Single-elimination tournament

A single-elimination, knockout, or sudden death tournament is a type of elimination tournament where the loser of each match-up is immediately eliminated from the tournament. Each winner will play another in the next round, until the final match-up, whose winner becomes the tournament champion. Each match-up may be a single match or several, for example two-legged ties in European football or best-of series in American pro sports. Defeated competitors may play no further part after losing, or may participate in "consolation" or "classification" matches against other losers to determine the lower final rankings; for example, a third place playoff between losing semi-finalists. In a shootout poker tournament, there are more than two players competing at each table, and sometimes more than one progressing to the next round. Some competitions are held with a pure single-elimination tournament system. Others have many phases, with the last being a single-elimination final stage, often called playoffs.

Nomenclature

In English, the round in which only eight competitors remain is generally called (with or without hyphenation) 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.

The round before the quarterfinals has multiple designations. Often it's called the round of sixteen, last sixteen, or (in South Asia) pre quarter-finals. In many other languages the term used to describe these eight matches translates to eighth-final in four European languages : ("huitième de finale" in French, ' "octavos de final" in Spanish, "achtelfinale" in German, and "ottavi di finale" in Italian), though this term is rare in English itself.

The round before the "round of sixteen" sometimes called "round of thirty-two" in English, in some languages, its translation to English is "sixteenth final".

Earlier rounds are typically numbered counting forwards from the first round, or by the number of remaining competitors. If some competitors get a bye, the round at which they enter may be named the first round, with the earlier matches called a preliminary round, qualifying round, or the play-in games".

Examples of the diverse names given to concurrent rounds in various select disciplines:

By competitors	Fraction of final	Grand Slam tennis^[1]	FA Cup football	Coupe de France^[2]	NCAA Men's Basketball	North American Debating Ch'ship
Round of 2	Final	Final	Final	Final	National Championship	Final
Round of 4	Semifinals	Semifinals	Semi-finals	Semifinals	Final Four (National semifinals)^{[t 1]}	Semifinals
Round of 8	Quarterfinals	Quarterfinals	quarter-finals^{[t 2]}	Quarterfinals	Elite Eight (Regional finals)^{[t 3]}	Quarterfinals
Round of 16	Eighth-finals	4th round (Wimbledon^[4]) Round of 16 (US Open^[5])	5th round^{[t 4]}	8th-finals	Sweet Sixteen (Regional semifinals)^{[t 5]}	Octofinals
Round of 32	16th-finals	3rd round	4th round^{[t 4]}	16th-finals	3rd/2nd round^{[t 6]}^{[t 7]}	Double-octofinals or Decimosexto-finals
Round of 64	32nd-finals	2nd round	3rd round^{[t 4]}	32nd-finals	2nd/1st round^{[t 6]}^{[t 7]}	Triple-octofinals
Round of 128	64th-finals	1st round	2nd round^{[t 8]}^{[t 4]}	8th qualifying round^{[t 9]}	First Four^{[t 6]}	Quad-octofinals^{[t 10]}

Notes:

^ The NCAA also uses the "Final Four" terminology in the Division I women's tournament, as well as the Division III tournaments for both men and women. In Division II for both sexes, this round is called the "semifinals"; both championship events in that division consist of eight teams instead of four.
^ The quarter-finals were called the "sixth round" until 2016–17, the first in which replays were discontinued for this round.^[3]
^ In the Division II men's and women's tournaments, the Elite Eight is the championship event, with all qualifying teams participating at a single site. The NCAA does not use "Elite Eight" in Division III, simply calling this round the "regional finals".
^ ^a ^b ^c ^d The first to fifth rounds are often called the "first/second/etc. round proper", to distinguish them from the "first/second/etc. qualifying round".
^ The NCAA only uses the term "Sweet Sixteen" in the Division I tournaments.
^ ^a ^b ^c Starting in 2011, 68 teams played in the Championship, with four play-in games, nicknamed the First Four, before the top 60 teams enter at the round of 64. (From 2001 to 2010, there was a single "Opening Round" game before the round of 64.) The NCAA originally called the First Four the first round, making the rounds of 64 and 32 the second and third rounds respectively; in 2014 it announced that from 2016 it would revert to calling the rounds of 64 and 32 the first and second rounds.^[6]
^ ^a ^b Since the NCAA Division I women's tournament expanded beyond 32 teams in 1986, the round of 32 has always been called the "second round", and the preceding round the "first round". The women's tournament has involved 64 teams since 1994, but has never expanded beyond that number.
^ The FA Cup 2nd round involves 40 teams, of which 20 qualify for the 3rd round, to which the top 44 teams will have received byes.
^ The 8th qualifying round involves 88 teams, of which 44 qualify for the 32nd-finals, to which the top 20 teams will have received byes.
^ The number of eligible teams is typically less than 128, but more than 64, so not all teams play this round.

Example

The final three rounds of the 2014 Australian Open – Women's Singles knock-out tournament:

Quarterfinals					Semifinals				Final


14	Ana Ivanovic	7	5	2

30	Eugenie Bouchard	5	7	6
					30	Eugenie Bouchard	2	4

					4	Li Na	6	6
4	Li Na	6	6

28	Flavia Pennetta	2	2
									4	Li Na	7⁷	6

									20	Dominika Cibulková	6³	0
11	Simona Halep	3	0

20	Dominika Cibulková	6	6
					20	Dominika Cibulková	6	6

					5	Agnieszka Radwańska	1	2
5	Agnieszka Radwańska	6	5	6

2	Victoria Azarenka	1	7	0

Classification

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 called third place playoffs, the winner therein placing third and the loser fourth. Many Olympic single-elimination tournaments feature the bronze medal match if they do not award bronze medals to both losing semifinalists. The FIFA World Cup has long featured the third place match (since 1934), though the UEFA Euro has not held one since the 1980 edition.

Sometimes, contests are also held among the losers of the quarterfinal matches to determine fifth to eighth places – this is most commonly encountered in the Olympic Games, with the exception of boxing, where both fighters are deemed to be third place. 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; those are used often in qualifying tournaments where only the top five teams advance to the next round; 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.

The number of distinct ways of arranging a single-elimination tournament (as an abstract structure, prior to seeding the players into the tournament) is given by the Wedderburn–Etherington numbers.^[7] Thus, for instance, there are three different arrangements for five players:

The players may be divided into brackets of two and three players, the winners of which meet in the final game
The bottom four players may play a two-round tournament, the winner of which plays the top player
The bottom two players may meet, after which each subsequent game pairs the winner of the previous game with the next player

However, the number of arrangements grows quickly for larger numbers of players and not all of them are commonly used.

Seeding

Opponents may be allocated randomly (such as in the FA Cup); 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 both advance that far), none of the top four can meet prior to the semifinals, and so on. If no seeding is used, the tournament is called a random knockout tournament.

One version of seeding is where brackets are 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. A good example of this occurring was when World No. 33 Florian Mayer was drawn against (and eventually defeated by) then-World No. 1 Novak Djokovic in the first round of the 2013 Wimbledon Championships,^[8] in what was also a rematch of a quarterfinal from the previous year.^[9] 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, the NFL and WNBA employ this tactic, but MLS and the NBA do not (and neither does the NCAA college basketball tournament). MLB does not have enough teams (10) in its playoff tournament where re-seeding would make a large difference in the matchups; only the NFL's at the minimum, which is six from each conference [or league in MLB] for a total of 12). 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. MLS's format is identical, except that the top-seed gets a conference-quarterfinal bye to play 4th-5th seed winner.

In some situations, a seeding restriction may be implemented; from 1975 until 1989 in the NFL, and from 1994 until 2011 in MLB there was a rule where at the conference or league semifinal, should the top seed and last seed (wild card) be from the same division, they could not play each other; in that case, the top seed plays the worst division champion; the second-best division champion plays the wild card team. This is due to the scheduling employed for the regular season, in which a team faces any given divisional opponent more often than any given non-divisional opponent – the tournament favors matchups that took place fewer times in the regular season (or did not take place, in some cases).^{[citation needed]}

In international fencing competitions, it is common to have a group stage. Participants are divided in groups of 6–7 fencers who play a round-robin tournament, and a ranking is calculated from the consolidated group results. Single elimination is seeded from this ranking.

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 association football, games ending in a draw may be settled in extra time and eventually by a penalty shootout 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. However, losing one game requires the competitor to win more games in order to win the tournament.

In a random knockout tournament (single-elimination without any seeding), awarding the second place to the loser of the final is unjustified: any of the competitors knocked out by the tournament winner might have been the second strongest one, but they never got the chance to play against the losing finalist. In general, it is only fair to use a single-elimination tournament to determine first place. To fairly determine lower places requires some form of round-robin in which each player/team gets the opportunity to face every other player/team.

Also, if the competitors' performance is variable, that is, it depends on a small, varying factor in addition to the actual strength of the competitors, then not only will it become less likely that the strongest competitor actually wins the tournament, in addition the seeding done by the tournament organizers will play a major part in deciding the winner.^[10]^[11] As a random factor is always present in a real-world competition, this might easily cause accusations of unfairness.

Other tournament systems

Variations of the single-elimination tournament include:

Double-elimination tournament
McIntyre System, a group of tournament formats that combine features of single- and double-elimination tournaments. Varieties of this system include:
- Page playoff system (four teams)
- Top five play-offs
- Top six play-offs
- McIntyre Final Eight System
- Super League play-offs, which formerly used a McIntyre Final Eight variant
- AFL final eight system, another variant of the McIntyre Final Eight, currently used by the Australian Football League

Other common tournament types include:

Round-robin tournament
Swiss-system tournament

^ In singles only (the other disciplines have less number of rounds)
^ "Coupe de France : football, résultats, calendrier, reportage, photos" (in French). French Football Federation. Archived from the original on 19 July 2012. Retrieved 16 July 2012.
^ "Premier League clubs want the FA Cup moved to midweek and replays scrapped". talkSPORT. 31 May 2018 [2017]. Retrieved 31 March 2019.; Emirates FA Cup (17 February 2017). "We do now! The sixth round has been renamed as the quarter-finals from this season". @EmiratesFACup. Twitter. Retrieved 31 March 2019.
^ "Gentlemen's Singles – 2012 Official Site by IBM". Wimbledon Championships Website. Retrieved 16 July 2012.
^ "2012 Tournament Schedule". US Open. Archived from the original on 17 July 2012. Retrieved 16 July 2012.
^ Cary, Tim (21 November 2014). "What's in a Name? March Madness First Round Is the 'First Round' Again". The Cheat Sheet. Retrieved 6 June 2015.
^ Maurer, Willi (1975), "On most effective tournament plans with fewer games than competitors", The Annals of Statistics, 3: 717–727, doi:10.1214/aos/1176343135, JSTOR 2958441, MR 0371712.
^ Wimbledon: Novak vs Mayer in R1; Andy, Roger, Rafa all in bottom half, Novak Djokovic official website, 21 June 2013
^ Wimbledon 2013: Men's matches to watch out for, The Roar, 22 June 2013
^ Ryvkin, Dmitry (March 2005). "The Predictive Power of Noisy Elimination Tournaments" (PDF). CERGE-EI. Retrieved 21 October 2010.
^ Kim, Michael P.; Suksompong, Warut; Vassilevska Williams, Virginia (2017). "Who Can Win a Single-Elimination Tournament?". SIAM Journal on Discrete Mathematics. 31 (3): 1751–1764. arXiv:1511.08416. doi:10.1137/16M1061783.

[3] The NCAA also uses the "Final Four" terminology in the Division I women's tournament, as well as the Division III tournaments for both men and women. In Division II for both sexes, this round is called the "semifinals"; both championship events in that division consist of eight teams instead of four.

[5] The quarter-finals were called the "sixth round" until 2016–17, the first in which replays were discontinued for this round.^[3]

[6] In the Division II men's and women's tournaments, the Elite Eight is the championship event, with all qualifying teams participating at a single site. The NCAA does not use "Elite Eight" in Division III, simply calling this round the "regional finals".

[proper-9] The first to fifth rounds are often called the "first/second/etc. round proper", to distinguish them from the "first/second/etc. qualifying round".

[10] The NCAA only uses the term "Sweet Sixteen" in the Division I tournaments.

[ncaabb-11] Starting in 2011, 68 teams played in the Championship, with four play-in games, nicknamed the First Four, before the top 60 teams enter at the round of 64. (From 2001 to 2010, there was a single "Opening Round" game before the round of 64.) The NCAA originally called the First Four the first round, making the rounds of 64 and 32 the second and third rounds respectively; in 2014 it announced that from 2016 it would revert to calling the rounds of 64 and 32 the first and second rounds.^[6]

[NCAAW-12] Since the NCAA Division I women's tournament expanded beyond 32 teams in 1986, the round of 32 has always been called the "second round", and the preceding round the "first round". The women's tournament has involved 64 teams since 1994, but has never expanded beyond that number.

[13] The FA Cup 2nd round involves 40 teams, of which 20 qualify for the 3rd round, to which the top 44 teams will have received byes.

[14] The 8th qualifying round involves 88 teams, of which 44 qualify for the 32nd-finals, to which the top 20 teams will have received byes.

[16] The number of eligible teams is typically less than 128, but more than 64, so not all teams play this round.

[1] In singles only (the other disciplines have less number of rounds)

[2] "Coupe de France : football, résultats, calendrier, reportage, photos" (in French). French Football Federation. Archived from the original on 19 July 2012. Retrieved 16 July 2012.

[4] "Premier League clubs want the FA Cup moved to midweek and replays scrapped". talkSPORT. 31 May 2018 [2017]. Retrieved 31 March 2019.; Emirates FA Cup (17 February 2017). "We do now! The sixth round has been renamed as the quarter-finals from this season". @EmiratesFACup. Twitter. Retrieved 31 March 2019.

[7] "Gentlemen's Singles – 2012 Official Site by IBM". Wimbledon Championships Website. Retrieved 16 July 2012.

[8] "2012 Tournament Schedule". US Open. Archived from the original on 17 July 2012. Retrieved 16 July 2012.

[15] Cary, Tim (21 November 2014). "What's in a Name? March Madness First Round Is the 'First Round' Again". The Cheat Sheet. Retrieved 6 June 2015.

[17] Maurer, Willi (1975), "On most effective tournament plans with fewer games than competitors", The Annals of Statistics, 3: 717–727, doi:10.1214/aos/1176343135, JSTOR 2958441, MR 0371712.

[18] Wimbledon: Novak vs Mayer in R1; Andy, Roger, Rafa all in bottom half, Novak Djokovic official website, 21 June 2013

[19] Wimbledon 2013: Men's matches to watch out for, The Roar, 22 June 2013

[ryv-20] Ryvkin, Dmitry (March 2005). "The Predictive Power of Noisy Elimination Tournaments" (PDF). CERGE-EI. Retrieved 21 October 2010.

[kim-21] Kim, Michael P.; Suksompong, Warut; Vassilevska Williams, Virginia (2017). "Who Can Win a Single-Elimination Tournament?". SIAM Journal on Discrete Mathematics. 31 (3): 1751–1764. arXiv:1511.08416. doi:10.1137/16M1061783.

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