What Is the Correct Way to Seed a Knockout Tournament?

1. WHY ARE TOURNAMENTS SEEDED? In an elimination tournament, also called a knockout tournament, teams (or individual competitors) play head-to-head matches with the loser eliminated from further competition and the winner progressing to the next round of competition. This form of competition is widespread, and includes such popular tournaments as NCAA basketball; professional playoffs in football, baseball, basketball, and hockey; individual competitions such as Olympic boxing and wrestling; and Wimbledon and other tennis tournaments. It is also used in top-level bridge tournaments. In the purest form of knockout tournament, the number of teams entered is precisely a power of two. The NCAA basketball tournament (recently having 64 teams) is an excellent example. But the general method can be used with other numbers by having first round byes as needed to reach a power of two in the second round. For example, in the National Football League, 12 teams qualify for the playoffs. In the first weekend, four division winners are given byes, and the other eight compete so that the field is reduced to eight, the four winners and the four byes. Similarly, in a tennis tournament it would be exceedingly lucky to have the number of entrants just happen to be a power of two. First round byes solve the problem, but they have the disadvantage of leaving some players idle during the first round. In competitive bridge, such as national team titles (Reisinger, Vanderbilt, and Spingold Championships) and the international team championship (the Bermuda Bowl), it is possible to arrange some three-way matches in the first round. This approach has the advantage of not leaving anyone idle during the first round, but of course it just doesn't work for boxing or wrestling. Depending upon the number of original teams, either one or two teams from each three-way match survive. For this approach, suppose the number of teams is n = 2k + r with 0 < r < 2k. When r < 2k-1, the schedule uses r three-way matches (with one survivor from each) and 2k-1 r two-way (head-to-head) matches to reach 2k-1 survivors entering the second round. When 2k-1 < r < 2 k, we need 2k r three-way matches (with two survivors each) and 2r 2k head-to-head matches, again reaching 2k-1 survivors entering the second round. When r is precisely 2k1, we can have 2k 1 three-way matches perfectly covering the entire field. Here we can choose to allow either one or two survivors from each match, so that the second round field size is either 2k-1 or 2k. When time permits, it seems that the larger field of size 2k is preferred.