Single-elimination tournament
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A single-elimination tournament, also called a knockout or sudden death tournament, is a type of tournament where the loser of each match is immediately eliminated from winning the championship or first prize in the event. (However, it does not always mean that the defeated competitor will not participate further in the tournament (in some such tournaments, consolation or "classification" contests are subsequently held among those already defeated to determine the awarding of lesser places, for example, a Third place playoff.)
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[edit] Format
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Often, the number of participants in a single-elimination tournament is fixed as a power of two; for example, the Wimbledon singles championships are tournaments of 128 players. This ensures all competitors will face opponents who have previously played the same number of matches. The full schedule of pairings across all rounds (the bracket) may be allocated before the start of the tournament; or each round may be allocated at the end of the preceding round. Each successive round halves the number of competitors remaining (assuming there are no byes — see below). The round in which only eight remain at the start is generally called the quarter-final round; this is followed by the semi-final round in which only four are left, the two winners of which then meet in the final, or championship round.
In cases where the number of competitive entities at the start of the tournament is not a power of two, some competitors may receive a bye in the first round, which entitles these competitors to advance to the second round automatically without playing. Often, these byes will be awarded to the highest-rated competitors in the event as a reward for some previous accomplishment; indeed, in some American team sports - most notably football - the number of teams qualifying for the postseason tournament will be intentionally set at a number which is not a power of two, in order to provide such an advantage to a high-achieving team in the just-completed regular season.
Multiple rounds of byes are also possible: in the FA Cup, the teams in the top two league divisions enter in the third round "proper" (of eight); the two next-highest divisions' teams will have entered in the first round; lower-division teams in one of 6 preliminary rounds.
When matches are held to determine places or prizes lower than first and second (the loser of the final-round match gaining the latter position), these typically include a match between the losers of the semifinal matches, the winner therein placing third and the loser fourth; sometimes contests are also held among the losers of the quarterfinal matches to determine fifth through eighth places (this is most commonly encountered in the Olympic Games). In one scenario, two "consolation semifinal" matches may be conducted, with the winners of these then facing off to determine fifth and sixth places and the losers playing for seventh and eighth; or some method of ranking the four quarterfinal losers might be employed, in which case only one round of additional matches would be held among them, the two highest-ranked therein then playing for fifth and sixth places and the two lowest for seventh and eighth.
In a tournament where more than two opponents or teams can play together at once, such as in a shootout poker tournament, players are removed when they can no longer play until one player remains from the group. This player moves on to the next round.
SOP for tournament brackets.
N = the number of teams in the tournament
Start with x = 1; Repeat x = 2*x (i.e. keep doubling x) until x exceeds N.
The number of teams with a bye for the play-in bracket, B = x – N;
The number of teams in the play-in bracket, N – B (which also equals 2*N-x) will be an even number. It is possible for B to equal N, in which case there will be no play-in games.
For the play-in bracket, the team in position B+1 plays team N with the winner taking position B+1, team B+2 plays team N-1 with the winner taking position B+2, etc.
After the play-in games are complete, the number of teams left, F = x/2 (N- (2*N-x)/2), will be a number for a normal bracket (2, 4, 8, 16, etc)
In the next round, team 1 plays team F with the winner taking position 1, team 2 play team F-1 with the winner taking position 2, etc.
Subsequent rounds proceed likewise.
[edit] Seeding
Opponents may be allocated randomly; however, since the "luck of the draw" may result in the highest-rated competitors being scheduled to face each other early in the competition, seeding is often used to prevent this. Brackets are set up, so that the top two seeds could not possibly meet until the final round (should either or both advance that far), none of the top four can meet prior to the semifinals, and so on.
Ideally, the brackets would be set up so that the quarterfinal pairings (barring any upsets) would be the 1 seed vs. the 8 seed, 2 vs. 7, 3 vs. 6 and 4 vs. 5; however, this is not the procedure that is followed in most tennis tournaments, where the 1 and 2 seeds are placed in separate brackets, but then the 3 and 4 seeds are assigned to their brackets randomly, and so too are seeds 5 through 8, and so on. This may result in some brackets consisting of stronger players than other brackets, and since only the top 32 players are seeded at all in Tennis Grand Slam tournaments, it is conceivable that the 33rd-best player in a 128-player field could end up playing the top seed in the first round. While this may seem unfair to a casual observer, it should be pointed out that rankings of tennis players are generated by computers, and players tend to change ranking positions very gradually, so that a more equitable method of determining the pairings might result in many of the same head-to-head matchups being repeated over and over again in successive tournaments.
Sometimes the remaining competitors in a single-elimination tournament will be "re-seeded" so that the highest surviving seed is made to play the lowest surviving seed in the next round, the second-highest plays the second-lowest, etc. This may be done after each round, or only at selected intervals. In American team sports, for example, both the NFL and NHL employ this tactic, but the NBA does not (and neither does the NCAA college basketball tournament). The NBA's format calls for the winner of the first-round series between the first and eighth seeds (within each of the two conferences the league has) to face the winner of the first-round series between the fourth and fifth seeds in the next round, even if one or more of the top three seeds had been upset in their first-round series; critics have claimed that this gives a team fighting for the fifth and sixth seeding positions near the end of the regular season an incentive to tank (deliberately lose) games, so as to finish sixth and thus avoid a possible matchup with the top seed until one round later.
[edit] Evaluation
The single-elimination format enables a relatively large number of competitors to participate. There are no "dead" matches (perhaps excluding "classification" matches), and no matches where one competitor has more to play for than the other.
The format is less suited to games where draws are frequent. In chess, each fixture in a single-elimination tournament must be played over multiple matches, because draws are common, and because white has an advantage over black. In football, games ending in a draw may be settled in extra time and eventually by a penalty shootout, viewed by many fans as an unsatisfactory conclusion to a fixture, or by replaying the fixture.
Another perceived disadvantage is that most competitors are eliminated after relatively few games. Variations such as the double-elimination tournament allow competitors a single loss while remaining eligible for overall victory.
