# Round-robin tournament

A round-robin tournament (or all-play-all tournament) is a competition "in which each contestant meets all other contestants in turn".12 It contrasts with an elimination tournament.

## Terminology

The term round-robin is derived from the term ruban, meaning "ribbon". Over a long period of time, the term was corrupted and idiomized to robin.34

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).

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.567 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 a "quad".8

## 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 Cup and the respective continental tournaments (e.g. UEFA European Championship, CONCACAF Gold Cup, etc.). There are also round-robin 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.

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 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.

## 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 football, 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 round less than 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 32 team tournament can be completed in just 5 rounds (i.e. 31 matches) in a knockout format. However if the same teams are put through a single round-robin it would require 31 rounds (i.e. 496 matches) 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 match-up 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). 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.

Swiss system tournaments 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 $n$ is the number of competitors, a pure round robin tournament requires $\begin{matrix} \frac{n}{2} \end{matrix}(n - 1)$ games. If $n$ is even, then in each of $(n - 1)$ rounds, $\begin{matrix} \frac{n}{2} \end{matrix}$ games can be run in parallel, provided there exist sufficient resources (e.g. courts for a tennis tournament). If $n$ is odd, there will be $n$ rounds, each with $\begin{matrix} \frac{n - 1}{2} \end{matrix}$ 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 (the pivot, 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 $(n-1)$ 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.10 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).11

Alternatively Berger tables,12 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,1314 with due reference to its inventor Richard Schurig.1516

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 has a fixed position, and all other players are rotated clockwise $\begin{matrix} \frac{n}{2}\end{matrix}$ 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 $\begin{matrix} \frac{n}{2}\end{matrix}$ to the previous row, with the exception of player $n$. When the result of the addition is greater then $(n-1)$, then subtract $(n-1)$.

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
× 2 3 4 5 6 7 8 9 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 10 11 12 13
Round Robin Schedule
× 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 2 3 4 5 6 7 8 9 10 11 12 13 1
3 3 4 5 6 7 8 9 10 11 12 13 1 2
4 4 5 6 7 8 9 10 11 12 13 1 2 3
5 5 6 7 8 9 10 11 12 13 1 2 3 4
6 6 7 8 9 10 11 12 13 1 2 3 4 5
7 7 8 9 10 11 12 13 1 2 3 4 5 6
8 8 9 10 11 12 13 1 2 3 4 5 6 7
9 9 10 11 12 13 1 2 3 4 5 6 7 8
10 10 11 12 13 1 2 3 4 5 6 7 8 9
11 11 12 13 1 2 3 4 5 6 7 8 9 10
12 12 13 1 2 3 4 5 6 7 8 9 10 11
13 13 1 2 3 4 5 6 7 8 9 10 11 12

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

Round Robin Schedule Span Diagram

This method was discovered by Felix Walecki and reported by Lucas (1883) as a solution to the Walking schoolgirl problem (Les Promenades du Pensionnat).17

### Original construction of pairing tables by Richard Schurig (1886)

For 7 or 8 players, Schurig16 builds a table with $n/2$ vertical rows and $n-1$ 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 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. The first or second position are alternating substituted by a bye, if $n$ is not even, or by $n$.

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.18