Sunday, 7 December 2003

The real scandal: coaches don't pay attention to football

James Joyner is right that, if things play out the way they look, the BCS is in big trouble. However, the hidden story in this is why the BCS is in trouble. Let’s look at the ESPN/USA Today top five (i.e. the Coaches’ poll):

  1. USC: 37 first-place votes, 1542 points.
  2. LSU: 18 first-place votes, 1516 points.
  3. Oklahoma: 8 first-place votes, 1449 points.
  4. Michigan: 0 first-place votes, 1393 points.
  5. Texas: 0 first-place votes, 1272 points.

As I’ve mentioned before, the major polls are compiled using a Borda count. The media poll (AP) has 65 voters, while the coaches’ poll has 63 voters. The Borda count procedure is fairly simplistic: the #1 team on each ballot gets 25 points, #2 gets 24… all the way down to #25, which gets 1 point. (In math terms, the points added for each team are 26 minus the ranking.) The Borda count, incidentally, happens to be a very rotten way of aggregating preferences, but it has the benefit over other methods (like Condorcet voting) of not requiring a lot of thought to apply.

With that aside out of the way, let’s stare at the numbers. The full ballots aren’t released (a glaring oversight in the system), but we do know that 8 people ranked Oklahoma #1. Assume, for the sake of argument, the “objectively correct” ranking of Oklahoma is no higher than #3; in other words, no voter should have ranked OU #1 or #2.* Oklahoma thus recieved 8×2 or 16 more points than it should have, reducing its total to 1433 points.

Let’s also assume that these 8 poll voters all ranked LSU, USC, and Michigan (a fair assumption); we don’t even need to know which team was ranked #2 on these ballots. Bumping Oklahoma to #4 bumps each of these teams up by 8×1 points. This gives USC 1550, LSU 1524, OU 1433 (per above), and Michigan 1393. Now, Michigan is within 40 points of being #3 (down from 56).

Now, let’s return to the original numbers. We know that some voters ranked OU above #3. Why? Well, for starters, they got 8 first-place votes. It also turns out that OU’s total of 1449 is exactly the total that they would have received had they been ranked #3 on all 63 ballots (63×(26-3)=1449). Now we have an interesting problem: reconstructing the position of OU on the ballots.

We know OU was #1 on eight ballots. They received 1249 points from 55 other ballots. Let’s assume the only reasonable rankings for OU on those ballots is 2, 3, 4, and 5. So we have an integer programming problem: (26-2)a+(26-3)b+(26-4)c+(26-5)d=24a+23b+22c+21d=1249, where a–d are all non-negative integers, and a+b+c+d=55.

Solving this problem iteratively, there are two possible ballot configurations: 17 second-place votes, 15 third-place votes, 13 fourth-place votes, and 10 fifth-place votes; or 18 second-place, 14 third-place, 12 fourth-place, and 11 fifth-place. Now, let’s also drop OU to third on these ballots.

Assuming there are 17 second-place votes for OU, dropping them to third will reduce their point total by 17. Assuming Michigan was ranked third or below on all of these ballots (a trivial assumption; we know they never were ranked #1, and we know they couldn’t have been #2 because OU was), OU loses 17 and Michigan gains 17. This makes the point totals: USC 1550, LSU 1524, OU 1416, and Michigan 1410.

If there were 18 second-place votes for OU, the “correct” point totals—if they’d actually ranked OU third—would have been USC 1550, LSU 1524, OU 1415, and Michigan 1411.

Now, we can broaden the analysis a little. Assume that some voters ranked OU as low as 6th. If that is the case, more than 18 voters must have voted OU #2 for them to get 1449 points. If 20 voters ranked OU #2, when they should have been #3, OU and Michigan would have been tied. And it’s possible, although unlikely, that as many as 24 voters ranked OU #2.

The moral of the story: the Borda count sucks. And so do the coaches.

* I personally think Michigan is a more worthy #3. However, I think it’s disputable that OU should be ranked below both LSU and USC: that the best team in the country doesn’t lose by four touchdowns (28) to an inferior team at a neutral site. USC lost its game to Cal by a field goal, while LSU lost by a touchdown to Florida. By way of comparison, 9–3 Ole Miss—a team ranked in the teens in both polls—lost three games by a combined total of 17 points (10 points @Memphis, 4 points versus Texas Tech, and 3 points versus LSU).