Round-robin tournament

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A round-robin tournament or all-play-all tournament is a type of group 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).

The term round robin is derived from the French term ruban meaning ribbon. Over a long period of time the term got corrupted and idiomized to robin. see: round robin.

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; the World Chess Championship was decided in 2005, and will again be decided in 2007, in 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 and UEFA Cup (since 2004–05) in soccer, the National Provincial Championship of rugby union in New Zealand, the Cricket World Cup and many American Football college conferences, such as the Mountain West Conference. 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.

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[edit] 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 competitor's 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 opponents 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).

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

[edit] 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 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 (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 and has a bye. 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.

[edit] Number of Possible Results

Let us have n competitors, and a scoring system where no ties are permitted. Let si be the number of wins scored by competitor i, and let us order them such that s_1 \le s_2 \le \cdots \le s_n. Thus we define a score vector, (s_1, s_2, \cdots, s_n), satisfying the following three conditions:

  1. 0 \le s_1 \le s_2 \le \cdots \le s_n
  2. s_1 + s_2 + \cdots + s_i \ge {i \choose 2}, \mbox{for }i = 1, 2, \cdots, n - 1
  3. s_1 + s_2 + \cdots + s_n = {n \choose 2}

Let s(n) be the number of different score vectors of size n. Then it is possible to show, for sufficiently large n:

s(n) > c_1 4^n n^{-{5 \over 2}}

Also:

s(n) < c_2 4^n n^{-{5 \over 2}}

Where: c_1 = 0.049,\,\, c_2 < 4.858

Together these show that:

s(n) \in \Theta (4^n n^{-{5 \over 2}}).

Here Θ signifies an asymptotically tight bound.

[edit] References

  • Takacs, Lajos (1991). "A Bernoulli Excursion and Its Various Applications". Advances in Applied Probability 23 (3): 557-585.

[edit] External links

  • SCHED link to a DOS program to quickly create round robin schedules
  • Tournament 16 link to a Windows XP program to create round robin schedules and export to HTML.
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