A round-robin tournament or all-play-all tournament is a type of tournament in which each participant plays every other participant an equal number of times. In a pure 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 all of the major United States professional sports).
In sports with a large number of competitive matches per season, double round-robins are common. Almost all football (soccer) 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. There are also round-robin chess tournaments.
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 and UEFA Cup (since 2004–05) in soccer, the National Provincial Championship of rugby union in New Zealand, and many American Football college conferences, such as the Mountain West Conference. The group phase of the 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 the Euroleague.
Ranking
In sports where ties are rare or impossible, competitors typically are ranked by number of wins, with ties counting half. Where ties are more common, this may be 2 points for a win and 1 for a tie, which is mathematically equivalent but avoids having too many half-points in the listings.
Other sports may have more complex ranking criteria. In football (soccer), where draws are relatively common, many leagues give 3 points for a win and 1 for a draw to encourage attacking play. In rugby union, bonus points may be awarded for scoring a certain number of tries, which are a more crowd-pleasing form of score than goals.
When competitors are level on points, the tie-breaker criteria may be:
- head-to-head: considering only results of games between the deadlocked competitors. If more than a single game is involved, a subtable may be used recursively for the ranking.
- scoring average: the ratio of points (goals, etc.) scored to those conceded.
- scoring differential: the difference between points (goals,etc) scored and those conceded.
- points scored, irrespective of points conceded.
- disciplinary record (fouls conceded, etc).
Ties remaining on one of these criteria may be resolved by resorting to another one.
Rather than using statistics from the previous matches, there may instead be a play-off, with extra matches between the tied competitors. If there are more than two such, this may take the form of either another round-robin, or a knockout tournament.
Evaluation
In a round-robin format, the element of luck is seen to be reduced, given that all competitors face the same opponents, and a few bad performances need not cripple a competitorīs chances of ultimate victory. In English football, although the FA Cup was founded before the Football League, the (round-robin) League champions have always been regarded as the "best" team in the land, rather than the (knockout) Cup winners.
Disadvantages include the existence of games 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. This asymmetry means that playing the same opponents is not necessarily equitable: the same opponenents in a different order may play harder or easier matches. There is also no showcase final match. The ability to recover from defeats, while rewarding overall consistency, may also be seen as a crutch for competitors who lack the temperament to handle the pressure of a knockout tournament.
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).
Scheduling Algorithm
If n is the number of competitors, a pure round robin tournament requires n/2 * (n-1) games. If n is even, then in each of (n − 1) rounds, n/2 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 with (n-1)/2 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 …
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. The upper and lower rows can indicate home/away in sports, white/black in chess, etc (this must alternate between rounds since competitor 1 is always on the first row). If, say, competitors 3 and 8 were unable to fulfill 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.
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