Asymmetric individual-pairs tournament design for 6 players

Julian D. A. Wiseman, April 2004

This is a 6-player individual-pairs tournament design which was last updated in April 2004.

Properties of this tournament design:

     i     ii    
 1  C:A  D+E:F+B 
 2  E:D  B+C:A+F 
 3  B:C  D+F:E+A 
 4  A:E  C+F:B+D 
 5  C:F  A+D:E+B 
 6  D:B  F+E:C+A 
 7  F:E  A+B:D+C 

• This is an individual pairs for 6 players. Because this number of players is 2 modulo 4, the design has to be somewhat asymmetric. Hence it is strongly recommended that, if at all possible, individual pairs tournaments of this size are avoided.

• Each round consists of one game of one-versus-one (played at venue i), and one game of two-versus-two (at venue ii). There are no byes.

• Each pair of players partner exactly once, except C and E who never partner.

• Each pair of players oppose between once and four times, with only B & D opposing four times.

• Each player self-partners exactly twice, except C and E who each do so three times.

• No player plays singles twice consecutively.

• In each doubles game, each team has the same number of players who were at the same venue in the previous round. The total of this number of players has been minimised.

• The left-right totals for each player are as even as possible, at single, at doubles, and in total.

• All pairs of distinct players who oppose at least twice (except the twice-opposing A & F) do so at least once from the left and at least once from the right.

• All pairs of distinct players (except A & F and the four-times-opposing B & D) who oppose at least twice do so approximately evenly (that is, an even number of opposition split exactly equally, an odd number nearly so).

• All pairs of distinct players (except A & B and A & D) who oppose at least twice at doubles do so at least once playing immediately before and at least once immediately after. This is as balanced as possible.

• The remaining symmetry, in which 1^st+3^rd:2^nd+4^th can be exchanged for 3^rd+1^st:4^th+2^nd (equivalent to a 180° mat rotation in tiddlywinks), is then used to ensure that the number of times a player is ever consecutively at the same position at the same venue is minimised (to once, in round 4 at venue ii).

• If players are ranked, from A the best to F the worst, this tournament has an uncubed unfairness measure of 137.28.

• This design was found by an automated search, and has no known underlying group theoretic construction.

Usual disclaimer and copyright terms apply.