Scheduling Combinatorics

© Copyright 2007, Paul Kislanko

The number of possible schedules for a 120-team field1 that plays a 12-game schedule in 14 weeks is a number too large to write down. It's not even productive to describe how to calculate it, because all the possibilities that don't have the right conference matchups in the right week need to be discarded and there's no mathematical way to know what those are.

1: For the purpose of analyzing schedules, we consider all 1-AA teams to be equal, and count them all as the 120th team in 1-A.

What we can do is bring out the DataMaster to slice and dice the actual schedule graph. In all of the following, rankings are the computer consesus compiled by Kenneth Massey as of January 8, 2007.

State of Affairs

Home→
↓Visitor 		1-20 21-40 41-60      61-80 81-100 101-120
1-20 21 27 18   12 16 7
21-40 25 26 15   16 12 12
41-60 29 18 17   16 22 12
               
61-80 24 10 24   16 20 19
81-100 14 25 17   25 16 19
101-120 26 28 36   42 38 35
There are 755 games between Division 1 opponents this year.

That last item might be considered a "mis-match quotient", and as fans we note that it includes fewer than half the scheduled games.
Home→
↓Visitor 		1-20 21-40 41-60      61-80 81-100 101-120
1-20 4 7 5   0 6 2
21-40 7 3 3   4 3 6
41-60 10 6 5   7 4 5
               
61-80 13 3 5   5 4 7
81-100 6 11 5   7 7 3
101-120 24 24 26   26 19 14

That's not the case when we switch to just non-conference games, of which there are 296.

Not surprisingly, 117 of the 154 "mismatches" are at the higher-ranked team's stadium.

SEC teams are stereotyped as weak non-conference schedulers, but if we summarize the data by conference we find that pretty much all the BCS conferences are about the same:
NC
Games 		Teams Conf 1-20 21-40 41-60 — 61-80 81-
100 		101-
120
40 8 SBC 11 8 8   5 1 7
12 1 ND 2 4 2   2 1 1
31 10 P102 10 4 5   2 3 7
36 9 MW 7 4 8   3 4 10
22 2 Ind 3 4 3   3 3 6
37 9 WAC 5 3 8   6 2 13
58 13 MAC 7 8 10   14 6 13
48 12 CUSA 10 8 2   4 11 13
48 12 ACC 9 6 5   10 3 15
48 12 SEC 4 7 4   4 7 22
48 12 B12 2 5 8   7 11 15
44 11 B10 4 2 5   6 7 20
40 8 BigE 2 3 5   6 8 16

2 - The PAC 10 is a bit of an anomaly because they play nine conference games. If one subtracted 17 games from the 12-team conferences' schedules all from the bottom, there'd be little difference.

There's a dynamic at work that makes this ordering pretty much inevitable. The top half of the field is predominantly made up of teams from the BCS autobid confierences and the bottom half by the others. So, except for teams from the Football Championship Subdivision, there's a better than 50-50 chance that any non-conference opponents for the non-BCS conference come from the top half of the field. Consider CUSA: there are 107 FBS teams not in C-USA, and 57 of those (53 percent) finished in the top 60 last year. Still, the Sun Belt teams are way over-scheduled.

Compare CUSA's situation to the SEC's. Again there are 107 possible non-conference opponents, but only 52 (48 percent) in the top 60. The stronger the conference, the harder it is to find strong non-conference opponents just because your conference mates use up too many of the desired ranks.

Another dynamic that's even more interesting (in the sense it complicates such measurements to a greater degree) is that it's a zero-sum game between rankings success and strength of schedule. The ACC comes out slightly ahead of the SEC in SOS because the ACC teams the SEC played got kicked down a notch by losing their games to the SEC. The SEC teams' whose power ratings moved up into the 1-20 range and the ACC teams' down into the 21-40 range because of those games shows how immeasurable "SOS" is when summarized at anything other than a team level.

Harder than Sudoku

Here's a fun game you can play solitaire-style, as if you were king of division 1, or you can invite 119 of your closest friends over and assign them the role of AD at all your potential opponents (be prepared for half of them to hate you, though!)

The goal is to improve the division 1 Football Bowl Subdivision schedules from the fans' perspective. That's not a very high bar, but there are some constraints that make it less than easy. The rules of the game:

  1. Standard scheduling rules apply
    1. Each team can have no more than 12 games within the 14-week period
    2. Each team must have at least 6 home games
    3. Teams in the ACC, Big 12, MAC, and SEC may not have games scheduled in week 14
  2. Conference games cannot be altered (conference games are flagged with †)
  3. Non-conference games between top-60 teams cannot be altered (These are flagged with ‡)
  4. When a bottom-half opponent is replaced by a top-half opponent, a replacement game for the bottom-half opponent must be provided.
    1. If two bottom-half opponents are displaced by matching up their top-half opponents, the bottom-half teams may only be matched as replacements if they are from different conferences.
    2. Championship Subdivision (1AA, flagged with *) teams may not be added to the schedule, but may be moved from a top-half matuchup to a bottom-half matchup that is scheduled for the same week
  5. The total number of bottom-half vs bottom-half games may not exceed 50% of all games

The solitaire version is a lot easier if you have a spreadsheet program that can handle Excel workbooks like this one. You can sort on each week to find matching open dates.