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Tournament Designs: Individual-Pairs

Julian D. A. Wiseman, June 2003 and April 2004

Abstract: Some number of players wish to play an individual-pairs tournament, in which each partners each other once and hence opposes each other twice. How can this be arranged?

Contents: Introduction; Presentations and file formats; Technical notes (unfairness measure, permuting players); Also see.

Publication history: Earlier versions of some of the designs here have previously been made available in paper form by Dr Nicholas F. J. Inglis, for the use of ETwA and CUTwC tournament organisers. This is believed to be the only publication on the web. Usual disclaimer and copyright terms apply.

Quick individual-pairs links: complete list; pure 4 5 8 9 12 13 16 16* 17 20 21 24 25; mixed 3 7 9 11 15 19 23; asymmetric 6 10 14 18 22 26.


       i       ii       iii   
 1  E+G:I+J  L+A:B+H  C+F:D+K 
 2  L+D:E+K  F+I:G+C  H+J:A+B 
 3  I+A:J+F  K+B:D+E  L+G:H+C 
 4  C+E:G+H  L+J:K+F  A+D:B+I 
 5  L+B:C+I  D+G:E+A  F+H:J+K 
 6  G+J:H+D  I+K:B+C  L+E:F+A 
 7  A+C:E+F  L+H:I+D  J+B:K+G 
 8  L+K:A+G  B+E:C+J  D+F:H+I 
 9  E+H:F+B  G+I:K+A  L+C:D+J 
10  J+A:C+D  L+F:G+B  H+K:I+E 
11  L+I:J+E  K+C:A+H  B+D:F+G 

Twelve players (or teams) are all to play a pairs-style tournament, in which each game consists of one partnership against another. This can be done by having each player partner each of the others exactly once, and hence oppose each of the others exactly twice. Each player has to play eleven games, so this tournament requires at least eleven rounds. With three games happening simultaneously, there must be at least three venues. On the right can be found an example 12-player individual-pairs at three venues venues over eleven rounds.

Individual pairs are not possible for all numbers of players. To see this start by calculating, for any given number of players, the number of possible partnerships. Each game in a tournament uses two of these partnerships, so if the number of partnerships is odd, an individual pairs cannot be produced for that number of players.

The calculation can be altered by allowing self-partnering. But self-partnering makes sense only in non-aerobic games in which turns are taken by players who may confer. So it works in the likes of tiddlywinks, croquet, snooker, and billiards. But self-partnering doesn’t work in games such as bridge and doubles tennis. An individual pairs tournament in which every game is two-versus-two is said to be ‘pure’, and one which includes self-partnering is said to be ‘mixed’.

In a pure individual pairs tournament with n players, there are n(n–1)/2 partnerships. This is even iff n is 0 or 1 modulo 4. In a mixed individual pairs tournament with n players, in which each player self-partners once, there are n(n–1)/2+n partnerships. This is even iff n is 0 or 3 modulo 4. This result continues to hold if each player self-partners any odd number of times. If players self-partner an even number of times, then n must be either 0 or 1 modulo 4.

So pure individual pairs are possible if n is 0 or 1 modulo 4; mixed individual pairs are possible if n is 0 or 1 or 3 modulo 4; and if n is 2 modulo 4 then neither type is possible.

Pure individual-pairs tournament designs are published for various numbers of players from four to twenty-five: 4, 5, 8, 9, 12, 13, 16, (and in another version, 16*), 17, 20, 21, 24, and 25. Mixed individual-pairs tournament designs are published for various numbers of players from three to twenty-three: 3, 7, 9, 11, 15, 19, and 23. Asymmetric individual-pairs tournament designs are published for various numbers of players from six to twenty-six: 6, 10, 14, 18, 22 and 26. Also see the complete list of individual-pairs links.

Meaning of terms

The precise meaning of some of the terms can vary according to the game being played.

Presentations and file formats

Each of these individual-pairs tournaments is published in a number of different presentations and file formats.

Technical notes: symmetric tournaments

The construction of the symmetric individual pairs relies on a branch of mathematics called design theory. The following paragraph, by Matt Fayers of The Department of Pure Mathematics and Mathematical Statistics at The University of Cambridge, assumes some knowledge of design theory.

An individual pairs tournament is referred to in the mathematical literature as a whist tournament, and the existence of a whist tournament for any number of player congruent to 0 or 1 mod 4 was proved by Baker in 1975[*1] using a variety of construction techniques. For practical purposes, it is often easiest to find lots of possible whist tournaments by seeking cyclic formats. For 4n+1 players, this means indexing the players and the rounds by the integers modulo 4n+1 (so they start at zero), and choosing the games in round 0 and then decreeing that if A+B:C+D is a game in round 0, then (A+i)+(B+i):(C+i)+(D+i) is a game in round i. Of course, the first round must be chosen in such a way that this produces a whist tournament, but the necessary and sufficient condition on the first round is a simple one based on the differences (mod 4n+1) between the partners and opponents in the various games. For 4n players, a similar procedure is adopted, but here we cycle mod 4n–1 and label one of the players ∞ (with the usual convention that ∞+i=∞). In fact, an analogous procedure can be used for any group of order 4n+1 or 4n–1; in the case of 9 players, this is necessary, since there is no cyclic whist tournament for 9 players, although there is one based on the elementary abelian group of order 9. To assign games to venues is a rather different problem. In the case of 4n+1 players, we might dictate that the game (A+i)+(B+i):(C+i)+(D+i) happens at the same venue as A+B:C+D; this guarantees that each player plays exactly four times at each venue, but also guarantees that three players stay at the same venue between consecutive rounds, even if the rounds are re-ordered. For 4n players, adopting this procedure would keep player ∞ at the same venue throughout, and so is only useful when ∞ is a ‘dummy’ player used in a tournament for 4n–1 players (whoever partners the dummy actually self-partners). When there really are 4n players, some shuffling of games between venues is necessary. The number of possibilities is unwieldy, so the approach used here is to label the venues 0,…,n–1, and then assign games to venues for round 0 and dictate that if A+B:C+D happens in round 0 at venue j, then (A+i)+(B+i):(C+i)+(D+i) happens in round j at venue j+i (reduced mod n). In short, games were shifted one place to the right each round. This guarantees that player ∞ plays three times at one venue and four times at each of the others, and it is then a matter of trying various arrangements of the games in round 1 to try to prevent players staying at the same venue between games and to try to ensure that each of the other players has a reasonably even distribution between venues.

This done, it is still possible to permute the players, with the objective of minimising that which is here called ‘unfairness’. In the symmetric tournaments each player partners everybody once, and opposes everybody twice, so a player would consider it a terrible waste to partner the best-ranked player, A, who is the current world champion, against the two worst ranked players. The partnering of A would be, in some sense, wasted on easy opponents. And likewise, it would be highly desirable to partner the worst ranked player against A+B: it gets out of the way a terrible partner and one of the two oppositions against the two top players, for the loss of one game rather than three. So we would like the combinations of opponents and partners to be chosen fairly, and we start by assigning a numerical score to each player: the worst is 1, the second-worst 2, and so on up to A who scores n. For each player, the average of the values of the other players is calculated, and for player Z this is written (Z). The relative advantage in game W+X:Y+Z for player Z is therefore (Z)+Y-X-W, and for each player the total of these is zero. So the total of the cubes of the relative advantages is calculated, knowing that if a player’s games are fairly distributed, then its sum of cubes will be near zero. The variable that is minimised is then the weighted sum of the squares of these sums of cubes; giving a greater weight to better players by weighting the outer summation by the players’ numeric values. For large numbers of players (above 16), it is possible that the absolute minimum has not been found. When there is self-partnering, there is a slight adjustment to the scoring, so that in W+X:Z+Z the relative advantage of Z is 2(Z)-X-W.

Technical notes: asymmetric tournaments

      i     ii       iii   
  1  A:H  D+F:E+J  B+C:G+I 
  2  E:G  J+I:H+F  C+A:B+D 
  3  C:F  I+E:J+B  H+A:D+G 
  4  G:I  B+H:C+J  E+D:A+F 
  5  J:C  F+I:A+G  B+E:H+D 
  6  H:E  A+D:B+I  C+F:J+G 
  7  D:J  E+A:I+C  G+H:F+B 
  8  C:A  D+J:G+B  F+E:I+H 
  9  F:B  E+G:J+A  D+I:H+C 
 10  I:D  G+C:E+H  A+B:F+J 
 11  B:E  F+G:C+D  J+H:I+A 

Much of the work of the construction of the symmetric tournaments was done by the mathematics within design theory. The asymmetric tournaments do not have this mathematical base, and for ≥10 players these designs are built by a cascade of optimisations.

Also see

Designs for other sizes of tournaments—though requiring much more work from the user—can be found at Durango Bill's Bridge Probabilities and Combinatorics.

*1 R. Baker, ‘Whist Tournaments’, Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory and Computing, Congressus Num. 14, 89-100.

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