Design a tournament that avoids byes

Question

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:

1. If a stronger team always beats a weaker one, then the strongest team is certain to win.
2. All teams have equal opportunity to win: if every match is a coin toss, then every team would have the same probability of winning.
3. There is an exciting final.
4. There are relatively few matches.
5. 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.

Prior research:

• 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?

Answer 1

• 3 groups of 3 teams
• Round robin play per group
• Winners of the group advance to championship bracket
• Runner-ups placed in wild card group
• Losers in consolation group

Round robin play for both wild card and consolation groups

Winner of runner-up group moves on to championship bracket as fourth team

After that you could play around with it have a losers bracket etc

Just have tiebreakers ready since group play isn’t always clear cut

Answer 2

Pools leading into knockout are perfectly possible with an odd number of teams; you simply have one more team in one of the pools. This provides a direct competition loser, with the remaining places deciding appropriate final positions for all other teams (final is A1 v B1, third/fourth is A2 v B2, etc.).

If so desired, the top four teams can be crossed over in semifinals, or the top eight in quarterfinals. This has a known number of matches, requires only half the number of byes and half the number of rounds of a complete round robin, and provides for finals, but assigns a bye to every team in one pool, plus one additional bye.

---

Specifically for nine teams, divide them by either random sorting into, or by seeding in some manner that gives a balanced set of, three primary pools of three.

Each primary pool plays a complete round robin, giving three winners, three runners-up and three losers who form the corresponding secondary pools. Each secondary pool plays an additional complete round robin. This gives a ranking for each team, who are paired 1 v 2, 3 v 4, etc. in the finals. All teams have played seven matches except for the last-place team who receive the only bye in the finals round.

---

More generally, designing tournaments to satisfy these specific conditions requires significant thought in advance and, when dealing with larger numbers of teams, can involve decision-making that affects the structure in fundamental ways. This is the subject of lengthy documents used by a range of sporting organisations and would be inappropriate to replicate here.