0.12	+1.00	+0.04
28	Algeria	+1.08	-0.04	+0.64	-0.24
29	Iran	+0.81	-0.21	+0.90	-0.03
30	United States	+1.07	+0.00	+0.64	-0.10
31	Paraguay	+0.65	-0.19	+1.06	+0.25
32	Scotland	+0.82	-0.12	+0.83	-0.07
33	South Korea	+0.81	-0.21	+0.81	-0.04
34	Czech Republic	+0.92	-0.07	+0.57	-0.10
35	Ghana	+0.57	-0.08	+0.88	+0.28
36	Cape Verde	+0.63	+0.13	+0.72	+0.19
37	South Africa	+0.56	-0.03	+0.76	+0.15
38	Bosnia and Herzegovina	+0.67	+0.02	+0.50	-0.11
39	Jordan	+0.74	-0.11	+0.38	-0.16
40	Panama	+0.55	-0.15	+0.56	+0.10
41	Tunisia	+0.66	-0.07	+0.45	-0.47
42	Saudi Arabia	+0.40	-0.15	+0.62	-0.07
43	Uzbekistan	+0.61	-0.09	+0.35	-0.63
44	Haiti	+0.56	+0.01	+0.16	-0.13
45	New Zealand	+0.49	+0.16	+0.20	-0.41
46	Iraq	+0.36	-0.17	+0.32	-0.38
47	Qatar	+0.51	-0.17	+0.04	-0.30
48	Curaçao	+0.22	-0.06	+0.05	-0.12

The first thing to notice is what didn’t move. Brazil finished the World Cup at -0.04 attack, -0.00 defence from where they started it, despite being knocked out in the round of 16. Six matches cannot outshout a decade of evidence, and WC_BOOST = 3 is exactly how loud the tournament is permitted to shout. This is a feature: a model that re-ranked its elite after every upset would be a mood ring, not a model.

The champions are the exception that proves the weighting. Spain’s defence rose +0.50, the biggest single move at the top of the table, and it was earned the hard way: one goal conceded in eight matches. In model terms, every attack that now faces Spain gets divided by an extra $\exp(0.50) \approx 1.65$ . England drifted the other way, attack +0.19 but defence -0.43, which is what reaching a semi-final while conceding in nearly every match looks like on a log scale. The 6-4 third-place game against France did neither team’s defence rating any favours.

The genuine re-learnings live in the middle of the table, where six matches can change the verdict. Norway rose (a second-place group finish behind France, then Brazil removed in the round of 16), Senegal’s attack climbed +0.16 on the strength of a 5-0 over Iraq, and Cape Verde improved on both axes for holding the champions to that 0-0. Going the other way, Tunisia’s defence collapsed -0.47 after shipping twelve goals in three defeats, and Uzbekistan’s fell further still.

This chart is also the answer to the changelog’s most persistent question: why did the prediction for X move when X didn’t even play? Because there is no X in isolation. Every re-fit re-centres all ~200 teams against each other, so Paraguay holding Germany changes what every past result against Paraguay was worth. The model is one organism, not 48 independent dials.

Which raises the only question that actually matters: were the numbers it published any good?

The Report Card

Part 1 ended by asking whether a robot that promised itself 60 points in the group stage would actually collect them. Here’s the homework, graded.

The amber curve is simple: the points genuinely banked, 3 for an exact scoreline, 1 for a correct outcome, accumulating match by match. The grey curve is the unusual one, and it’s the model’s promise to itself. Before each fixture the model knows both its pick and its own probabilities, so it can price what the pick is worth using the expected-points formula from Part 1:

$EP(s) = 3 \cdot P(s) + 1 \cdot \big(P(\text{outcome of } s) - P(s)\big)$

A pick with a 12% chance of being exact and a 55% chance of the right result is worth $3(0.12) + 1(0.55 - 0.12) = 0.79$ points, on average, before a ball is kicked. Sum those per-fixture worths across the group stage and you get grey.

The verdict, over 70 scored fixtures: 44 correct outcomes, of which 8 were exact scores, so $8 \times 3 + 36 \times 1 = 60$ points banked, against 60 promised. It said its picks would come off 58% of the time; they came off 63%.

It’s worth being precise about what that does and doesn’t prove, because it separates two ideas that are easy to conflate. Accuracy is how many you got right, and it is substantially luck. Calibration is whether reality matched your stated confidence, and it’s the thing a model actually owes you. If the probabilities had been bravado, amber would have sagged below grey; had they been falsely modest, it would have ridden above. Promising yourself sixty points and collecting exactly sixty is the real flex; clairvoyance was never on offer.

And the 8 exact scores from 70 (11%) is not the failure it might look like. It’s the humility the scoreline grid promised back in Part 1: the best available exact pick lands about one time in six, and it did.

Provenance, for the suspicious (click to expand)

Every prediction on this chart was frozen the moment its fixture was played, read back from timestamped snapshots written after each model run, so there is no retro-fitting: the grey curve could not be quietly improved after the fact. Two of the 72 group fixtures were played before the very first snapshot existed and are excluded rather than reconstructed. Knockout fixtures aren’t included because the snapshots only stored per-fixture predictions for the group stage, an oversight I’d fix next time.

Calibration is one half of how the model updates. The other half is blunter: reality simply overwrites the dice.

Reality Overwrites the Dice

Once a match is played it stops being simulated at all. Every one of the 100,000 tournaments thereafter contains Mexico 2-0 South Africa as historical fact; those points can never be un-won. Formally, every number the model published was always $P(\text{event} \mid \text{results so far})$ , and the conditioning bar is doing more of the work with every matchday: by the semi-finals a “simulation” is mostly transcript with a little dice at the end, which is why group probabilities march to 100% and champion odds squeeze towards 0 or 100. The twelve panels below are that argument, drawn twelve times: the model’s confidence in its group-winner pick (solid) and runner-up pick (faint) after each day’s run.

Learn one cell and you’ve learned all twelve. Flat segments are rest days, every step is a matchday being fitted, and every line is one-way, because information only accumulates; confidence never retreats. The hollow circles are the most honest moments a forecaster has: not “more sure about the same pick” but “it’s not them any more”. Group D carries the cleanest example, the pick flipping to the United States on the 13th of June after their 4-1 over Paraguay, and from there it locked first of all twelve groups. It was also, for balance, the only group where my six scoreline picks banked exactly nothing: the model knew who was going through before anyone else and still couldn’t price a single match there. Certainty about the destination, useless on the route. Group K is the other extreme, a mid-stage flip and no lock until the final matchday, with Colombia only decided by the last round of fixtures. And in nearly every cell the faint line locks later than the solid one, because second place is genuinely the harder call.

The very last run of the season, after the final, was a simulation with no dice left in it: 100,000 identical tournaments, one of them real.

What It Never Knew

A final list, stated with some pride. The robot knew no injuries, no suspensions, no team news, no tactics, no weather, and no narratives. It read a decade of scorelines and not one news article, and it still won the league: 120 points, twenty-one clear of second place, with 72 of the tournament’s 104 fixtures called right in some form and 16 of them on the exact scoreline. It also placed 225th on the platform’s global leaderboard - 197th if you rank by points alone, since a crowd of us tied on 120, and after a summer of shootout coins I know better than to respect a tiebreak.

I should be honest about my own contribution to all this: I typed in whatever the robot said and got on with my week. Part 1 admitted that football offers me little entertainment value personally, and a full World Cup has done nothing to change that. It turns out you can win a football leaderboard without acquiring the slightest interest in football, and I’d be lying if I said that wasn’t my favourite result in this whole post.

Things I’d try next time, one sentence each. The Dixon-Coles low-score correction², because four of the five biggest shocks being cheap draws is exactly the bias it fixes. Strength-weighted penalty shootouts, although “I tested it and nothing changed” remains the pre-registered reply, even after watching the coin delete Germany and the Netherlands in a single evening. Fitting the ratings on shot-based xG rather than goals, which is the credible upgrade but needs data that simply doesn’t exist for most international sides. And per-venue altitude and travel effects, for a tournament spread across three countries.

So what did I actually add? The honest answer is: not the maths. The model is proudly 1982 (Maher’s attack/defence Poisson), the recency weighting is 1997 (Dixon & Coles), and the simulate-it-100,000-times architecture is exactly what the academic consortium of Groll, Zeileis and colleagues runs, down to the same simulation count³. Their pre-tournament forecast had Spain on top at 14.5%, and the favourite duly won; the robot’s list on the opening weekend also had Spain first, at 15.7%, before it spent a fortnight infatuated with Argentina. On the one scoreable comparison of the summer, the professionals and the hobbyist agreed more than either would like to admit.

What’s genuinely mine is the two layers around the maths. The decision layer, because no academic has to hand one scoreline to a league admin by Friday: picks priced under a 3/1 scoring rule, a joker spent by expected points, draws avoided not by opinion but by geometry. And the ops layer: a fixed seed, a timestamped snapshot after every run, a changelog line attributing every moved number to data or a knob, parameters locked on the 13th of June, and pre-match predictions frozen so the report card couldn’t cheat. Neither layer is a better model. Both are what let a 1982 model be accountable in 2026.

That, I think, is the meta-lesson of the whole exercise: the engineering mattered more than the maths. Calibration, not clairvoyance, is what a good model owes you, and it’s a debt you can only prove you’ve paid if you wrote everything down at the time.

The robot is switched off now. It retires with a league title, exactly the group-stage points it promised itself, one perfect final it was never allowed to double, and no opinions whatsoever about Euro 2028.

On that, for once, we agree.

I Built a Robot Instead of Having Opinions ↩
Dixon & Coles (1997), “Modelling Association Football Scores and Inefficiencies in the Football Betting Market”, JRSS C 46(2) ↩
https://www.r-bloggers.com/2026/06/football-meets-machine-learning-forecasting-the-2026-fifa-world-cup/ ↩

Some figures in this post may not survive your feed reader - read it on the site.

I Built a Robot Instead of Having Opinions

Wed, 08 Jul 2026 12:30:00 GMT

Updated 2026-07-08: monte carlo simulation visualisation added

The Pitch

The World Cup has rolled around once again, and as a workplace tradition, there’s a prediction league¹ that most of us got involved in for a bit of fun, not to mention the honourable bragging rights of topping our little leaderboard. This was the first one for me, and although football offers little entertainment value personally, the possibility of predicting the outcome based purely on historical data seemed rather enticing.

The only issue? I know nothing about football. Or rather, I know nothing of the state of play when it comes to teams’ performances. So, I set out to create a robot to figure it all out for me.

But we have to rewind a little as to how we got there.

As a somewhat cathartic exercise in humouring my current interests, I was looking into physics and radioactive decay. One rabbit hole later and I stumbled across the Poisson distribution² and wondered whether or not this could have some application to our little prediction league. Sure enough, my excitement at discovering a new application for this formula was swiftly curtailed after a little research showed this was a well-trodden field. After some research into what others more well versed had produced, I set out to implement what I hoped to be a winning formula.

Before moving onto the good stuff, a little aside on how the prediction league works:

Correct Prediction	Points Scored
Outcome (win/draw/loss)	1
Score	3

In addition to the above, for each group/knockout stage there’s the use of a Joker. The application of which can only be used once at each stage and has the effect of doubling your points for the specific fixture you use it on (2 points for correct outcome, 6 for correct results). Your chosen team to lift the World Cup trophy needs to be locked in by the knockout stages but alterations to the scoreline can be made until right before kickoff.

The first stage was acquiring the data, of which there’s a surprisingly large amount publicly available. The problem was making sure I had the right kind of data. After some research into how this prediction engine might work, all I really needed was the outcomes of as many football matches between international teams. After a little digging I found 49,499 (at the time of initial collection) international football results since 1872, courtesy of martj42’s public dataset³, perfect. Alongside the results file, I used the shootouts file to provide the outcomes of any matches that went to penalties.

And that’s it.

Every prediction the robot makes starts from the same two numbers, more on that below.

I Care About Nothing But Net

It seems a little counterintuitive at first, but the predictive model cares nothing for how many games you won, lost or drew. The only factor it considers is the goal breakdown, how many were scored, how many were conceded, and importantly, against whom.

The model assumes that each side’s goal rate, $\lambda$ , is multiplicative (Maher 1982⁴):

$\lambda$ (i vs j) = (baseline) × (i’s attack multiplier) × (j’s defence multiplier)

The adjustments of these variables are the whole trick: scoring 3 past Brazil says more than scoring 3 past a minnow. However, this calculation is circular: to rate the att of a team you must know the def of the opposition to calculate its impact.

To fix this chicken and egg problem, every att and def starts at 0. From here we freeze the defence scores for all teams and fit the attack score given the current defence value of 0. We then freeze the attack score and refit the defence score. In addition to these two, both mu (the average) and home_adv are refit in the same way. Each pass round these freeze and refit cycles looks to improve the fit, so it can never oscillate forever.

A fictitious fitting (click to expand)

To help visualise this fitting loop I’ve generated a small sample which uses the same maths behind the ~200 team results data:

Four fake football teams, Astoria, Borland, Cresta and Dovia, and their Sunday league.

Matches
Astoria 2-0 Dovia
Astoria 2-1 Borland
Astoria 0-0 Cresta
Borland 3-2 Dovia
Borland 1-1 Cresta
Cresta 1-0 Dovia

League totals:

Team	Scored	Conceded
Astoria	4	1
Borland	5	5
Cresta	2	1
Dovia	2	6

Every team starts with att and def at 0. mu starts at the log of the average goals per side, 13 goals / 12 matches = 1.083, mu $= \log(1.083) = 0.080$ .

With all defences frozen at 0, each side’s “expected if average” is the same for every match. $\sum r = \text{total matches} \times \exp(\text{mu} - 0) = 3.25$ .

team      Σk (scored)   Σr      ratio    att = log(ratio)
Astoria   4             3.25    1.231    +0.208
Borland   5             3.25    1.538    +0.431
Cresta    2             3.25    0.615    −0.486
Dovia     2             3.25    0.615    −0.486

Note Cresta and Dovia get identical attacks: same goal count, and the model doesn’t yet know their opponents apart. Remember that, it’s about to change.

Defence update (using the fresh attacks):

team      att              def                mu = −0.409 (baseline 0.664)
Astoria   +0.029 (×1.03)   +0.928 (÷2.53)
Borland   +0.936 (×2.55)   −1.191 (÷0.30)
Cresta    −0.672 (×0.51)   +1.056 (÷2.87)
Dovia     −0.293 (×0.75)   −0.792 (÷0.45)

Cresta and Dovia’s attacks split apart (−0.67 vs −0.29) despite identical goal totals - the loop worked out who the goals came against and priced the schedules differently. This is the opponent adjustment indicated with Brazil above, visible in the wild.

Astoria’s attack is… average (+0.03), even though it out-scored everyone but Borland. The model looked at its 4 goals, noticed they came against the two leakiest defences in the universe (Borland ÷0.30, Dovia ÷0.45), weighed the 0-0 against Cresta, and concluded: goals explained by opposition, not by attack. Astoria’s dominance lives almost entirely in its defence (+0.93). With three matches of evidence, one blank is loud.

Borland shows attack ×2.55, defence ÷0.30 (concedes more than three times the average).

Model Parameters (“the tuning knobs”)

The elephant in the room however, was that every historical match had an equal impact on the fitting of these values. It seems a little unfair that a football match that took place before any of the current team players were born should have the same weighting as their own personal contribution.

As part of the model simulation there were 4 distinct constants. Each of these had a particular impact on the teams’ ratings:

Recency decay - 2 years: a match 2 years ago counts half as much as one today, if it were 4 years ago then a quarter of what it counts today, etc. This meant that anything pre-2015 had less than a 2% impact on any ratings.
Friendlies - 0.5: any friendly matches held only 50% of their potential impact.
World Cup boost - 3.0: a weight multiplier for 2026 tournament matches. Any matches played in the tournament had a 3 times weighting for the rating.
Host bonus - 0.2: a log-scale boost to hosts’ expected goals (USA/Canada/Mexico). This was calibrated against previous World Cups.

Each of these tuning knobs was tested in isolation and was tested against a fixed seed and run against 4 historical World Cups.

All tuning was locked in by 13th June to avoid any overfitting/look-ahead bias.

Tuning for Improved Fit (click to expand)

Two of the tuning knobs, half-life and World Cup boost, were a little more nuanced to determine. The best course of action was to backtest these against historical tournaments to see what value helped to fit results that we already know the outcome of.

Each cell replays the knockout rounds of four past World Cups (2010, 2014, 2018, 2022) with ratings refitted as of each tournament’s group-stage end under that setting, then scores the predictions two ways:

Pooled log-loss (lower = better; punishes confident wrong answers hardest)
Correct advance picks.

Darker cell = better log-loss, the outlined cells are the live settings used for the simulation.

The knob with genuine signal is the memory (half-life) one. Although the log-loss was lower at 1 year, at year 2, an additional correct knockout outcome was predicted for minor log-loss increase.

So, two grades per team (`att` and `def`). But how do two become one (outcome/score)?

The Poisson Bit

In 1898, Russian economist and statistician Ladislaus Bortkiewicz published a book titled The Law of Small Numbers, in which he observed that rare events in a large population follow a Poisson distribution even when the underlying probabilities vary. To show that the distribution was more prominent than originally thought, he turned to Prussian army records: the number of soldiers killed by horse kicks each year, in each of the 14 cavalry corps, over a 20-year period. Those counts followed a Poisson distribution.

Goals follow this same species of event, namely rare, roughly independent and take place within a fixed amount of time.

Before we get into the Poisson distribution formula, it’s important we understand how the goal rate, $\lambda$ , is calculated based on the teams’ attack and defence multipliers we discussed in the previous section:

$\lambda$ (i vs j) = (baseline) × (i’s attack multiplier) × (j’s defence multiplier)

These attack and defence multipliers are exponential ( $\exp$ ) values, so if we take the $\log$ of these, we can turn the multiplication into addition:

$\log \lambda = \text{mu} + \text{home\_adv} \cdot (i \text{ at home}) + \text{att}_i - \text{def}_j$

mu - log of the average team’s goals against an average team
att_i - teams i’s attack on a $\log$ scale. att = +0.10 → $\exp(0.10)$ → ×1.105, i.e. +10.5% goals in every fixture, where average = 0.
def_j - team j’s defence, same scale as the attack but the sign is flipped so that higher is better (concedes fewer goals). A team facing a defence of +0.50 divides the team’s attack by $\exp(0.5) \approx 1.65$ .
home_adv - this is a shared bump in stats for the home side. The fitted value for this came out at 0.226 which is a 1.25 multiplier, not bad given the well known (apparently) ~30% home effect. At the World Cup, almost everything is neutral ground, so this mostly mattered for fitting against the historical data. The three hosts in this year’s World Cup instead got a fixed HOST_BONUS of 20% at prediction time.

The use of exponentials is important. Firstly, rates ( $\lambda$ ) must be positive, regardless of what each team’s ratings are, $\exp$ guarantees this. Secondly, the effect should compound, a good attack helps proportionally against a weak defence.

Some real numbers from the current fit (as at 2026-07-02):

Team	att	def
Spain	+1.573	+1.558
France	+1.549	+1.416
Argentina	+1.485	+1.773
England	+1.420	+1.402
Panama	+0.608	+0.587

mu = -0.007 (baseline $\approx$ 0.99 goals) home_adv = 0.226

Plugging in these values for Argentina’s rate against France:

$\log \lambda = -0.007 + 1.485 - 1.416 = 0.062$
$\lambda = \exp(0.062) = 1.065$

And France’s rate against Argentina:

$\log \lambda = -0.007 + 1.549 - 1.773 = -0.231$
$\lambda = \exp(-0.231) = 0.794$

Argentina’s edge in this fixture is entirely down to its superior defence. However, an average of 1.065 is not a prediction of 1.065 goals, it’s a scoring menu.

This is where the Poisson distribution comes in. If something happens at an average rate of $\lambda$ per window, and each occurrence is roughly independent of the last, then the probability of seeing exactly k occurrences is:

$P(k) = \exp(-\lambda) \cdot \lambda^k / k!$

Where the formula comes from without calculus

If you were to chop the football match into 90 one-minute slices, a team averaging $\lambda$ = 1.4 goals per match scores in any given minute with probability ~1.4/90 $\approx$ 1.5%. The chance of no goals in 90 straight minutes is $0.985^{90} \approx 25\%$ - and $\exp(-1.4) \approx 0.247$ . That’s not a coincidence, the Poisson formula is exactly the limit of this minute-by-minute counting, and the slices get infinitely thin.

The current fit gives Argentina $\lambda$ = 1.065 against France. The goal menu:

P(0) = exp(−1.065)              = 0.345   (34%)
P(1) = exp(−1.065) · 1.065      = 0.367   (37%)
P(2) = exp(−1.065) · 1.065²/2   = 0.195   (20%)
P(3) = exp(−1.065) · 1.065³/6   = 0.069   ( 7%)

Those are literally the bars in the goal-menu figure below. Note the headline lesson: an average of ~1 goal means scoring zero is nearly as likely as scoring one.

The visualisation below shows the probabilities of xG (expected goals) for each team, i.e. Argentina to score 0 goals against France 35%, 37% to score 1. It’s these menus that will be used in the simulations, which we’ll look into in the next section.

The Poisson distribution has a single parameter, its mean $\lambda$ . This is why the entire modelling problem can be reduced down to a single good $\lambda$ per team per fixture.

That’s just one side of the coin though, there’s also a goal menu for the opposing side and by multiplying the two menus for each xG⁵, every scoreline has a price. This is what will be used to determine the exact score used in the prediction.

The scoreline grid offers a humble realisation. The most likely score is only 16-17% with the current team ratings, not exactly a clear answer. To emphasise this point a little more, have a look at the interactive scoreline grid below. Drag both sliders to 0.80 for both teams and watch how 0-0 and 1-0 dominate the predicted outcomes; this is why knockout football can be such a coin flip. Drag one of those lines to 3.0 and the favourite still only wins ~82%.

Play with the machine
drag each side's expected goals; every scoreline reprices instantly (starts at the model's Argentina v France: 1.06 v 0.80)
home expected goals: away expected goals: 
away goals
home goals
0
0

1
1

2
2

3
3

4
4

5
5

<div style="font-size:0.8rem;color:var(--dim, #8a8273)">(the sliders need JavaScript - the static figures above show the same idea)</div>

A known flaw (click to expand)

The scoreline grid assumes the two scores are entirely independent. Dixon & Coles (1997)⁶ showed low-scoring draws are slightly more common than independence implies. I have deliberately not corrected the model given its simplicity offsets the minor log-loss improvement.

The honest output of all this is that a ‘clear favourite’ prices out a W/D/L near 50/28/22. The better team draws or loses all the time, and the model refuses to pretend otherwise.

Now that we’ve a price for every scoreline of every fixture, it’s time to do something with it. After all, a World Cup is not 104 independent fixtures.

Monte Carlo Simulations

There are two distinct parts to this model: questions to be answered.

What score do I predict for this fixture?
Who wins the group / makes the final / wins it all?

The second of these is the macro question and requires the chaining of many possibilities to come to a conclusion. The Poisson distribution answers the first question given it sums the 81 cells (8 goals as the maximum scoreline, 9×9 grid) and can calculate the optimal prediction.

The Monte Carlo⁷ rolls from those same Poisson menus only this time the same distributions are executed with dice instead of summed with algebra.

The Monte Carlo rolls the dice 100,000 times for match outcomes based on a sampling of the Poisson menus, rolling up and counting the scoreline for each match. The most common of which becomes the predicted outcome (note this is not the predicted score used for point scoring). This builds a picture for what the model believes will be group winners, knockout results and the overall winner.

The visualisation below shows the reason for running so many simulations. The x-axis is logarithmic, so at 10 simulations the “estimate” swings by tens of points, at 100 it is still wobbling by several, by a few thousand the lines are flat. The mini simulation below runs to 4,000 simulations; however, a similar story is shown at 100,000 simulations.

The only place that the model refuses an opinion is for penalty shootouts. For the knockout stages and beyond, any draws that were predicted had extra time added at a third of the slices for Poisson distribution and penalties were a 50/50 coin flip.

Now that our simulations have provided probabilities, we need scorelines.

The Decision-Theory Layer

Determining the probability of a match is just the first part of the problem though. To win the league I’d need to optimise my scorelines to ensure that I give myself the best chance of winning. So far we’ve produced the probability of scorelines coming to fruition, and the initial reaction would be to choose this as our scoreline. After all, it’s the most likely one.

However, choosing the scoreline is a separate, non-statistical problem. What we want to do is maximise our expected payoff under the league’s scoring rule as indicated in The Pitch section. Three points for exact score, one for outcome and none for an incorrect outcome.

For this we need another equation to calculate the expected points (EP) for a scoreline:

\begin{aligned} EP(s) &= 3 \cdot P(s) + 1 \cdot \big(P(\text{outcome of } s) - P(s)\big) \\ &= 2 \cdot P(s) + P(\text{outcome of } s) \end{aligned}

A scoreline pick’s value is twice its chance of being exactly right, plus once its chance of being directionally right. Every one of the 81 grid cells for a fixture gets this score, and the model takes the maximum value.

Importantly, the most likely scoreline does not automatically make it the best pick as the modal score only maximises the first term, whereas this calculation lets the outcome term vote too.

If we again cast our minds back to the scoreline grid, geometrically, a draw pick banks only on the diagonal; a win pick banks the whole triangle to one side of it.

To explain this a little clearer, consider the scoreline grid below for Mexico v United States:

We can extract two important probabilities from this table (although 1-0 shows 12%, this was rounding precision and is in fact 11.9%):

pick	P(exact)	P(outcome)
1-1 (most likely score)	12.0%	25.1%
1-0 (EV-optimal)	11.9%	50.8%

Taking these probabilities, we can then plug each cell into the equation to get the following:

Giving up one-tenth of a percentage point of exact-hit chance and I can double the outcome mass of the pick - that’s +52% expected points, for free. The optimal pick is a scoreline of the favoured outcome except in genuine coin flips; this is because the draw diagonal is nearly always the smallest of the three regions. As a result, the robot almost never predicted draws, not because they are rare - in fact the model predicted 25-32% on average - but because the scoring rule punishes draw picks.

The use of the joker was fairly simple: whichever fixture in the group/knockout stage had the highest EP, that’s the one it went on. The idea being, spend it on the most boring mismatch available and hope there’s not an upset, which of course, there will be.

The knockout correction (click to expand)

In the knockout stages, there must be a winner, and so over optimising the 90-minute grid is subtly wrong. A 1-1 pick can’t cash if the tie ends 2-1 in the 105th minute. For this, an additional corrected grid was created, and the logic was just to follow each route to the final score. Each predicted draw in the knockout stages spawns a fan of possibilities given the additional 30 minutes of extra time. This is its own Poisson grid using $\lambda/3$ for each side.

Perhaps Mexico v USA ends 1-1 at 90-minutes; however, using the knockout correction full-time becomes 2-1 if the ET goes 1-0, which the additional grid prices at 22%:

P(full-time 2-1) = P(2-1 at 90')  +  P(1-1 at 90') × 22%
                   └ route 1: no ET ┘   └ route 2: via extra time ┘

For the two non-overlapping routes, probabilities are summed.

So the robot knows what to predict and how much each prediction is worth. Time to see what it actually said.

The Cliffhanger

As a preface to the bracket graphic below and all of the visualisations above, they were all generated using simulation snapshots up to July 2nd 2026. At the time of writing this, there have been a number of upsets, some exact scorelines and quite a few correctly identified outcomes.

The above is the single most likely remaining tournament based on the simulated model. Every tie shows the best-expected-points score and the winner’s chance of advancing. However, this exact script is wildly unlikely; it is just more likely than any other outcome.

The model's two finals (click to expand)

As an interesting aside, the model simulation was run on 7th July (visualisations are frozen at July 2) and it produced two different finals, and both are right. The simulation predicted that the most frequent pairing across 100,000 simulations was in fact Spain v England (15.7%). The bracket, however, was built match by match and instead ended with Spain v Argentina. This divergence is due to how the model builds its pathway to the final. In the 7th July simulation, England needed only a single win against Norway to advance to the semi-finals, whereas Argentina are yet to play and must first get past Egypt and then play the quarter-final match. As a result, summed over every simulated tournament, England are in the final more often than Argentina (~35% vs ~29%) even though they usually lose the head-to-head.

The most common destination and the most likely route are different questions, and they split apart when the head-to-head favourite has the harder road (at that point in time of the simulation). The two finals may re-converge or swap as the knockouts continue to play out, but that’ll become apparent soon enough.

The final is to be played on the 19th July and I intend to follow this up with a part 2, a post-mortem:

What did the tournament do to the model's hidden numbers?

Did a robot that promised itself 60 points in the group stages actually collect them?

A cruel poetic irony has been wrought. I write this article to showcase my creation's knowledge of football, a robot I built to avoid having opinions. And yet, the robot has decided to alter its opinion of the predicted World Cup winner. Alas, I'm locked in and await the verdict.

https://worldcupfootballpredictor.co.uk ↩
https://en.wikipedia.org/wiki/Poisson_distribution ↩
https://github.com/martj42/international_results ↩
Maher (1982), “Modelling association football scores”, Statistica Neerlandica 36(3), 109–118 ↩
https://en.wikipedia.org/wiki/Ladislaus_Bortkiewicz ↩
Here, the xG (expected goals) means the model’s pre-match average, not the TV shot statistic ↩
Dixon & Coles (1997), “Modelling Association Football Scores and Inefficiencies in the Football Betting Market”, JRSS C 46(2) ↩