Sunday, 7 December 2003

The allegory of the cave

Today’s big exercise—besides watching the Indianapolis Colts defeat the Nashville Carpetbaggers—has been dealing with job applications. I’ve divided the pile of job listings, representing about 30 jobs, into four stacks based on the sort of packet they’ll get (Research/American, Research/Methodology, Teaching, and Post-Doc), updated and polished the vita, and printed out my old teaching evaluations.

One of the more peculiar requirements of many academic job applications, particularly in the “Teaching” stack, is that they require a statement of teaching philosophy—sometimes coupled with a statement of research interests. I have a broad idea of what I’d like to write, but these exercises, like the related need to write cover letters, always seem to call for a degree of introspection that makes me uncomfortable—I’ve never been a huge fan of the “self-marketing” exercise. I’ll get through it, but it still bugs me.

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).

My top five

James Joyner notes the upcoming Charlie-Foxtrot in the BCS standings resulting from Oklahoma’s drubbing by Kansas State in the Big XII title game and the convincing wins by LSU (over Georgia) and USC (over Oregon State). If I had a ballot, here’s how I’d rank the teams:

  1. Southern Cal (11-1). Their only loss was in overtime, which—given the funky OT rules in college—is understandable.
  2. LSU (12-1). Really, only three teams have even given LSU problems this year: Florida, Georgia (in the first game), and Ole Miss. Everyone else, LSU has basically destroyed.
  3. Michigan (10-2). Probably the scariest team in the country today, even if John Navarre isn’t your prototypical great quarterback. Then again, he doesn’t have to be; he’s got Chris Perry in the backfield.
  4. Oklahoma (12-1). The team folded like a cheap kite at the first sign of real adversity this season (the 5-8 Crimson Tide—with losses to NIU and Hawaii, along with practically everyone in the SEC—don’t count as “adversity” in this discussion). They’re lucky I haven’t dropped them below the TCU Horned Frogs or the Blue Turf Warriors of Boise State.
  5. Texas (10-2). Really, just the best of a bunch of two-loss teams lurking below Michigan.