I'm designing a football (soccer) tournament for youth sides, and there are nine teams playing. When this was done before, byes were given to 7 teams, in a single elimination tournament. This is probably what we will do again, but it previously caused complaints from the two teams that didn't get a bye in the first round.
I am looking for a tournament design that keeps the good qualities of single elimination, but avoids byes, and is more manifestly "fair" to all teams.
The advantages of a single elimination tournament with 2^n teams are:
- If a stronger team always beats a weaker one, then the strongest team is certain to win.
- All teams have equal opportunity to win: if every match is a coin toss, then every team would have the same probability of winning.
- There is an exciting final.
- There are relatively few matches.
- The number of rounds is known in advance
Points 1 and 2 make the tournament "fair", 3 makes it "exciting", 4 and 5 make it "convenient". If byes are used, then a single elimination tournament fails point 2.
- A league structure in which every team plays every other team is fair, but there is no final and there are a large number of matches.
- A Swiss-style competition still has trouble with an odd number of teams.
- A groups + knockout is possible for 12 teams (four groups of 3, followed by knockout of the four winners), but not for an odd number.
- I thought of a tournament in which the teams play in a circle: A v. B, B v. C ... H v. I, I v. A. (so every team plays twice in the first round) Those teams that win both matches would then play in a second round and third round if required. This avoids byes, but there might be no "final" and the number of rounds wouldn't be known in advance (it is also possible for no team to win both matches, but this would be unusual).
Is it possible to design a tournament that is fair, exciting and convenient for a number of teams that is not a power of two?