TL;DR

Where we continue our quest for a tournament scheduling for matches with 6 players.

In Allocating games in tournaments - 6 players matches we ended up with an arrangement that is compliant with the requirements of the Social Golfer Problem, i.e. that participants never compete against each other more than once.

How these golfers can really be called social beats me though. If they’re this social, they shouldn’t mind playing twice with the same person every now and then, should they? Anyway.

What’s missing

To understand what we are missing, let’s remember what we did to get the schedule:

we started from an affine-plane based schedule for matches with 7 players each
then removed the first 7 players, and
the whole first round.

Let’s look at this removed round, then, with the re-numbering we already did the last time (i.e. $8..49$ mapped onto $1..42$):

removed round:
  ()
  (  1,   7,  13,  19,  25,  31,  37)
  (  2,   8,  14,  20,  26,  32,  38)
  (  3,   9,  15,  21,  27,  33,  39)
  (  4,  10,  16,  22,  28,  34,  40)
  (  5,  11,  17,  23,  29,  35,  41)
  (  6,  12,  18,  24,  30,  36,  42)

Each non-empty match here shows us which players will not compete against each other. As an example, players 1 and 7 appear in the same match, so we are sure that they do not compete against each other in any of the rounds that we kept (because of how the matches have been designed in the first place).

What to do about it?

There are a few things that might be done about it…

Ignore the round

The first possible thing to do is to just ignore the round, and fall back to the solution we discussed in the previous post. Nothing more to see here.

Relax: number of participants in a match

How about a tournament where matches are normally arranged for 6 people, but one of them contains 7? Many games would allow this, so why not?

This would be the eight round then:

round 8:
  (  1,   7,  13,  19,  25,  31,  37)
  (  2,   8,  14,  20,  26,  32,  38)
  (  3,   9,  15,  21,  27,  33,  39)
  (  4,  10,  16,  22,  28,  34,  40)
  (  5,  11,  17,  23,  29,  35,  41)
  (  6,  12,  18,  24,  30,  36,  42)

This has the advantage that each player would compete against each other exactly once (yay!) and would not need to relax anything only for a subset of players, which might be considered… less fair (e.g. what if people competing twice or more strike a cheating deal?!?).

Relax: some pairs can face twice

If it’s nothing too official, and the risk of cheating is low (with risk assumed to be the product of probability and damage of cheating), then we can arrange one more round where only a bare minimum of players will face each other for the second time:

round 8:
  (       7,  13,  19,  25,  31,  37)
  (       8,  14,  20,  26,  32,  38)
  (  3,   9,  15,       27,  33,  39)
  (  4,  10,       22,  28,  34,  40)
  (  5,  11,  17,  23,       35,  41)
  (  6,  12,       24,  30,  36,  42)
  (  1,   2,  21,  16,  18,  29     )

In short, we removed one participant from each 7-players match and put all of them in an additional match (the last one). In this way, only those players (i.e. 1, 2, 21, …) will face each other for the second time in the tournament, while the others will still comply with the Social Golfer Problem requirements.

The choice of players for the last match is not random. We could of course have taken the first player in each of the previous ones (i.e. players 1 to 6), but this would have been a sheer repetition of the very first match in the very first round, which is admittedly boring. So, we opted for making at least sure that no three players in that match ever played at the same table at the same time in a previous round, which you can easily verify.

Characteristics:

still there are pairs of players that didn’t play against each other. As an example, in this arrangement player 1 never gets to play against players 7, 13, …, i.e. all players in the first match of the additional eighth round;
this lack of completeness is unbalanced, too: while player 7 does not get to ever play with player 1 only, player 1 does not get to ever play with six other players;
the asymmetry also shows up in that only a few players get to face each other twice;
on the good side, we have another round of 6-players matches!

Summary

This post contains three possible alternatives for tournaments of matches with 6 players… Now the choice is yours! Stay tuned, though: we still have to elaborate a bit about premium players…

If you want to take a look at all posts, here’s the list:

ETOOBUSY 🚀 minimal blogging for the impatient

Allocating games in tournaments - 6 players matches, again