All-play-all tournament design for 23 players

Julian D. A. Wiseman, January 2007

This 23-player all-play-all tournament design, last updated in January 2007, is based on an original by Matt Fayers of The Department of Mathematics at Queen Mary, University of London (formerly of The Department of Pure Mathematics and Mathematical Statistics at The University of Cambridge). It has been discussed in a thread on the CUTwC bulletin board.

Properties of this tournament design:

    Bye  i   ii   iii  iv    v   vi   vii viii  ix    x   xi  
  1  I  K:S  M:F  D:P  G:H  O:T  B:R  C:V  N:Q  U:J  E:W  L:A
  2  R  J:C  T:I  M:V  K:B  L:S  F:E  N:G  U:P  Q:A  O:D  W:H
  3  M  T:A  W:G  N:D  H:U  J:K  I:S  R:P  E:C  O:B  V:L  F:Q
  4  P  E:K  Q:C  R:H  J:O  D:W  N:B  F:T  M:L  A:S  G:U  I:V
  5  L  B:W  G:R  E:O  V:A  T:C  S:F  U:D  Q:K  P:H  M:I  J:N
  6  O  G:M  F:U  A:R  P:I  Q:E  T:J  L:H  S:B  N:K  W:C  V:D
  7  G  U:E  S:O  V:K  N:T  A:I  Q:W  J:R  L:F  C:M  D:H  B:P
  8  C  R:V  H:T  Q:G  W:L  U:M  A:O  D:E  B:J  S:P  I:N  K:F
  9  T  I:D  B:L  J:M  E:V  K:R  C:P  S:Q  G:A  W:N  H:F  O:U
 10  E  V:G  D:S  C:N  Q:J  P:L  H:M  K:O  F:W  B:T  U:A  R:I
 11  B  C:F  L:N  H:Q  T:P  E:G  V:U  M:W  O:I  J:D  S:R  A:K
 12  Q  H:I  V:B  P:A  U:C  R:F  W:K  O:N  D:M  G:L  T:E  S:J
 13  K  F:N  E:J  G:S  L:R  W:A  D:T  P:M  I:U  H:V  Q:B  C:O
 14  A  P:J  O:V  I:F  M:K  H:B  L:C  Q:U  W:T  E:R  N:S  D:G
 15  N  D:L  U:K  S:E  O:W  F:P  G:I  H:A  C:R  T:Q  J:V  M:B
 16  D  M:O  P:E  T:U  A:F  N:V  K:L  W:S  J:H  I:C  B:G  Q:R
 17  J  L:U  R:M  W:I  D:Q  S:H  P:V  E:B  A:N  F:O  C:K  G:T
 18  H  O:Q  J:W  K:T  C:S  B:U  R:D  V:F  P:G  L:I  A:M  N:E
 19  U  W:R  N:P  F:B  I:E  C:D  O:H  A:J  V:S  K:G  L:Q  T:M
 20  W  S:T  A:D  L:J  F:G  V:Q  U:N  B:I  R:O  M:E  K:P  H:C
 21  V  N:H  I:Q  B:C  S:M  G:O  E:A  T:L  K:D  R:U  F:J  P:W
 22  F  Q:P  C:A  O:L  B:D  M:N  J:G  I:K  H:E  V:W  R:T  U:S
 23  S  A:B  K:H  U:W  R:N  I:J  M:Q  G:C  T:V  D:F  P:O  E:L

• Each player plays each of the others exactly once.

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

• Each player plays exactly once on each of side of each venue.

• Hence each player plays exactly twice at each venue.

• It is seeded, so that important games come late in the tournament if player A is the best, B the second best, etc.

• It has a permutation score of 235318.661082, which is believed to be maximal given the underlying construction.

• It has a left-right asymmetry measure of 26.948328, which is minimal given the assignment of games to rounds and venues.

• This design was generated as follows.

  Define a terrace to be an ordering a₀, a₁, …, a₂₂ of the numbers 0, 1, …, 22 such that the values a₀+a₁, a₁+a₂, a₂+a₃, &hellip, a₂₂+a₀ give all the different mod 23 values. (This is not necessarily the usual group theorist’s definition of a terrace.) We regard a terrace as a cyclic ordering, so that a₀, a₁, …, a₂₂ is the same as a₁, a₂, …, a₂₂, a₀. Note that if a₀, a₁, …, a₂₂ is a terrace, then so is xa₀, xa₁, …, xa₂₂, for any non-zero x. Given a terrace, we construct a format as follows: number the venues 0, 1, …, 10. To venue 0, assign the games a₀:a₁, a₁:a₂, …, a₂₂:a₀. To venue 1 assign the games 2a₀:2a₁, 2a₁:2a₂, …, 2a₂₂:2a₀; to venue 2 assign 4a₀:4a₁, 4a₁:4a₂, …, 4a₂₂:4a₀; and so forth. In general, assign all the games 2ⁱa_j: 2ⁱa_j+1 to venue i. (The significance of the number 2 here is that it is the square of a primitive root mod 23, i.e. it has multiplicative order 11.) Now, which games go in which rounds? Because the a_j form a terrace, then for each k and each i we can find exactly one game b:c at venue i which satisfies b+c = k. This game is played in round k.

  This gives a format, but not necessarily an all-play-all — some games might be repeated or omitted. But if it is an all-play-all, then the construction will guarantee that each player plays once in each round and twice at each venue. Then we can consider re-ordering the rounds such that no player plays two consecutive games at the same venue.

  So how can terraces be generated?

  • Scaling: replace a_j with xa_j. Alas if the old terrace doesn’t produce an all-play-all, then neither will the new one. So scaling is useless.

  • Shifting: replace a_j with x+a_j. This can be helpful, but there are at most 23 different shifts of any terrace.

  • Breaking: break the terrace into three chunks A, B and C. Reverse chunk A and switch B and C. This doesn’t always produce a terrace, but it is easy to write down a pair of equations which are necessary and sufficient for this to produce a terrace. Happily, there are many ways to break a terrace into suitable chunks.

  So a random walk was generated from repeated breakage, with testing of the new terrace and all shifts of it. Six hours on Dr Fayers’ computer proved sufficient to find a design, which was then, on the author’s computer, put through the usual optimisations of constrained row permutation and unconstrained player permutation.

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