Football World Cup

It’s World Cup season, and it keeps delivering: dubious referee decisions, fans marching the streets celebrating and sometimes disrupting opposing teams’ sleep and, of course, underdogs standing their ground against the great football nations’ teams. The Norwegian team beat the exquisite Brazilian team, Cape Verde’s selection put up a tremendous fight against the world champion Argentina, and, of course, the Nationalmannschaft lost the penalty shootout against the team from Paraguay.

This was bad news for coach Julian Nagelsmann, who was fired just a couple of days after. Because if the 2014 world champion loses aginst the 22nd team in the world, clearly the coaching was below par, right?

Unfair Tournament Format

The world’s best chess player, Magnus Carlsen, famously retreated from the FIDE world championship in 2023. Among other points of criticism, Carlsen stated in an interview that he found the competition’s format inadequate:

So I think 12 games, or now 14 games, that there is for the world championship is a fairly low sample size. If you want to determine who the best player is, or at least the best player in that particular matchup, you need more games. And I think, to some extent, if you’re going to have a world champion and call him the best player, you’ve got to make sure that the format increases the chance of finding the best player. So I think having more games, and if you’re gonna have a lot more games, then you need to decrease the time control of it, […]

But where does his criticism of small sample size come from? In the 2018 World Chess Championship, Carlsen, rated 2835 at the time, fought a war of attrition against his contender, Fabiano Caruana, rated 2832. After 12 consecutive draws, it was in the tie breaks that Carlsen beat his opponent 3-0 and was declared world champion following 50 hours of play and over 750 moves.

Using their Elo ratings and the formula for calculating the expected outcome of one game

$E_A = \frac{1}{1+10^{(R_B - R_A) /400}}$

we see that the expected outcome is pretty much a draw, with a .4% edge for Carlsen.

But what are the odds that there is no clear winner of the course of a 12-game match? Let’s simulate 100.000 matches between Caruana and Carlsen and find out where Carlsen’s opinion comes from.

So even after 12 games, Carlsen’s chance of winning the competition was a mere 40%. In a world where they played 200 games instead, his chances of winning would have grown to about 52%.

Back to Football

Over at eloratings.net, they maintain a list of world football Elo ratings that allow us to estimate outcomes at the World Cup¹. The German team is rated 1932, and Paraguay 1834. We can reuse the chess formula:

They played a single match, and the Nationalmannschaft had a 64% probability of winning it. Maybe you share my opinion that this probably matches how the match went – the German team mostly dominating. However, about one in three times, Paraguay is expected to win. This was one of those times.

What if Germany and Paraguay played a FIDE-style world championship match though?

Winning the World Cup is About Luck

Aren’t you rationalizing a bit too hard just because your team lost a game of football?, I hear my dear reader ask. Maybe I am. But honestly, I’m not that big a football guy. I’m a data-guy, and what drives me mad is some German media calling the loss a “disgrace” (Bild) or “humiliation” (Sportschau), although the odds of losing were one out of three all along. You might find yourself disgraced and humiliated quite frequently if this is your bar, no? So I really dislike the discourse. A more honest opinion could have stated that the team’s performance during the course of Nagelsmann’s work has been sub-par, and that the world-cup performance verified this.

But once a team reaches the elimination stages, winning the World Cup is about luck. Let’s explore this by running another simulation. Taking the actual round of 32 bracket and each team’s Elo rating at World Cup kickoff, the widget below simulates the entire knockout stage to see how often each team walks away with the trophy.

Even with a nearly 400 point Elo gap between the favorites and the underdogs, no team clears much more than a coin flip’s worth of certainty by the time you multiply out four straight knockout rounds. That’s the point: a single-elimination bracket is a phenomenal way to crown a winner, and a mediocre way to prove who’s actually the best team in the world.

Of course, the real World Cup only runs the bracket once. Hit the button below and find out who takes home the trophy in this single roll of the dice.