Asymmetric individual-pairs tournament design for 10 players

Julian D. A. Wiseman, April 2004

This is a 10-player individual-pairs tournament design which was last updated in April 2004.

Properties of this tournament design:

      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 

• This is an individual pairs for 10 players. Because this number of players is 2 modulo 4, the design has to be somewhat asymmetric.

• Each round consists of one game of one-versus-one (played at venue i), and two games of two-versus-two (at venues ii and iii). 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 thrice, with as many pairs as possible opposing twice.

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

• 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.

• Each player plays between three and six times at each doubles venue (J playing six times at venue ii). Subject to this, the total of the distances from four has been minimised.

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

• All except 5 pairs of distinct players who oppose at least twice do so at least once from the left and at least once from the right. The number of exceptions has been minimised.

• All except 3 pairs of distinct players who oppose at least twice at doubles do so at least once playing immediately before and at least once immediately after. The number of exceptions has been minimised.

• 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 3 at venue iii). Subject to this, the distribution with which players appear at each position at each venue is then balanced, and subject to that, at each position totalled over venues.

• If players are ranked, from A the best to J the worst, this tournament has an uncubed unfairness measure of 61.4074.

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

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