Round-robin tournament

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A round-robin tournament (or all-play-all tournament) is a competition in which each contestant meets all other contestants in turn. [1] [2] A round-robin contrasts with an elimination tournament, in which participants are eliminated after a certain number of losses.

Competition arises whenever at least two parties strive for a goal which cannot be shared: where one's gain is the other's loss.

A contestant is someone who takes part in a competition, usually a professional competition or a game show on television. The participants competing against each other have to go through rounds. The winners may have to compete in later stages or rounds until there is just one winner.

Terminology

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. [3] [4]

A ribbon or riband is a thin band of material, typically cloth but also plastic or sometimes metal, used primarily as decorative binding and tying. Cloth ribbons are made of natural materials such as silk, velvet, cotton, and jute and of synthetic materials, such as polyester, nylon, and polypropylene. Ribbon is used for innumerable useful, ornamental, and symbolic purposes. Cultures around the world use ribbon in their hair, around the body, and as ornamentation on non-human animals, buildings, and packaging. Some popular fabrics used to make ribbons are satin, organza, sheer, silk, velvet, and grosgrain.

Folk etymology or reanalysis – sometimes called pseudo-etymology, popular etymology, analogical reformation, or etymological reinterpretation – is a change in a word or phrase resulting from the replacement of an unfamiliar form by a more familiar one. The form or the meaning of an archaic, foreign, or otherwise unfamiliar word is reanalyzed as resembling more familiar words or morphemes. Rebracketing is a form of folk etymology in which a word is broken down or "bracketed" into a new set of supposed elements. Back-formation, creating a new word by removing or changing parts of an existing word, is often based on folk etymology.

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, [1] 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 (1940–41) and All-America Football Conference for exceptions).

American Football League, also known retrospectively as the AFL III to distinguish it from earlier organizations of that name, was a major professional American football league that operated from 1940–1941. It was created when three teams, the original Cincinnati Bengals, the Columbus Bullies, and the Milwaukee Chiefs, were lured away from the minor-league American Professional Football Association and joined three new franchises in Boston, Buffalo, and New York City in a new league. It competed against the National Football League (NFL), the oldest existing professional football league, established 1920 and reorganized 1922.

The All-America Football Conference (AAFC) was a professional American football league that challenged the established National Football League (NFL) from 1946 to 1949. One of the NFL's most formidable challengers, the AAFC attracted many of the nation's best players, and introduced many lasting innovations to the game. However, the AAFC was ultimately unable to sustain itself in competition with the NFL. After its folding, three of its teams were admitted to the NFL: the San Francisco 49ers, the Cleveland Browns and the original Baltimore Colts.

In the United Kingdom, a round-robin tournament is often called an American tournament in sports such as tennis or billiards which usually have knockout tournaments. [5] [6] [7] In Italian it is called girone all'italiana (literally "Italian-style circuit"). In Serbian it is called the Berger system (Бергеров систем, Bergerov sistem), after chess player Johann Berger. A round-robin tournament with four players is sometimes called "quad" or "foursome". [8]

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.

Serbian is the standardized variety of the Serbo-Croatian language mainly used by Serbs. It is the official language of Serbia, co-official in the territory of Kosovo, and one of the three official languages of Bosnia and Herzegovina. In addition, it is a recognized minority language in Montenegro, where it is spoken by the relative majority of the population, as well as in Croatia, North Macedonia, Romania, Hungary, Slovakia, and the Czech Republic.

Johann Nepomuk Berger was an Austrian chess master, theorist, endgame study composer, author and editor.

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 in qualification for major tournaments such as the FIFA World Cup and the continental tournaments (e.g. UEFA European Championship, CONCACAF Gold Cup). There are also round-robin bridge, chess, draughts, go, curling and Scrabble 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.

Association football, more commonly known as football or soccer, is a team sport played with a spherical ball between two teams of eleven players. It is played by 250 million players in over 200 countries and dependencies, making it the world's most popular sport. The game is played on a rectangular field called a pitch with a goal at each end. The object of the game is to score by moving the ball beyond the goal line into the opposing goal.

The FIFA World Cup, often simply called the World Cup, is an international association football competition contested by the senior men's national teams of the members of the Fédération Internationale de Football Association (FIFA), the sport's global governing body. The championship has been awarded every four years since the inaugural tournament in 1930, except in 1942 and 1946 when it was not held because of the Second World War. The current champion is France, which won its second title at the 2018 tournament in Russia.

The UEFA European Championship is the primary association football competition contested by the senior men's national teams of the members of the Union of European Football Associations (UEFA), determining the continental champion of Europe. 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 name in 1968. Starting with the 1996 tournament, specific championships are often referred to in the form "UEFA Euro [year]"; this format has since been retroactively applied to earlier tournaments.

Group tournaments rankings usually go by number of matches won and drawn, with any of a variety of tiebreaker criteria.

In a group tournament, unlike a knockout tournament, there is no scheduled decisive final match. Instead, all the competitors are ranked by examining the results of all the matches played in the tournament. Typically, points are awarded for each match, with competitors ranked based either on total number of points or average points per match. Usually each competitor finishes with an equal number of matches, in which case rankings by total points and by average points are equivalent at the end of the tournament, though not necessarily while it is in progress. Examples with unequal numbers of matches include the 1895 County Championship in English cricket, and the U.S. National Football League prior to 1972, when tie games were excluded from the winning percentage used for regular-season standings.

Frequently, pool stages within a wider tournament are conducted on a round-robin basis. Examples with single round-robin scheduling include the FIFA World Cup, UEFA European Football Championship, and UEFA Cup (2004–2009) in football, Super Rugby (rugby union) in the Southern Hemisphere during its past iterations as Super 12 and Super 14 (but not in its later 15- and 18-team formats), the Cricket World Cup along Pakistan Super League & Indian Premier League, the two major Twenty-20 Cricket tournaments, [The International (Dota 2)]] and many American Football college conferences, such as the Big 12 (which currently has 10 members). The group phases of the UEFA Champions League and Copa Libertadores de América are 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.

Season ending tennis tournaments also use a round robin format prior to the semi on stages

Evaluation

The champion, in a round-robin tournament, is the contestant that wins the most games.

In theory, a round-robin tournament is the fairest way to determine the champion from among a known and fixed number of contestants. Each contestant, whether player or team, has equal chances against all other opponents because there is no prior seeding of contestants that will preclude a match between any given pair. The element of luck is seen to be reduced as compared to a knockout system since one or two bad performances need not ruin a competitor's chance of ultimate victory. Final records of participants are more accurate, in the sense that they represent the results over a longer period against the same opposition.

The system is also better for ranking all participants, not just determining the winner. This is helpful to determine the final rank of all competitors, from strongest to weakest, for purposes of qualification for another stage or competition as well as for prize money.

In team sport the (round-robin) major league champions are generally regarded as the "best" team in the land, rather than the (elimination) cup winners.

Moreover, in tournaments such as the FIFA or ICC world cups, a first round stage consisting of a number of mini round robins between groups of 4 teams guards against the possibility of a team travelling possibly thousands of miles only to be eliminated after just one poor performance in a straight knockout system. The top one, two, or occasionally three teams in these groups then proceed to a straight knockout stage for the remainder of the tournament.

In the circle of death (see below), it is possible that no champion emerges from a round-robin tournament, even if there is no draw. However, most sports have tie-breaker systems which resolve this.

Round-robins can suffer from being too long compared to other tournament types, and with later scheduled games potentially not having any substantial meaning. They may also require tiebreaking procedures.

Swiss system tournaments attempt to combine elements of the round-robin and elimination formats, to provide a worthy champion using fewer rounds than a round-robin, while allowing draws and losses.

Tournament length

The main disadvantage of 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 round less than the number of participants multiplied by half the number of participants if the number of participants is even, and as many rounds as participants if the number of participants is odd. For instance, a tournament of 16 teams can be completed in just 4 rounds (i.e. 15 matches) in a knockout (single elimination) format; a double elimination tournament format requires 30 (or 31) matches, but a round-robin would require 15 rounds (i.e. 120 matches) to finish if each competitor faces each other once. 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, who might have been quickly eliminated from title contention, are forced to play out their remaining games. Thus games are played late in the 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. It may also be possible for a competitor to play the strongest opponents in a round robin in quick succession while others play them intermittently with weaker opposition. This asymmetry means that playing the same opponents is not necessarily completely equitable: the same opponents in a different order may play harder or easier matches, while other teams are presented with more adversity during periods of the competition.[ clarification needed ] There is also no scheduled showcase final match. Only by coincidence would two competitors meet in the last match of the tournament where the result of that match determined the championship. A notable instance of such an event was the May 26, 1989 match between Arsenal and Liverpool.

Qualified teams

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). Four pairs in the 2012 Olympics Women's doubles badminton, having qualified for the next round, were disqualified for attempting to lose in the round robin stage to avoid compatriots and better ranked opponents. [9] The round robin stage at the Olympics were a new introduction and potential problems were readily known prior to the tournament.

Circle of death

Another disadvantage, especially in smaller round-robins, is the "circle of death," where teams cannot be separated on a head-to-head record. In a three-team round-robin, where A defeats B, B defeats C, and C defeats A, all three competitors will have a record of one win and one loss, and a tiebreaker will need to be used to separate the teams. [10] This famously happened during the 1994 FIFA World Cup Group E, where all four teams finished with a record of one win, one draw, and one loss.

Scheduling algorithm

If ${\displaystyle n}$ is the number of competitors, a pure round robin tournament requires ${\displaystyle {\begin{matrix}{\frac {n}{2}}\end{matrix}}(n-1)}$ games. If ${\displaystyle n}$ is even, then in each of ${\displaystyle (n-1)}$ rounds, ${\displaystyle {\begin{matrix}{\frac {n}{2}}\end{matrix}}}$ games can be run concurrently, provided there exist sufficient resources (e.g. courts for a tennis tournament). If ${\displaystyle n}$ is odd, there will be ${\displaystyle n}$ rounds, each with ${\displaystyle {\begin{matrix}{\frac {n-1}{2}}\end{matrix}}}$ games, and one competitor having no game in that round.

The circle method is the standard algorithm to create a schedule for a round-robin tournament. All competitors are assigned to numbers, and then paired in the first round:

 1 2 3 4 5 6 7 14 13 12 11 10 9 8

Next, one of the contributors in the first or last column of the table is fixed (number one in this example) and the others rotated clockwise one position

 1 14 2 3 4 5 6 13 12 11 10 9 8 7
 1 13 14 2 3 4 5 12 11 10 9 8 7 6

This is repeated until you end up almost back at the initial position:

 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 ${\displaystyle (n-1)}$ 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. [11] This schedule is applied in chess and draughts tournaments of rapid games, where players physically move round a table. In France this is called the Carousel-Berger system (Système Rutch-Berger). [12]

The schedule can also be used for "asynchronous" round-robin tournaments where all games take place at different times (for example, because there is only one venue). The games are played from left to right in each round, and from the first round to the last. When the number of competitors is even, this schedule performs well with respect to quality and fairness measures such as the amount of rest between games. On the other hand, when the number of competitors is odd, it does not perform so well and a different schedule is superior with respect to these measures. [13]

Alternatively Berger tables, [14] named after the Austrian chess master Johann Berger, are widely used in the planning of tournaments. Berger published the pairing tables in his two Schachjahrbucher, [15] [16] with due reference to its inventor Richard Schurig. [17] [18]

 Round 1. Round 2. Round 3. ... Round 13. 1–14 2–13 3–12 4–11 5–10 6–9 7–8 14–8 9–7 10–6 11–5 12–4 13–3 1–2 2–14 3–1 4–13 5–12 6–11 7–10 8–9 ... 7–14 8–6 9–5 10–4 11–3 12–2 13–1

This constitutes a schedule where player 14 has a fixed position, and all other players are rotated clockwise ${\displaystyle {\begin{matrix}{\frac {n}{2}}\end{matrix}}}$ 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. Arithmetically, this equates to adding ${\displaystyle {\begin{matrix}{\frac {n}{2}}\end{matrix}}}$ to the previous row, with the exception of player ${\displaystyle n}$. When the result of the addition is greater than ${\displaystyle (n-1)}$, then subtract ${\displaystyle ({\begin{matrix}{\frac {n}{2}}\end{matrix}}-1)}$.

This schedule can also be represented 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.

Diagonal Scheme
×234567891011121312345678910111213
112345678910111213
212345678910111213
312345678910111213
412345678910111213
512345678910111213
612345678910111213
712345678910111213
812345678910111213
912345678910111213
1012345678910111213
1112345678910111213
1212345678910111213
131 2 3 4 5 6 7 8 9 10111213
Round Robin Schedule
×12345678910111213
112345678910111213
223456789101112131
334567891011121312
445678910111213123
556789101112131234
667891011121312345
778910111213123456
889101112131234567
991011121312345678
1010111213123456789
1111121312345678910
1212131234567891011
1313123456789101112

The above schedule can also be represented by a graph, as shown below:

Both the graph and the schedule were reported by Édouard Lucas in [19] as a recreational mathematics puzzle. Lucas, who describes the method as simple and ingenious, attributes the solution to Felix Walecki, a teacher at Lycée Condorcet. Lucas also included an alternative solution by means of a sliding puzzle.

Original construction of pairing tables by Richard Schurig (1886)

For 7 or 8 players, Schurig [18] builds a table with ${\displaystyle n/2}$ vertical rows and ${\displaystyle n-1}$ horizontal rows, as follows:

 1 Round 1 2 3 4 2 , 5 6 7 1 3 , 2 3 4 5 4 , 6 7 1 2 5 , 3 4 5 6 6 , 7 1 2 3 7 , 4 5 6 7

Then a second table is constructed (with counting from the end) as shown below:

 1 Round . 1 . 7 . 6 . 5 2 , . 5 . 4 . 3 . 2 3 , . 2 . 1 . 7 . 6 4 , . 6 . 5 . 4 . 3 5 , . 3 . 2 . 1 . 7 6 , . 7 . 6 . 5 . 4 7 , . 4 . 3 . 2 . 1

By merging above tables we arrive at:

 1 Round 1, 1 2, 7 3, 6 4, 5 2 , 5, 5 6, 4 7, 3 1, 2 3 , 2, 2 3, 1 4, 7 5, 6 4 , 6, 6 7, 5 1, 4 2, 3 5 , 3, 3 4, 2 5, 1 6, 7 6 , 7, 7 1, 6 2, 5 3, 4 7 , 4, 4 5, 3 6, 2 7, 1

Then the first column is updated: if ${\displaystyle n}$ is even, player number ${\displaystyle n}$ is alternatingly substituted for the first and second positions, whereas if ${\displaystyle n}$ is odd a bye is used instead.

The pairing tables were published as an annex concerning the arrangements for the holding of master tournaments. Schurig did not provide a proof nor a motivation for his algorithm. For more historical details, see Ahrens. [20]