Round-robin tournament
Encyclopedia
A round-robin tournament (or all-play-all tournament) is a competition "in which each contestant meets all other contestants in turn".
A round-robin tournament
with four players is sometimes called a "quad". In croquet
and tennis
clubs in the United Kingdom, a round-robin tournament is often called an American tournament. The term round-robin
is derived from the French term ruban, meaning "ribbon". Over a long period of time, the term was corrupted and idiomized to robin.
and the respective continental tournaments (e.g. UEFA European Championship, CONCACAF Gold Cup
, etc.). There are also round-robin chess
and go tournaments. The World Chess Championship
decided in 2005 and in 2007 on an eight-player double round-robin tournament where each player faces every other player once as white and once as black.
Group tournaments rankings
usually go by number of matches won and drawn, with any of a variety of tiebreaker criteria.
Frequently, pool stages within a wider tournament are conducted on a round-robin basis. Examples with pure round-robin scheduling include the FIFA World Cup
, UEFA European Football Championship
and UEFA Cup (2004–2005) in football, Super Rugby (rugby union
) in the Southern Hemisphere during its past incarnations as Super 12 and Super 14 (but not in the current 15-team format), the Cricket World Cup
, Indian Premier League
Twenty-20 Cricket and many American Football
college conferences, such as the Pacific-10. The group phase of the UEFA Champions League
is contested as a double round-robin, as are most basketball
leagues outside the United States, including the regular-season and Top 16 phases of the Euroleague; the United Football League has used a double round-robin for both its 2009
and 2010
seasons.
, the Football League, the (round-robin) League champions are generally regarded as the "best" team in the land, rather than the (knockout) FA Cup
winners.
The primary disadvantage to a round robin tournament is the time needed to complete it. Unlike a knockout tournament where half of the participants are eliminated after each round, a round robin requires one less than the number of participants. For instance, a 32 team tournament can be completed in just 5 rounds in a knockout format. However if the same teams are put through a round robin it would require 31 rounds to finish. Other issues stem from the difference between the theoretical fairness of the round robin format and practice in a real event. Since the victor is gradually arrived at through multiple rounds of play teams who perform poorly can be eliminated from title contention rather early on, yet they are forced to play out their remaining games. Thus games occur late in competition between competitors with no remaining chance of success. Moreover, some later matches will pair one competitor who has something left to play for against another who does not. This asymmetry means that playing the same opponents is not necessarily equitable: the same opponents in a different order may play harder or easier matches. Teams may also suffer injuries to their star players during competition and thus a matchup may have a completely different complexion than it would have if the order of play was different. There is also no showcase final match.
Further issues arise where a round-robin is used as a qualifying round within a larger tournament. A competitor already qualified for the next stage before its last game may either not try hard (in order to conserve resources for the next phase) or even deliberately lose (if the scheduled next-phase opponent for a lower-placed qualifier is perceived to be easier than for a higher-placed one).
Swiss system tournament
s attempt to combine elements of the round-robin and elimination formats, to provide a reliable champion using fewer rounds than a round-robin, while allowing draws and losses. Also if the tournament is not held at a true neutral location and is instead at a team's home field or away the system of Double Round Robin is an effective equalizer. In this format each team plays each other twice, once away and once at home, in an effort to account for meetings of teams where homefield could sway the results.
is the number of competitors, a pure round robin tournament requires 
games. If 
is even, then in each of 
rounds, 
games can be run in parallel, provided there exist sufficient resources (e.g. courts for a tennis
tournament). If
is odd, there will be 
rounds, each with 
games, and one competitor having no game in that round.
The standard algorithm
for round-robins is to assign each competitor a number, and pair them off in the first round …
Round 1. (1 plays 14, 2 plays 13, ... )
1 2 3 4 5 6 7
14 13 12 11 10 9 8
then fix one competitor (number one in this example) and rotate the others clockwise one position
Round 2. (1 plays 13, 14 plays 12, ... )
1 14 2 3 4 5 6
13 12 11 10 9 8 7
Round 3. (1 plays 12, 13 plays 11, ... )
1 13 14 2 3 4 5
12 11 10 9 8 7 6
until you end up almost back at the initial position
Round 13. (1 plays 2, 3 plays 14, ... )
1 3 4 5 6 7 8
2 14 13 12 11 10 9
If there are an odd number of competitors, a dummy competitor can be added, whose scheduled opponent in a given round does not play and has a bye
. The schedule can therefore be computed as though the dummy were an ordinary player, either fixed or rotating. Instead of rotating one position, any number relatively prime to
will generate a complete schedule.
The upper and lower rows can indicate home/away in sports, white/black in chess
, etc.; to ensure fairness, this must alternate between rounds since competitor 1 is always on the first row. If, say, competitors 3 and 8 were unable to fulfil their fixture in the third round, it would need to be rescheduled outside the other rounds, since both competitors would already be facing other opponents in those rounds. More complex scheduling constraints may require more complex algorithms.
This schedule is applied in chess and draughts tournaments of rapid games, where players physically move round a table.
Alternatively Berger tables, due to its inventor Johann Berger
, are widely used in the planning of tournaments.
Round 1. 1-14 2-13 3-12 4-11 5-10 6-9 7-8
Round 2. 14-8 9-7 10-6 11-5 12-4 13-3 1-2
Round 3. 2-14 3-1 4-13 5-12 6-11 7-10 8-9
…
Round 13. 7-14 8-6 9-5 10-4 11-3 12-2 13-1
This constitutes a schedule where player 14 is fixed or constitutes a bye, and all other players are rotated
positions. This schedule alternates colours and is easily generated manually. To construct the next round, the last player, number 8 in the first round, moves to the head of the table, followed by player 9 against player 7, player 10 against 6, until player 1 against player 2.
This schedule can also be expressed as a (n-1, n-1) table, expressing a round in which players meets each other. For example player 7 plays against player 11 in round 4. If a player meets itself, then this shows a bye or a game against player n. All games in a round constitutes a diagonal in the table.
Terminology
In a single round-robin schedule, each participant plays every other participant once. If each participant plays all others twice, this is frequently called a double round-robin. The term is rarely used when all participants play one another more than twice, and is never used when one participant plays others an unequal number of times (as is the case in almost all of the major United States professional sports leagues - see AFL III and All America Football Conference for exceptions).A round-robin tournament
Tournament
A tournament 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:...
with four players is sometimes called a "quad". In croquet
Croquet
Croquet is a lawn game, played both as a recreational pastime and as a competitive sport. It involves hitting plastic or wooden balls with a mallet through hoops embedded into the grass playing court.-History:...
and tennis
Tennis
Tennis is a sport usually played between two players or between two teams of two players each . Each player uses a racket that is strung to strike a hollow rubber ball covered with felt over a net into the opponent's court. Tennis is an Olympic sport and is played at all levels of society at all...
clubs in the United Kingdom, a round-robin tournament is often called an American tournament. The term round-robin
Round-robin
The term round-robin was originally used to describe a document signed by multiple parties in a circle to make it more difficult to determine the order in which it was signed, thus preventing a ringleader from being identified...
is derived from the French term ruban, meaning "ribbon". Over a long period of time, the term was corrupted and idiomized to robin.
Use
In sports with a large number of competitive matches per season, double round-robins are common. Most association football leagues in the world are organized on a double round-robin basis, in which every team plays all others in its league once at home and once away. This system is also used during qualification for major tournaments such as the FIFA World CupFIFA World Cup
The FIFA World Cup, often simply the World Cup, is an international association football competition contested by the senior men's national teams of the members of Fédération Internationale de Football Association , the sport's global governing body...
and the respective continental tournaments (e.g. UEFA European Championship, CONCACAF Gold Cup
CONCACAF Gold Cup
The CONCACAF Gold Cup is the main association football competition of the men's national football teams governed by CONCACAF, determining the regional champion of North America, Central America, and the Caribbean.The Gold Cup is held every two years and when it does not fall the same year as an...
, etc.). There are also round-robin chess
Chess
Chess is a two-player board game played on a chessboard, a square-checkered board with 64 squares arranged in an eight-by-eight grid. It is one of the world's most popular games, played by millions of people worldwide at home, in clubs, online, by correspondence, and in tournaments.Each player...
and go tournaments. The World Chess Championship
World Chess Championship
The World Chess Championship is played to determine the World Champion in the board game chess. Men and women of any age are eligible to contest this title....
decided in 2005 and in 2007 on an eight-player double round-robin tournament where each player faces every other player once as white and once as black.
Group tournaments rankings
Group tournament ranking system
In a group tournament, unlike a knockout tournament, there is no decisive final match. Instead, all the competitors are ranked by examining the results of all the matches played in the tournament. Points are awarded for each fixture, with competitors ranked based either on total number of points...
usually go by number of matches won and drawn, with any of a variety of tiebreaker criteria.
Frequently, pool stages within a wider tournament are conducted on a round-robin basis. Examples with pure round-robin scheduling include the FIFA World Cup
FIFA World Cup
The FIFA World Cup, often simply the World Cup, is an international association football competition contested by the senior men's national teams of the members of Fédération Internationale de Football Association , the sport's global governing body...
, UEFA European Football Championship
UEFA European Football Championship
The UEFA European Football Championship is the main football competition of the men's national football teams governed by UEFA . Held every four years since 1960, in the even-numbered year between World Cup tournaments, it was originally called the UEFA European Nations Cup, changing to the current...
and UEFA Cup (2004–2005) in football, Super Rugby (rugby union
Rugby union
Rugby union, often simply referred to as rugby, is a full contact team sport which originated in England in the early 19th century. One of the two codes of rugby football, it is based on running with the ball in hand...
) in the Southern Hemisphere during its past incarnations as Super 12 and Super 14 (but not in the current 15-team format), the Cricket World Cup
Cricket World Cup
The ICC Cricket World Cup is the premier international championship of men's One Day International cricket. The event is organised by the sport's governing body, the International Cricket Council , with preliminary qualification rounds leading up to a finals tournament which is held every four years...
, Indian Premier League
Indian Premier League
The Indian Premier League is a professional league for Twenty20 cricket competition in India. It was initiated by the Board of Control for Cricket in India , headquartered in Mumbai, and is supervised by BCCI Vice President Rajeev Shukla, who serves as the league's Chairman and Commissioner...
Twenty-20 Cricket and many American Football
American football
American football is a sport played between two teams of eleven with the objective of scoring points by advancing the ball into the opposing team's end zone. Known in the United States simply as football, it may also be referred to informally as gridiron football. The ball can be advanced by...
college conferences, such as the Pacific-10. The group phase of the UEFA Champions League
UEFA Champions League
The UEFA Champions League, known simply the Champions League and originally known as the European Champion Clubs' Cup or European Cup, is an annual international club football competition organised by the Union of European Football Associations since 1955 for the top football clubs in Europe. It...
is contested as a double round-robin, as are most basketball
Basketball
Basketball is a team sport in which two teams of five players try to score points by throwing or "shooting" a ball through the top of a basketball hoop while following a set of rules...
leagues outside the United States, including the regular-season and Top 16 phases of the Euroleague; the United Football League has used a double round-robin for both its 2009
2009 UFL season
The 2009 United Football League season -- referred to by the professional American football league as the UFL Premiere Season -- was the inaugural season of the United Football League...
and 2010
2010 UFL season
The 2010 United Football League season was the second season of the United Football League. The regular season ran from September 18 to November 20 and featured five teams playing eight games each over a 10-week span. The 2010 season was a relatively competitive one as no team won more than five...
seasons.
Evaluation
In theory a round robin tournament is the fairest way to determine a champion among a known and fixed number of participants. Each player or team has an equal chance against all other participants. The element of luck is seen to be reduced as compared to a knockout system since a few bad performances need not cripple a competitor's chances of ultimate victory. A participant's final record is thus seen to be more accurately represented in the results since it was arrived at over a prolonged period against equal competition. This can also be used to determine which teams are the poorest performers and thus subject to relegation if the format is used in a multi-tiered league. In English footballFootball in England
Association football is a national sport in England, where the first modern set of rules for the code were established in 1863, which were a major influence on the development of the modern Laws of the Game...
, the Football League, the (round-robin) League champions are generally regarded as the "best" team in the land, rather than the (knockout) FA Cup
FA Cup
The Football Association Challenge Cup, commonly known as the FA Cup, is a knockout cup competition in English football and is the oldest association football competition in the world. The "FA Cup" is run by and named after The Football Association and usually refers to the English men's...
winners.
The primary disadvantage to a round robin tournament is the time needed to complete it. Unlike a knockout tournament where half of the participants are eliminated after each round, a round robin requires one less than the number of participants. For instance, a 32 team tournament can be completed in just 5 rounds in a knockout format. However if the same teams are put through a round robin it would require 31 rounds to finish. Other issues stem from the difference between the theoretical fairness of the round robin format and practice in a real event. Since the victor is gradually arrived at through multiple rounds of play teams who perform poorly can be eliminated from title contention rather early on, yet they are forced to play out their remaining games. Thus games occur late in competition between competitors with no remaining chance of success. Moreover, some later matches will pair one competitor who has something left to play for against another who does not. This asymmetry means that playing the same opponents is not necessarily equitable: the same opponents in a different order may play harder or easier matches. Teams may also suffer injuries to their star players during competition and thus a matchup may have a completely different complexion than it would have if the order of play was different. There is also no showcase final match.
Further issues arise where a round-robin is used as a qualifying round within a larger tournament. A competitor already qualified for the next stage before its last game may either not try hard (in order to conserve resources for the next phase) or even deliberately lose (if the scheduled next-phase opponent for a lower-placed qualifier is perceived to be easier than for a higher-placed one).
Swiss system tournament
Swiss system tournament
A Swiss-system tournament is a commonly used type of tournament where players or teams need to be paired to face each other for several rounds of competition. This type of tournament was first used in a Zurich chess tournament in 1895, hence the name "Swiss system". The Swiss system is used when...
s attempt to combine elements of the round-robin and elimination formats, to provide a reliable champion using fewer rounds than a round-robin, while allowing draws and losses. Also if the tournament is not held at a true neutral location and is instead at a team's home field or away the system of Double Round Robin is an effective equalizer. In this format each team plays each other twice, once away and once at home, in an effort to account for meetings of teams where homefield could sway the results.
Scheduling algorithm
If is the number of competitors, a pure round robin tournament requires games. If is even, then in each of rounds, games can be run in parallel, provided there exist sufficient resources (e.g. courts for a tennisTennis
Tennis is a sport usually played between two players or between two teams of two players each . Each player uses a racket that is strung to strike a hollow rubber ball covered with felt over a net into the opponent's court. Tennis is an Olympic sport and is played at all levels of society at all...
tournament). If
is odd, there will be rounds, each with games, and one competitor having no game in that round.The standard algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
for round-robins is to assign each competitor a number, and pair them off in the first round …
Round 1. (1 plays 14, 2 plays 13, ... )
1 2 3 4 5 6 7
14 13 12 11 10 9 8
then fix one competitor (number one in this example) and rotate the others clockwise one position
Round 2. (1 plays 13, 14 plays 12, ... )
1 14 2 3 4 5 6
13 12 11 10 9 8 7
Round 3. (1 plays 12, 13 plays 11, ... )
1 13 14 2 3 4 5
12 11 10 9 8 7 6
until you end up almost back at the initial position
Round 13. (1 plays 2, 3 plays 14, ... )
1 3 4 5 6 7 8
2 14 13 12 11 10 9
If there are an odd number of competitors, a dummy competitor can be added, whose scheduled opponent in a given round does not play and has a bye
Bye (sports)
A bye, in sports and other competitive activities, most commonly refers to the practice of allowing a player or team to advance to the next round of a playoff tournament without playing...
. The schedule can therefore be computed as though the dummy were an ordinary player, either fixed or rotating. Instead of rotating one position, any number relatively prime to
will generate a complete schedule.The upper and lower rows can indicate home/away in sports, white/black in chess
Chess
Chess is a two-player board game played on a chessboard, a square-checkered board with 64 squares arranged in an eight-by-eight grid. It is one of the world's most popular games, played by millions of people worldwide at home, in clubs, online, by correspondence, and in tournaments.Each player...
, etc.; to ensure fairness, this must alternate between rounds since competitor 1 is always on the first row. If, say, competitors 3 and 8 were unable to fulfil their fixture in the third round, it would need to be rescheduled outside the other rounds, since both competitors would already be facing other opponents in those rounds. More complex scheduling constraints may require more complex algorithms.
This schedule is applied in chess and draughts tournaments of rapid games, where players physically move round a table.
Alternatively Berger tables, due to its inventor Johann Berger
Johann Berger
Johann Nepomuk Berger was an Austrian chess master, theorist, endgame study composer, author and editor.In September 1870, he won the first tournament in the Austro-Hungarian Empire at Graz...
, are widely used in the planning of tournaments.
Round 1. 1-14 2-13 3-12 4-11 5-10 6-9 7-8
Round 2. 14-8 9-7 10-6 11-5 12-4 13-3 1-2
Round 3. 2-14 3-1 4-13 5-12 6-11 7-10 8-9
…
Round 13. 7-14 8-6 9-5 10-4 11-3 12-2 13-1
This constitutes a schedule where player 14 is fixed or constitutes a bye, and all other players are rotated
positions. This schedule alternates colours and is easily generated manually. To construct the next round, the last player, number 8 in the first round, moves to the head of the table, followed by player 9 against player 7, player 10 against 6, until player 1 against player 2.This schedule can also be expressed as a (n-1, n-1) table, expressing a round in which players meets each other. For example player 7 plays against player 11 in round 4. If a player meets itself, then this shows a bye or a game against player n. All games in a round constitutes a diagonal in the table.
| × | 02 | 03 | 04 | 05 | 06 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | ||||||||||||
| 2 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | ||||||||||||
| 3 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | ||||||||||||
| 4 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | ||||||||||||
| 5 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | ||||||||||||
| 6 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | ||||||||||||
| 7 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | ||||||||||||
| 8 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | ||||||||||||
| 9 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | ||||||||||||
| 10 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | ||||||||||||
| 11 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | ||||||||||||
| 12 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | ||||||||||||
| 13 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
See also
- Group tournament ranking systemGroup tournament ranking systemIn a group tournament, unlike a knockout tournament, there is no decisive final match. Instead, all the competitors are ranked by examining the results of all the matches played in the tournament. Points are awarded for each fixture, with competitors ranked based either on total number of points...
, including details of tie-breaking systems - Tournament (graph theory)Tournament (graph theory)A tournament is a directed graph obtained by assigning a direction for each edge in an undirected complete graph. That is, it is a directed graph in which every pair of vertices is connected by a single directed edge....
, mathematical model of a round-robin tournament - Other tournament systems:
- Swiss system tournamentSwiss system tournamentA Swiss-system tournament is a commonly used type of tournament where players or teams need to be paired to face each other for several rounds of competition. This type of tournament was first used in a Zurich chess tournament in 1895, hence the name "Swiss system". The Swiss system is used when...
- Single-elimination tournamentSingle-elimination tournamentA single-elimination tournament, also called a knockout, cup or sudden death tournament, is a type of elimination tournament where the loser of each match or bracket is immediately eliminated from winning the championship or first prize in the event...
- Double-elimination tournamentDouble-elimination tournamentA double-elimination tournament is a type of elimination tournament competition in which a participant ceases to be eligible to win the tournament's championship upon having lost two games or matches...
- Swiss system tournament
- Chess:
- List of round-robin chess tournaments
- Scheveningen systemScheveningen systemThe Scheveningen system is a method of organizing a chess match between two teams. Each player on one team plays each player on the other team . The team with the highest number of games won is the winner. This system is a popular way to create title norm opportunities.The system was first used...
, where each member of one team plays each member of the other
External links
- Round Robin Discussion Board link to a discussion community and schedules (balanced,cyclic,first-fit,whist).
- Round Robin Schedule Generator link to an online tool to quickly create a round robin schedule, and export to MS Excel.
- Online free league-creation system: challonge.com
- Promoting Go in the UK and beyond