This weekend marks the second weekend of the NCAA basketball tournament (aka March Madness). For many, the NCAA tournament is the last chance to watch some of the best college basketball around. For others, it’s a time to fill out brackets, decide which 10-7 upset to pick, flip a coin on the 8-9 games, and cross your fingers.
If you’ve ever participated in an NCAA office pool, you’re familiar with at least one way of scoring correct bracket picks. For example, each correct pick is worth 1, 2, 4, 8, 16, and 32 points, in the 1st, 2nd, 3rd, 4th, 5th, and 6th rounds, respectively. This makes each round worth the same number of points, and thus gives equal weight to each round. But is that the best way to do things?
When I was still in graduate school, several of us (looking for another way to kill time so as to avoid doing real work) decided to examine different schemes for scoring brackets. Not that we were participating in an office pool – just in case we ever wanted to join one. Or start one. Or run one for several years.
With the above scheme (1, 2, 4, 8, 16, 32), there are 192 points available in the entire tournament, and each round is worth 1/6 of those points, giving each round equal weight. This places a great deal of weight on the final game, and for whatever reason, we found that unacceptable. Of course, being able to determine the overall winner out of 64 (now 65) teams ought to be significant, but we didn’t want an incorrect pick to essentially eliminate someone from the contest. So we started analyzing other natural-looking sequences of length 6, computing the relative weights of each round for each sequence:
|Points per game (Rds. 1-6)||Relative round weights (%)|
|1, 2, 4, 8, 16, 32||16.7, 16.7, 16.7, 16.7, 16.7, 16.7|
|1, 2, 3, 4, 5, 6||26.7, 26.7, 20, 13.3, 8.3, 5|
|1, 2, 3, 5, 8, 13||23.4, 23.4, 17.5, 14.6, 11.7, 9.5|
|1, 2, 3, 5, 10, 15||22.4, 22.4, 16.8, 14, 14, 10.4|
|2, 3, 5, 7, 11, 13||29.8, 22.3, 18.6, 13, 10.2, 6|
Certainly there are other “natural” sequences that could be analyzed. We ended up
choosing liking 1, 2, 3, 5, 10, 15 (although I no longer remember why). Perhaps it was a happy medium between schemes which weighted the final game too low (5 or 6%) and schemes that weighted it too high (16.7%). Regardless, we congratulated each other on an excellent nerding.
Later that spring, I discovered that Microsoft has an Excel template for scoring NCAA brackets, and I went berserk playing with it… but that’s a story for another day.