Wednesday, November 7, 2012

How To Pick A Tournament Bracket, Part 1

Pre-season is probably not the best time to be pondering the Tournament, but I've been recently thinking a bit more about the challenge of predicting the Tournament with the goal of winning something like the ESPN Tournament Challenge or the Machine Madness contest.  These sorts of contests are a dilemma to a machine predictor, because most predictors try to determine who is most likely to win a particular matchup.  But of course, that's exactly how the Tournament is seeded.  So the machine predictors end up predicting almost entirely "chalk" outcomes.

The only time the machines don't predict a win for the higher seed is when they believe the teams have been mis-seeded -- that is, when the Committee has made a mistake in their assessment of the relative strengths of the teams.  In last year's Machine Madness contest, Texas over Cincy and Purdue over St. Mary's were consensus upset picks by five of the six predictors -- strong evidence (to my mind, anyway) that those teams were mis-seeded.  But, for all the grumbling by fans, the Committee does a pretty good job at seeding the Tournament, and you can't expect to find many true mis-seedings. 

Neither chalk picks or mis-seedings are likely to win a Tournament challenge against a sizable field.  That's because (1) a lot of your competitors will have made the same picks, (2) there will be a significant number of true upsets where a weaker team beats a stronger team (historically, 22% in the first round, and 15% for the Tournament overall), and (3) someone out there will have picked those upsets.  So to win a Tournament challenge, the machine is going to have to pick some actual upsets -- and then hope that it gets lucky and those upsets are the ones that happen.

Knowing the historical frequency of upsets, my strategy last year was to force my predictor to pick 6 upsets in the first round and 5 more in the rest of the tournament.  But is that the right way to pick upsets?  How can we pick upsets to maximize (in some sense) our chance to win the Tournament challenge?

The first problem in answering this question is knowing how many points will be sufficient to win the challenge, because that will drive the selection of upsets.  Obviously, it's impossible to know this number a priori.  However, we could look at previous Tournament challenges and see how many points the competitors in the top (say) 1% had scored off correctly predicting upsets.  That would provide a reasonable goal G for our upset calculations.

Sadly, ESPN, Yahoo, etc., seem to remove the Tournament challenge information from the Internets fairly quickly, so I can't actually research this.  (If someone has some info on this, please let me know!)  However, we do have the results of the last two Machine Madness contests.  Last year, the winning entry scored 127 points and the "chalk" (baseline) entry scored 120 points, for G = 7.  The year before, the winning entry scored 69 points and the chalk entry scored 57 points, for G = 12.  (There's undoubtedly a correlation between the size of the field and G.   G = 12 might be sufficient most years to win the Machine Madness contest, but probably wouldn't be enough to win the ESPN Tournament Challenge.)

If we adopt the notation that V(u,v) is the value of a victory of Team U over Team V, then we will want to pick upsets such that:
G < V(u1,v1) + V(u2,v2) + V(u3,v3) ...
Because of the way the tournament is structured, the value of V(u,v) is determined by the seeding of the two teams.  The following table has seedings down both axises and shows how many points an upset is worth:

For example, a #8 seed beating a #1 seed is worth 2 points, because that matchup will necessarily occur in the second round.

If we adopt the notation that p(u1,v1) is the probability of u1 defeating v1, then the probability of G is:
p(u1,v1) * p(u2,v2) * p(u3,v3)  ...
(because we must get all of our upset picks correct to score G points).

Now imagine that we are predicting the tournament and we know that most games have a 0% chance of an upset.  However, four of the third round games are very likely upsets -- 49%.  And one of the semi-final games has a slight chance of an upset -- about 6%.  If G = 16, which upsets should we pick?

The (possibly surprising) answer is that we should pick the very unlikely semi-final upset!  To get 16 points we have to pick either the semi-final game, or all four of the third round games, and:

       .06 > .49*.49*.49*.49

The joint probability equation combined with the typical Tournament challenge scoring means that it will almost always be better to pick unlikely late round upsets that score highly than multiple likely early round upsets that score poorly.  Knowing this, it's easy to see that my strategy in previous years to force a certain number of upsets into my bracket was very non-optimal.

So, given a goal G and upset probabilities p(u,v) we have an approach for selecting upsets from our bracket.  We've seen how to calculate V(u,v) and how to estimate GHow can we estimate the upset probabilities?

Many predictors will produce something that can be used to estimate upset probabilities.  For example, in past years my predictor has used the predicted MOV to estimate upset probabilities -- the slimmer the predicted margin of victory, the more likely an upset.  But lacking any information of that sort, we could estimate the upset probabilities based upon historical performance of seeds within the tournament:

This table shows the upset percentage for each seeding matchup for the last ten years.  (I have left out matchups that have occurred 4 times or less.)  Each upset percentage is shaded to indicate the value of the matchup in typical Tournament challenge scoring.  For example, matchups between #1 seeds and #2 seeds have been won by the #2 seeds 52% of the time, and they are worth 8 points. With a few oddball exceptions (such as the 2-10 matchups), this table shows is that you should prefer to pick upsets of the #1 or #2 seeds by #2 or #3 seeds.  These matchups are worth the most points and -- because the teams are closely seeded -- are nearly tossups.

So if G = 16, filling out your bracket with all chalk picks and two #2 over #1 upsets would give you the best chance to win the Tournament challenge.

More thoughts to follow in Part 2 at some later date.

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