See studies by F. H. Cripps-Day (1918) and R. W. Barber and J. Barker (1989, repr. 2000).
A tournament (IPA [tərnəmənt]) is a competition involving a relatively large number of competitors, all participating in a sport or game. More specifically, the term may be used in either of two overlapping senses:
These two senses are distinct. All golf tournaments meet the first definition, but while match play tournaments meet the second, stroke play tournaments do not, since there are no distinct matches within the tournament. In contrast, football (soccer) leagues like the Premier League are tournaments in the second sense, but not the first, having matches spread across many stadia over a period of up to a year. Many tournaments meet both definitions; for example, the Wimbledon tennis championship.
A tournament-match (or tie or fixture or heat) may involve multiple game-matches (or rubbers or legs) between the competitors. For example, in the Davis Cup tennis tournament, a tie between two nations involves five rubbers between the nations' players. The team that wins the most rubbers wins the tie. In the later rounds of UEFA Champions League of football (soccer), each fixture is played over two legs. The scores of each leg are added, and the team with the higher aggregate score wins the fixture, with away goals used as a tiebreaker and a penalty shootout if away goals cannot determine a winner.
In a single-elimination tournament, only the top-ranked competitors in a fixture progress; in 2-competitor games, only the winner progresses. All other competitors are eliminated. This ensures a winner is decided with the minimum number of fixtures. However, most competitors will be eliminated after relatively few matches; a single bad or unlucky performance can nullify many preceding excellent ones. Some single-elimination tournaments such as NBA use a multiple-game format, in which teams would play each other more than one game (e.g. best-of-seven series in NBA) in order to determine who is the winner of this round. Other knockout formats provide a "second chance" for some or all losers.
A double-elimination tournament may be used in 2-competitor games to allow each competitor a single loss without being eliminated from the tournament. All losers from the main bracket enter a losers' bracket, the winner of which plays off against the main bracket's winner.
Some formats allow losers to play extra rounds before re-entering the main competition in a later round. Rowing regattas often have repechage rounds for the "fastest losers" from the heats. The winners of these progress, but are at a disadvantage in later rounds owing to the extra effort expended during the repechage.
In the playoffs of the Australian Football League, the teams with the best record before the playoffs are allowed to lose a game without being eliminated, whereas the lesser qualifiers are not. In athletics meetings, fastest losers may progress in a running event held over several rounds; e.g. the qualifiers for a later round might be the first 4 from each of 6 heats, plus the 8 fastest losers from among the remaining runners.
An extreme form of the knockout tournament is the step-ladder format where the strongest team is assured of a berth at the final round while the next strongest teams are given byes according to their strength/seeds; for example, in a four team tournament, the fourth and third seed figure in the first round, then the winner goes to the semifinals against the second seed, while the survivor faces the first seed at the final.
In a round-robin tournament, each competitor plays all the others an equal number of times, once in a single round-robin tournament and twice in a double round-robin tournament. This is often seen as producing the most reliable rankings. However, for large numbers of competitors it may require an unfeasibly large number of rounds. A Swiss system tournament attempts to determine a winner reliably, based on a smaller number of fixtures. Fixtures are scheduled one round at a time; a competitor will play another who has a similar record in previous rounds of the tournament. This allows the top (and bottom) competitors to be determined with fewer rounds than a round-robin, though the middle rankings are unreliable.
There may be other considerations besides reliability of rankings. In some professional team sports, weaker teams are given an easier slate of fixtures as a form of handicapping. This occurs in the National Football League in the USA. Sometimes schedules are weighted in favour of local derbies or other traditional rivalries. For example, NFL teams play two games against each of the other three teams in their division, one game against half of the other twelve teams in their conference, and one game against a quarter of the sixteen teams in the other conference. American sports are also unusual in providing fixtures between competitors who are, for ranking purposes, in different groups.
In 2-competitor games where ties are rare or impossible, competitors are typically ranked by number of wins, with ties counting half; each competitors' listings are usually ordered Wins-Losses(-Ties). Where draws are more common, this may be 2 points for a win and 1 for a draw, which is mathematically equivalent but avoids having too many half-points in the listings. These are usually ordered Wins-Draws-Losses. If there are more than 2 competitors per fixture, points may be ordinal (for example, 3 for first, 2 for second, 1 for third).
Sometimes, results from an earlier phase are carried over into a later phase. In the Cricket World Cup, the second stage, known as the Super Eight since 2007 and before that the Super Six, features two teams from each of four preliminary groups (previously three teams from two preliminary groups), who do not replay the teams they have already played, but instead reuse the original results in the new league table. Formerly in the Swiss Football League, teams played a double round-robin, at which point they were split into a top "championship" group and a bottom "relegation" group; each played a separate double round-robin, with results of all 32 matches counting for ranking each group. A similar system is also used in the Scottish Premier League since 2000. After 33 games, when every club has played every other club three times, the division is split into two halves. Clubs play a further 5 matches, against the teams in their half of the division. This can (and often does) result in the team placed 7th having a higher points total than the team placed 6th, because their final 5 games are considerably easier.
The top Slovenian basketball league has a unique system. In its first phase, 12 of the league's 13 clubs compete in a full home-and-away season, with the country's representative in the Euroleague (an elite pan-European club competition) exempt. The league then splits. The top seven teams are joined by the Euroleague representative for a second home-and-away season, with no results carrying over from the first phase. These eight teams compete for four spots in a final playoff. The bottom five teams play their own home-and-away league, but their previous results do carry over. These teams are competing to avoid relegation, with the bottom team automatically relegated and the second-from-bottom team forced to play a mini-league with the second- and third-place teams from the second level for a place in the top league.
In a "shootout" tournament, players do not change tables until every table has been reduced to one player.