Essay:The Math of Becoming Second at the Fifa World Cup and the UEFA Euro.

I want to take a short (mathematically inspired) look at the quality of the final match of the Fifa World Cup 2014 and the UEFA Euro 2016...

Fifa World Cup 2014
I know it's a much smaller dream that we all wanted, but we have to honour the shirt of the national team. We made the semi-finals, the top four teams in the world. And there were a lot of good teams that were knocked out before us.

Is Felipe Scolari right? Are these the top four teams of the world? Well, probably not. And if those are the top four, than the order may be wrong. Why? The FIFA uses a system combined from a group stage in which eight groups of four teams each play a round-robin tournament - an excellent way to determine the rank of each team in its group. But then, sixteen teams (the winners and the runner-ups of the groups) play single leg knockout matches. While a knockout tournament is an excellent way to find the best team (so, congratulations again, Germany), it's awful if you wish to look for the second-best - and even worse for the third and the fourth place. Indeed, a pure knockout tournament will have the second best team eliminated before the final match in roughly 50% of all cases.

How good is the system the FIFA is using? Under typical mathematical assumptions, it isn't that bad when it comes to the final game:



While under pure knockout rules, there is only a 51.6% chance that the 2nd best overall team reaches the final, this probability is up to 61.3% in the World Cup 2014. The rules guarantee that no team worse than 5th can play in the final match, while otherwise (k.o.), even the 17th best team (or 16th worst) would have had a - tiny - chance to play in the Maracanã.

Even if we may not get a perfect final match, pitting the best and the second best team against each other, we are quite sure (ca. 90%) that we get an interesting match between the best and the second or third best team.

The situation is little bit muddier when it comes to the Bronze Medals:



Perhaps (25.8%) it is the consolation prize for the second best team, which was knocked out during the quater-finals, more probably (32.9%), it goes to a team which doesn't deserve it. As for the fourth place, it is quite probably not the fourth best team which loses out to the Bronze winner:



Nevertheless, the match should be played by two quite good teams, not ranked worse than sixth - though it is wishful thinking to believe that the team which fared worst coming out of the quarter-finals is one of the top four teams of the world.

The probabilities for the events shown by the pictures above are obviously not independent (it is impossible - say - for the second best team to lose out in the finals and gain the Bronze medal), but we can calculate the probability that the first three places in the tournament are awarded to the top three teams of the world: it's 38.0% The probability of these being the three best teams overall is 57.0%, so a Dutch fan might find comfort in the thought that under the assumption that these are the three best teams, the odds are 1 in 2 (i.e., a probability of 33.3%), that it really should be Germany - Netherlands - Argentina.

tl;dr The World Cup 2014 gave us (quite probably) a worthy final match, and Brazil is (quite probably) not as good as Felipe Scolari thinks.

UEFA Euro 2012
The Euro 2012 was organised similarly to the World Cup 2014, but there were only 16 teams in four groups (instead of 32 teams in eight groups). Therefore, one of the knockout stages (round of the last sixteen) wasn't necessary.



Thus, the probability for a final match between the two top teams of Europe was quite high (73.3%), and at worst, the third best team would have played for the prize.

There is no game for third place at the Euro.

UEFA Euro 2016
The Euro is the biggest football event between the World Cups. The number of nations in Europe grew over the last decades, and the broadcasting rights per match became more and more lucrative. No wonder that the UEFA decided to inflate the number of nations in the group phase from 16 to 24. Only eight will be eliminated during the group tournament, and there will be 51 matches overall (instead of a measly 31 in 2012). An additional round in the knockout phase of the tournaments leads to a lower probability for the second best team to reach the final - but it gives the fourth best team a possibility to shine:



While we may get more quantity, I don't think that the new system improves the quality. The World Cup 2014 has shown that the matches during the group phase can be the most entertaining, while the knockout phase may harbour some disappointments. I'd prefer four groups of five teams each, with only the top two advancing to the knockout stage. This results in 47 games, but in a better probability for a final match of the best two teams...

tl;dr Looking only at the final match, the UEFA rules for 2016 are no improvement over those of 2012.