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# Selecting March Madness Brackets: Why Second-Favorite Teams Matter

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## Spoiler alert: This approach may help you in the office pool, but it's not going to get you Warren Buffett's billion-dollar prize.

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Well, as I write this, it's Selection Sunday 2014 -- time to get out and win some NCAA Basketball Bracket Pools. I've written three columns on the mathematics of the subject over the last two years; read those if you want the background. (Start with "March Madness: How to Gain the Betting Edge.") This year I'll content myself with a quick reminder.

The NCAA Men's Basketball Tournament (aka March Madness) starts with 68 college basketball teams and plays a single-elimination tournament to determine one winner. Since each game eliminates one team, that means there will be 67 games, and each year an amazing number of them turn out to be well-played and exciting.

A bracket pool is a game played -- mostly in offices, mostly illegally -- in which contestants fill out predictions for the 63 games (the first four are usually skipped) and pay a (usually) small amount of money. The top scorers according to some system are awarded shares of the pot.

Most people fill out their bracket selections according to team loyalty, emotion, superstition, rules of thumb or semi-randomly. Those who think about it usually concentrate on either basketball insight or analyzing the probabilities for the specific scoring system used. This is mostly a waste of time. Very few people have the ability to form better predictions of individual games than apps you can download free from any of a number of good quant analyst websites. Moreover, precise estimates of individual game probabilities are not very useful.

Analyzing the details of your bracket pool scoring system is somewhat more useful, but it's basically true that a good pick is a good pick, regardless of scoring system. Suppose, for example, that your office in Boulder is full of University of Colorado Buffalo fans, in love with Josh Scott's exciting play and undeterred by the recent weak game against USC and epic thrashing by Arizona. You think 80% or more of them will pick number 8 seed Colorado over number 9 seed Pittsburgh in the South regional (do not ask why two northern colleges play in the South regional). However, as a dispassionate numbers person, you know Pittsburgh is a significantly better team with a 69% chance of winning the game. Pittsburgh is a good pick. It doesn't matter how Round 1 games are weighted versus other games, or whether there are extra points for picking underdogs or anything else about the scoring system. And it doesn't matter if you think Pittsburgh is a 60% or 80% favorite.

Granted, that's an extreme case, but it's usually true that the details of the scoring system only come into play after you've winnowed your selections down to a few marginal last choices. And it doesn't always matter in the obvious way. For example, if the system rewards upsets, other people will pick more upsets, and often overreact, to the point where you should pick fewer upsets. It's more important to have a bracket selection sheet that is different from most other people's than to have a bracket selection sheet likely to get a high score.

The teams are divided into four regionals, and each team within the regional has a seed. Other than the first and last four games, the teams playing will have different seeds, with the lower-seeded team usually the more likely to win (Colorado-Pittsburgh is an exception). In a typical year, two-thirds of the slots are won by the lowest seeded team that can reach the slot, one-quarter by the second-lowest seeded team, and only 1/12 (usually five or six games a year) by even higher seeds.

One interesting regularity is that the chance of the second-lowest seed winning the slot is roughly constant in all rounds. In the first round the teams are usually mismatched, with the lower seed a strong favorite. But if the favorite loses, the second favorite has to win. In the late rounds the games are more evenly matched, but there is more opportunity for the second favorite to have been eliminated before the game takes place. Those two effects balance out pretty well.

You can write your number of games predicted correctly as: number of slots won by favorites + number of upset picks you got right - number of upset picks you got wrong. Since the first term is the same for everyone, all that matters is how many more upset picks you got right than wrong. Only upset picks matter.

A score of zero, the same number of correct versus incorrect upset picks, is not likely to win. For one thing, someone may get zero just by picking all favorites. For another, someone will almost certainly get more correct than incorrect upset picks. Therefore, you need to make enough upset picks to have a decent chance of getting more right than wrong. How many you need depends on the size and sophistication of your pool. The more people, and the smarter the people, the more upsets you're going to have to get right.

The trouble is, as you pick more upsets, you have to pick games where the second-favorite team has less chance of winning (except in the largest pools, it very rarely makes sense to pick anything but a favorite or a second-favorite). This, plus the law of large numbers, works against you. If you don't make enough upset picks, even if you win them all, you may not win the pool. If you make too many, your chance of getting more right than wrong is so small that you have less than an average chance of winning the pool. A rule of thumb that works reasonably well is to make upset picks equal to between one and two times the square root of the number of entries in the pool, with the lower number if the runner-up prizes are large or the other pickers unsophisticated, and the higher number if the runner-up prizes are small or the other pickers sophisticated.

Warren Buffett's billion-dollar prize does not fit this model, because in that case you don't care about getting closer than anyone else -- you care about getting every game correct. That's a different game, with different strategies.

So here's my personal list of second-favorites to pick, starting from the highest probability down to the lowest. I left out the first and last four; because the seed can be the same, you'll have to pick these on your own. Generally you should select the most likely second-favorites, except you should avoid ones likely to be popular in your pool, like local teams and well-known schools that aren't as good as usual this year, teams that were Cinderellas recently, and teams with big stars and a lot of press. You also should randomize, in case other people in your pool are looking at my list.

The table below shows the regional (East, Midwest, South, or West), the seed and team, the round in which it will be the second-favorite, and my estimate of its probability of winning that slot. For example, the most likely second-favorite to win is Pittsburgh in Round 1 over Colorado. Going down about halfway, you see I have the East's number 2 seed in Villanova with a 22% chance to win in the Elite 8 round (probably over Virginia), in which case Villanova would go to the Final Four.

So good luck to everyone, and don't forget the best part of March Madness is the games themselves. Enjoy a few weeks of great basketball -- I hope you win a little money, too.

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