Asymmetric individual-pairs tournament design for 18 players

Julian D. A. Wiseman, April 2004

This is a 18-player individual-pairs tournament design which was last updated in April 2004.

Properties of this tournament design:

      i     ii       iii      iv        v    
  1  O:G  A+K:H+I  Q+N:M+B  L+C:D+E  R+F:P+J 
  2  K:P  E+B:F+N  I+O:L+J  R+Q:H+G  M+C:A+D 
  3  E:C  M+D:Q+G  P+A:R+N  J+F:B+I  K+H:L+O 
  4  N:Q  E+L:O+R  C+I:H+F  P+K:G+M  B+D:J+A 
  5  R:M  Q+J:I+A  K+D:P+B  O+H:E+F  L+N:C+G 
  6  L:F  C+E:M+O  J+H:G+A  D+N:I+K  B+Q:P+R 
  7  H:J  N+A:P+L  K+M:E+R  F+B:Q+C  G+O:D+I 
  8  B:I  C+O:R+H  D+G:L+F  M+A:K+J  P+Q:N+E 
  9  D:H  L+G:K+B  N+P:A+E  C+R:I+Q  J+M:F+O 
 10  J:R  F+M:N+C  G+K:Q+H  B+A:O+E  I+P:D+L 
 11  Q:O  P+H:B+L  J+R:C+D  I+G:M+N  E+K:A+F 
 12  F:N  K+Q:J+D  M+E:I+L  A+O:C+P  H+B:R+G 
 13  G:E  I+M:O+P  R+A:F+K  N+B:H+D  Q+L:C+J 
 14  I:A  H+E:D+Q  B+J:M+L  G+P:F+C  K+R:O+N 
 15  M:B  K+C:N+J  P+D:O+Q  L+R:A+H  F+I:G+E 
 16  C:K  D+F:R+M  H+L:N+I  P+E:J+G  O+B:Q+A 
 17  P:I  J+E:H+N  O+K:B+C  A+L:M+Q  F+G:R+D 
 18  R:L  A+C:G+B  F+P:H+M  O+D:N+K  J+I:E+Q 
 19  A:D  Q+F:L+K  G+N:J+O  E+I:R+B  H+C:M+P 

• This is an individual pairs for 18 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 four games of two-versus-two (at venues ii, iii, iv and v). There are no byes.

• Each pair of players partner exactly once, except I and R 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 I and R who each do so thrice.

• No player plays two consecutive games at the same venue.

• Each player plays between three and five times at each doubles venue. 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 14 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 8 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 balance the distribution with which players appear at each position at each venue, and subject to that, at each position totalled over venues.

• If players are ranked, from A the best to R the worst, this tournament has an uncubed unfairness measure of 277.93.

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

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