With England through to FIFA World Cup semi-finals, I am wondering: how fair is the knockout system that determines the first, second and third-place winners?
To simplify things, let us assume that the teams can be ranked in order of their relative strengths, and that each team plays consistently to its ranking, so that a higher-ranked team will always beat a lower-ranked one. (Yes, I know this is a gross oversimplification.) Let us also assume that the seeding system used by the organizers is fair, so that the eight seeded teams (which include the host nation) are the top ranking eight teams (another big assumption).
These 8 seeded teams were then assigned randomly to 8 groups and based on our assumptions, each seeded team could be expected to win its group. It would then meet a different group’s (unseeded) second-place team in the round of 16, which it could also be expected to beat. So, based on our assumptions, the quarter-finalists should be the top-ranking eight teams.
In the knockout competition, the winning finalist is awarded first place, the losing finalist second place, and the losing semifinalists compete in a playoff for third place. So using this system, the #1 ranked entrant will always get first place. But, based on the random way the teams have been bracketed together, what are the chances of a team winning second or third place?
Well, I did the math, and here are the chances for each team’s possible outcomes:
|Team rank||1st place||2nd place||3rd place||Playoff loser||Eliminated in quarter-finals|
As you can see, the knockout system is only accurate for determining the first-place winner and there is a very good chance that the second and third places won’t go to the “right” teams. There’s even a chance that the second-place winner could be the #5 ranked team!
In fact, there’s only an 8-in-21 chance that the #2 and #3 teams will both be placed correctly, a 4-in-21 chance that their places will be reversed, and a 3-in-7 chance that one or other of them won’t get placed at all. The only certainty is that at least one of these two teams will get second or third place.
Note that this analysis doesn’t take account of the random factors that affect every team’s match-day performance: it only reflects the advantages and disadvantages that can arise from the “luck of the draw.”
Conclusion: the World Cup’s knockout system is not very fair at all when it comes to allocating second and third places.
DISCLAIMER: I’m posting this before England’s semi-final game is played, so don’t consider this to be sour grapes!