Looking for help with Schedule

[mgdad]

This is the scenario for a unique volleyball tournament... we have 14 players in total although this number varies from year to year. We play 5 players on each team thus for each game, 4 players sit out.  What is unique is that we want each player to ideally play with each other player the same number of times throughout the tournament as ultimately there is an individual winner, not a team winner.  The total number of games played in the tournament is usually equal to the total number of players in that given year.  We have suffered through this scheduling process manually for years but are hopeful that somewhere out there ... please.... a better solution exists.  Could anyone help us out on this?  If I have omitted any critical info, please let me know.... I'll watch the board closely.

[Ian Wakeling]

The requirement "want each player to ideally play with each other player the same number of times" can be interpretted in a number of different ways.

(1) play with each other as teammates the same number of times
(2) play with each other as opponents the same number of times
(3) play with each other either as teammates or opponents the same number of times. (optimal social mix).

I think the 14 player problem is unlikely to have a perfect solution where best possible balance can be achieved for all three of the criteria above, so it is important to know if there are different priorities.  If I assume that (2) is more important than (1) or (3), then I can get a schedule like the one below where most pairs of players oppose 4 times, and 14 pairs of players oppose 3 times, or from the perspective of one player, there are 2 other players opposed 3 times, and the remaining 11 players are opposed four times.  Is that useful, or do you have different priorities perhaps?

  ( 5  4  6 11  2 v 13  7  1 12 10)   ( 8  9  3 14)
  ( 6  5  7 12  3 v 14  1  2 13 11)   ( 9 10  4  8)
  ( 7  6  1 13  4 v  8  2  3 14 12)   (10 11  5  9)
  ( 1  7  2 14  5 v  9  3  4  8 13)   (11 12  6 10)
  ( 2  1  3  8  6 v 10  4  5  9 14)   (12 13  7 11)
  ( 3  2  4  9  7 v 11  5  6 10  8)   (13 14  1 12)
  ( 4  3  5 10  1 v 12  6  7 11  9)   (14  8  2 13)
  (12  5  2 13  4 v  3  9 11 14 10)   ( 8  1  7  6)
  (13  6  3 14  5 v  4 10 12  8 11)   ( 9  2  1  7)
  (14  7  4  8  6 v  5 11 13  9 12)   (10  3  2  1)
  ( 8  1  5  9  7 v  6 12 14 10 13)   (11  4  3  2)
  ( 9  2  6 10  1 v  7 13  8 11 14)   (12  5  4  3)
  (10  3  7 11  2 v  1 14  9 12  8)   (13  6  5  4)
  (11  4  1 12  3 v  2  8 10 13  9)   (14  7  6  5)

If you use the schedule above, consider rearranging the order of the 14 rounds to make the distribution of byes more balanced.

[mgdad]

You are amazing... Thank you.  We have always set the priority as trying to ensure that each player plays with each other player on the same team the same number of times.  Whether this is the right choice or not I am unsure of.  As you can imagine, we have varying skill levels so over the course of the tournament we want to ensure that a weak player is not partnered with a strong player a disproportionate number of times.  I would welcome your thoughts on what the best solution would be.  If it helps, this is how we score... Each member of the winning team gets 30 points.  Each member of the losing team gets the number of points they scored in the game (to maximum of 15 points).  To ensure more fairness, we run the schedule twice over two days with a redraw of schedule numbers after day one.

[Ian Wakeling]

I think we could argue both ways, the strength of your teammates and the strength of the opposition are both important factors.  The good news is that doubling the size of the schedule makes all the difference, and it is possible to get best possible balance for all three of my criteria above:

(1) pairs of players play together as teammates either 6 or 7 times.
(2) pairs of players play together as opponents either 7 or 8 times.
(3) pairs of players play together either as teammates or opponents either 13 or 14 times.

If you think about the above, then it must mean that pairs of players who oppose 8 times, must partner only 6 times (since 8 + 7 = 15).  Here is the schedule:

  (11  3  5  2  4 v 10 13 14 12  7)   ( 9  8  6  1)
  (12  4  6  3  5 v 11 14  8 13  1)   (10  9  7  2)
  (13  5  7  4  6 v 12  8  9 14  2)   (11 10  1  3)
  (14  6  1  5  7 v 13  9 10  8  3)   (12 11  2  4)
  ( 8  7  2  6  1 v 14 10 11  9  4)   (13 12  3  5)
  ( 9  1  3  7  2 v  8 11 12 10  5)   (14 13  4  6)
  (10  2  4  1  3 v  9 12 13 11  6)   ( 8 14  5  7)
  ( 7  3  5 12 14 v  9  6 11  4 10)   ( 2  1 13  8)
  ( 1  4  6 13  8 v 10  7 12  5 11)   ( 3  2 14  9)
  ( 2  5  7 14  9 v 11  1 13  6 12)   ( 4  3  8 10)
  ( 3  6  1  8 10 v 12  2 14  7 13)   ( 5  4  9 11)
  ( 4  7  2  9 11 v 13  3  8  1 14)   ( 6  5 10 12)
  ( 5  1  3 10 12 v 14  4  9  2  8)   ( 7  6 11 13)
  ( 6  2  4 11 13 v  8  5 10  3  9)   ( 1  7 12 14)

  (12 14  6  3  4 v  5 13  1  2 10)   ( 8  7 11  9)
  (13  8  7  4  5 v  6 14  2  3 11)   ( 9  1 12 10)
  (14  9  1  5  6 v  7  8  3  4 12)   (10  2 13 11)
  ( 8 10  2  6  7 v  1  9  4  5 13)   (11  3 14 12)
  ( 9 11  3  7  1 v  2 10  5  6 14)   (12  4  8 13)
  (10 12  4  1  2 v  3 11  6  7  8)   (13  5  9 14)
  (11 13  5  2  3 v  4 12  7  1  9)   (14  6 10  8)
  ( 5  8  1 11  9 v  3  4 13 10 14)   ( 7  2  6 12)
  ( 6  9  2 12 10 v  4  5 14 11  8)   ( 1  3  7 13)
  ( 7 10  3 13 11 v  5  6  8 12  9)   ( 2  4  1 14)
  ( 1 11  4 14 12 v  6  7  9 13 10)   ( 3  5  2  8)
  ( 2 12  5  8 13 v  7  1 10 14 11)   ( 4  6  3  9)
  ( 3 13  6  9 14 v  1  2 11  8 12)   ( 5  7  4 10)
  ( 4 14  7 10  8 v  2  3 12  9 13)   ( 6  1  5 11)

[mgdad]

Thanks Ian.  Just so I am clear, in your solution would we best not to redraw numbers after Day 1 such that for example Player 1 stays as Player 1 on both of the two days of the tournament?  In other words, we just play as you have suggested the first 14 games on day 1 and the second 14 games on day 2.

[Ian Wakeling]

Yes, keep the same player numbers for both days.

[mgdad]

Perfect.... Thanks again for all your help!!

[mgdad]

Ian, just sent you a PM wondering if you might assist again this year in helping to put together our schedule for our volleyball tournament.

Thanks so much.

[Ian Wakeling]

Here are some schedules similar to the one directly above which are for 15 or 16 players.  Hope they are useful to you.

15 players (team pairs 5 or 6 times, oppo pairs 7 or 8 times).

( 1 10  9  7  6 vs  8 12 13 11  3)   ( 2  5 14  4 15)
( 2 11 10  8  7 vs  9 13 14 12  4)   ( 3  6 15  5  1)
( 3 12 11  9  8 vs 10 14 15 13  5)   ( 4  7  1  6  2)
( 4 13 12 10  9 vs 11 15  1 14  6)   ( 5  8  2  7  3)
( 5 14 13 11 10 vs 12  1  2 15  7)   ( 6  9  3  8  4)
( 6 15 14 12 11 vs 13  2  3  1  8)   ( 7 10  4  9  5)
( 7  1 15 13 12 vs 14  3  4  2  9)   ( 8 11  5 10  6)
( 8  2  1 14 13 vs 15  4  5  3 10)   ( 9 12  6 11  7)
( 9  3  2 15 14 vs  1  5  6  4 11)   (10 13  7 12  8)
(10  4  3  1 15 vs  2  6  7  5 12)   (11 14  8 13  9)
(11  5  4  2  1 vs  3  7  8  6 13)   (12 15  9 14 10)
(12  6  5  3  2 vs  4  8  9  7 14)   (13  1 10 15 11)
(13  7  6  4  3 vs  5  9 10  8 15)   (14  2 11  1 12)
(14  8  7  5  4 vs  6 10 11  9  1)   (15  3 12  2 13)
(15  9  8  6  5 vs  7 11 12 10  2)   ( 1  4 13  3 14)

( 1  9 12 14  3 vs 10  4  7 15  8)   (11  6  5 13  2)
( 2 10 13 15  4 vs 11  5  8  1  9)   (12  7  6 14  3)
( 3 11 14  1  5 vs 12  6  9  2 10)   (13  8  7 15  4)
( 4 12 15  2  6 vs 13  7 10  3 11)   (14  9  8  1  5)
( 5 13  1  3  7 vs 14  8 11  4 12)   (15 10  9  2  6)
( 6 14  2  4  8 vs 15  9 12  5 13)   ( 1 11 10  3  7)
( 7 15  3  5  9 vs  1 10 13  6 14)   ( 2 12 11  4  8)
( 8  1  4  6 10 vs  2 11 14  7 15)   ( 3 13 12  5  9)
( 9  2  5  7 11 vs  3 12 15  8  1)   ( 4 14 13  6 10)
(10  3  6  8 12 vs  4 13  1  9  2)   ( 5 15 14  7 11)
(11  4  7  9 13 vs  5 14  2 10  3)   ( 6  1 15  8 12)
(12  5  8 10 14 vs  6 15  3 11  4)   ( 7  2  1  9 13)
(13  6  9 11 15 vs  7  1  4 12  5)   ( 8  3  2 10 14)
(14  7 10 12  1 vs  8  2  5 13  6)   ( 9  4  3 11 15)
(15  8 11 13  2 vs  9  3  6 14  7)   (10  5  4 12  1)

16 players (team pairs 5 or 6 times, oppo pairs 6 or 7 times).

( 1  3  8 14 11 vs 12  5 16  4  2)   (15  6 10 13  9  7)
( 2  4  9 15 12 vs 13  6  1  5  3)   (16  7 11 14 10  8)
( 3  5 10 16 13 vs 14  7  2  6  4)   ( 1  8 12 15 11  9)
( 4  6 11  1 14 vs 15  8  3  7  5)   ( 2  9 13 16 12 10)
( 5  7 12  2 15 vs 16  9  4  8  6)   ( 3 10 14  1 13 11)
( 6  8 13  3 16 vs  1 10  5  9  7)   ( 4 11 15  2 14 12)
( 7  9 14  4  1 vs  2 11  6 10  8)   ( 5 12 16  3 15 13)
( 8 10 15  5  2 vs  3 12  7 11  9)   ( 6 13  1  4 16 14)
( 9 11 16  6  3 vs  4 13  8 12 10)   ( 7 14  2  5  1 15)
(10 12  1  7  4 vs  5 14  9 13 11)   ( 8 15  3  6  2 16)
(11 13  2  8  5 vs  6 15 10 14 12)   ( 9 16  4  7  3  1)
(12 14  3  9  6 vs  7 16 11 15 13)   (10  1  5  8  4  2)
(13 15  4 10  7 vs  8  1 12 16 14)   (11  2  6  9  5  3)
(14 16  5 11  8 vs  9  2 13  1 15)   (12  3  7 10  6  4)
(15  1  6 12  9 vs 10  3 14  2 16)   (13  4  8 11  7  5)
(16  2  7 13 10 vs 11  4 15  3  1)   (14  5  9 12  8  6)

( 1 13  9 14  8 vs 12  7  6  5  3)   (10 15 16  2  4 11)
( 2 14 10 15  9 vs 13  8  7  6  4)   (11 16  1  3  5 12)
( 3 15 11 16 10 vs 14  9  8  7  5)   (12  1  2  4  6 13)
( 4 16 12  1 11 vs 15 10  9  8  6)   (13  2  3  5  7 14)
( 5  1 13  2 12 vs 16 11 10  9  7)   (14  3  4  6  8 15)
( 6  2 14  3 13 vs  1 12 11 10  8)   (15  4  5  7  9 16)
( 7  3 15  4 14 vs  2 13 12 11  9)   (16  5  6  8 10  1)
( 8  4 16  5 15 vs  3 14 13 12 10)   ( 1  6  7  9 11  2)
( 9  5  1  6 16 vs  4 15 14 13 11)   ( 2  7  8 10 12  3)
(10  6  2  7  1 vs  5 16 15 14 12)   ( 3  8  9 11 13  4)
(11  7  3  8  2 vs  6  1 16 15 13)   ( 4  9 10 12 14  5)
(12  8  4  9  3 vs  7  2  1 16 14)   ( 5 10 11 13 15  6)
(13  9  5 10  4 vs  8  3  2  1 15)   ( 6 11 12 14 16  7)
(14 10  6 11  5 vs  9  4  3  2 16)   ( 7 12 13 15  1  8)
(15 11  7 12  6 vs 10  5  4  3  1)   ( 8 13 14 16  2  9)
(16 12  8 13  7 vs 11  6  5  4  2)   ( 9 14 15  1  3 10)

[mgdad]

Thanks so much Ian... this is perfect.  

[mgdad]

This year we have 13 players... same scenario as prior years, 26 total games over two days.  Might you be able to assist with a schedule as before.

Thanks Ian.

[Ian Wakeling]

Here is a 13 player schedule as requested.   Team pairs 6 or 7 times, opposition pairs 8 or 9 times.

 ( 1  3  6 10 12 vs  5  8 13  4 11)   ( 9  7  2)
 ( 2  4  7 11 13 vs  6  9  1  5 12)   (10  8  3)
 ( 3  5  8 12  1 vs  7 10  2  6 13)   (11  9  4)
 ( 4  6  9 13  2 vs  8 11  3  7  1)   (12 10  5)
 ( 5  7 10  1  3 vs  9 12  4  8  2)   (13 11  6)
 ( 6  8 11  2  4 vs 10 13  5  9  3)   ( 1 12  7)
 ( 7  9 12  3  5 vs 11  1  6 10  4)   ( 2 13  8)
 ( 8 10 13  4  6 vs 12  2  7 11  5)   ( 3  1  9)
 ( 9 11  1  5  7 vs 13  3  8 12  6)   ( 4  2 10)
 (10 12  2  6  8 vs  1  4  9 13  7)   ( 5  3 11)
 (11 13  3  7  9 vs  2  5 10  1  8)   ( 6  4 12)
 (12  1  4  8 10 vs  3  6 11  2  9)   ( 7  5 13)
 (13  2  5  9 11 vs  4  7 12  3 10)   ( 8  6  1)

 ( 1 13  8  9  6 vs  4  2  5 12  3)   (11 10  7)
 ( 2  1  9 10  7 vs  5  3  6 13  4)   (12 11  8)
 ( 3  2 10 11  8 vs  6  4  7  1  5)   (13 12  9)
 ( 4  3 11 12  9 vs  7  5  8  2  6)   ( 1 13 10)
 ( 5  4 12 13 10 vs  8  6  9  3  7)   ( 2  1 11)
 ( 6  5 13  1 11 vs  9  7 10  4  8)   ( 3  2 12)
 ( 7  6  1  2 12 vs 10  8 11  5  9)   ( 4  3 13)
 ( 8  7  2  3 13 vs 11  9 12  6 10)   ( 5  4  1)
 ( 9  8  3  4  1 vs 12 10 13  7 11)   ( 6  5  2)
 (10  9  4  5  2 vs 13 11  1  8 12)   ( 7  6  3)
 (11 10  5  6  3 vs  1 12  2  9 13)   ( 8  7  4)
 (12 11  6  7  4 vs  2 13  3 10  1)   ( 9  8  5)
 (13 12  7  8  5 vs  3  1  4 11  2)   (10  9  6)

Note that the byes have an interesting property - that every pair of players has byes together exactly once.  Perhaps you could use this property to have a separate competition.

[mgdad]

Ian, thank-you for your timely response again this year!  Have a great weekend!

Michael