college football

For something (almost) completely different, I'd like to make some proposals regarding playoffs for the top division in college football.

My objection to an expansive playoff is primarily that the best team doesn't win every game, even insofar as there is such a thing as a best team; if #16 beats #1 in the first elimination game, I'm going to be wondering whether the winner of the tournament really is better than the team that lost that first match.  Of course, polls and selection committees aren't perfect, either, which is at least part of why playoff systems are popular; if there's a legitimate case to be made that #3 is likely to be better than #1 or #2, let's let them play each other and find out.^*

Part of the difficulty of comparing teams on the basis of their 12 games each is that there is so frequently little overlap among those teams' opponents.  Not only did Oklahoma State not play Alabama last year, but no team Oklahoma State played played Alabama last year.  There is an increasing tendency for teams to play almost all of their games in their own conference; this is sometimes hailed as a good thing, typically because the best teams tend not to play very good teams in their non-conference games — which makes it even harder to compare good teams in different conferences than if the occasional shared opponent were at least good. It would be better to have stronger schedules that weren't quite so incestuous.

On the other hand, in college basketball, some of the lower "mid-major" conference teams have complained that they have a hard time getting teams from the top conferences to play them precisely because strength-of-schedule is so emphasized; the top teams don't want to play a team that might hurt their strength-of-schedule metrics, so those teams never get a chance to develop the resumes that would make them look strong.  It would be nice if incentives induced teams to give the outsiders a chance to show what they have, but without creating too many rewards for beating up truly weak opponents.

Accordingly, here is my proposal:

1. A six-team tournament, comprising
   • the four teams believed by some reasonable process (e.g. a committee) to have the best claim to being the best team in the country
   • The next two teams selected by a process that emphasizes schedule strength and diversity
2. Teams 5 and 6 are selected according to the following ratings (probably modified for teams that play more than 12 games) that awards points to each team for each game between division I-A football teams:
   • For teams i and j, let n_ij be the smallest number of games required to link them; thus n_ii=0, n_ij=1 if i plays j, n_ij=2 if i did not play j (and i is not j) but they played a team in common, etc.
   • Team i gets 1/8^n_ij points for any game that team j wins
   Thus you get 1 point for a game you win, 1/8 for a game an opponent of yours wins, etc. — but you can only get credit for a game once

The rating system gives too much credit for strength of schedule, but only the two "incentive spots" in the playoffs are awarded on that basis; they won't be bad teams, and they won't get to the semifinals without beating a top 4 team, so they won't screw up the integrity of the playoffs, they just provide an inducement to athletic directors not to try to schedule too many weak teams.  On the other hand, if a mediocre team is being ignored by all the good teams, and is going to end up 8-4 against a weak schedule, then adding that team to your schedule is going to do you more good than adding a 6-6 team with a tougher schedule against teams that some of your other opponents are also playing; you move those 8 wins to a lower n_ij, as well as the wins of their opponents.  The general equilibrium effect, then, is actually to improve the ability of the committee to select the top four teams; when scheduling has been done knowing that, whatever else happens, a team with a good rating can snag one of these two incentive spots as a backup, that scheduling will result in more Oklahoma States scheduling decent common opponents with more Alabamas, increasing the likelihood that the teams that make the semifinals include the best one or two teams in the country.

---

^*The race isn't always to the swift nor the battle to the strong, but that's usually the way to bet.