I Ran 300,000 Simulations to Find the Third-Place Cutoff Score

The group stage of the FIFA World Cup is finally reaching its final matches.

There’s something that must have caught your attention while watching this tournament.

Starting this time, the tournament expanded to 48 teams. With 12 groups of 4 teams each, the top 2 teams from each group automatically advance. So far so good. The problem comes after that: among the third-place teams in each group, the “8 teams with the best records” also advance to the knockout stage.

Across the 12 groups, there are 12 third-place teams. Of those, 8 make it through.

In other words, even in third place, two-thirds of teams still go through. That’s a fairly generous rescue quota.

From 48 teams to 32 — the mechanics of “advancing as third place”

48 teams 12 groups × 4 teams

Top 2 in each group 24 teams Advance automatically

Third place in each group (12 teams) Only 8 teams Only the top performers advance

Knockout stage 32 teams

Even in third place, 8 out of 12 teams (roughly two-thirds) get picked up So the question became: what score marks the border for advancing as third place?

What caught my attention was where that third-place border would actually land in practice.

There’s no fixed passing score to begin with. What I wanted to know was the distribution

The structure itself is easy to see. Advancing as third place means lining up the third-place finishers from all 12 groups by record and taking the top 8. It’s an order statistic problem — taking the 8th out of 12. There’s no such thing as a fixed passing score; the border shifts every time depending on how the third-place teams in the other groups fared.

That’s not the issue, though. Seeing the structure and having the distribution appear in your head are two different things.

Around what point total does the 8th-ranked value tend to cluster? When a row of teams sit tied at the border, what ultimately separates their rankings? Within a group there’s correlation, since three teams are fighting over the same points; between groups it’s essentially independent. The distribution this mixture produces doesn’t resolve into a clear picture just by staring at the equations. It doesn’t collapse into a clean closed form.

For things like this, it’s faster to just run the numbers.

For now, let’s hammer it 300,000 times

The approach was simple.

Assume all teams have “equal” strength, and generate each match’s score randomly from a Poisson distribution. Simulate an entire group stage this way, and record how the top 8 among the third-place teams performed. Repeat this 300,000 times.

Making all teams equal in strength is deliberate. I wanted to see what kind of border the format itself produces, so I first erase differences in strength and extract the shape of the system alone.

Here’s what came out of running it.

The vast majority of the third-place teams that advanced had 3 points. Around 80-90% near the border cluster neatly at 3 points.

A team with 4 points is almost certainly through. Regardless of goal difference, they clear the bar first.

Conversely, with 2 points or fewer, things get considerably tough.

The real pass/fail line turned out to be “goal difference,” not “points”

This is the part where it really clicked into place.

The border clusters at 3 points. That means points aren’t what ultimately decides passage. After all, every team sitting at the border has the same 3 points.

The difference gets made after that — in goal difference, and total goals scored.

Even with the same 3 points, how much does goal difference change the outcome? Here’s roughly what the simulation showed.

When tied at 3 points, “goal difference” is what splits the pass rate

100% 50% 0%

95%

87%

65%

36%

Here it drops sharply

+1 0 −1 −2 Goal difference Just the difference between a goal difference of 0 and −1 shifts the pass rate by more than 20 points (a rough estimate from a simple symmetric model)

Just the difference between a goal difference of 0 and −1 shifts the pass rate by more than 20 percentage points.

In other words, “it’s already a dead rubber, might as well coast” is the most dangerous mindset of all. Preventing that one conceded goal, going for that one goal even against long odds — that’s exactly what determines whether you advance as third place. The cost of a heavy defeat in an earlier match comes due right at the very end.

Why hammering it with random numbers still produces a proper answer

A brief word on what’s happening underneath.

I wrote earlier that this doesn’t collapse into a closed form. The tricky part is inside each group. The three teams within a group are fighting over the same points, so a group’s third-place points and goal difference are entangled with each other, and you can’t write down a clean standalone distribution.

Cross the group boundary, though, and things change. Group A’s results have zero effect on Group B’s results. So overall, this is an order statistic problem: draw one third-place record from each of 12 independent boxes, and look at the 8th-ranked one from the top.

This independence between groups is quietly doing a lot of work. Since you’re just observing the average behavior of 12 independent boxes, the variance shrinks obediently even when you hammer it with random numbers. Run it 300,000 times, and the distribution around the 8th-place value barely moves anymore.

What’s messy is local, within a group; what’s well-behaved is global, across groups. This split between layers is the most satisfying part of this problem.

That said, this is only a story about “the shape of the system”

One honest caveat to close with.

This model treats all teams as equally strong, so it’s presumably underestimating the spread of outcomes compared to reality.

The real World Cup has groups of uneven strength. There are groups of death, and there are easy groups. So if you want to talk about what percent chance a particular team has of advancing, you’d need to rebuild the model with weights, using something like Elo or FIFA rankings to account for differences in strength.

Take the numbers here as a story about the skeleton, roughly what kind of border the system itself tends to produce. Fortunately, even at the level of just that skeleton, the conclusion that goal difference, not points, separates pass from fail looked like a fairly robust one.

A generous system, where even third place can advance.

Behind that generosity, a single goal in a dead rubber is quietly doing its work.

Next time you’re watching a match where “the standings probably won’t change anymore,” take a glance at the goal difference column. That one goal probably matters.

This piece was conceived and directed by Kuzuryu, with the writing done by AI.

Originally published in Japanese at https://clazytech.com/2026/06/1680/. Translated with LLM assistance and reviewed before publication.