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# March Madness: How to Gain the Betting Edge

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Last year I wrote two columns about using game theory to improve your NCAA basketball bracket play (see March Madness: Using Game Theory to Win the Office Pool and March Madness: Using Game Theory to Win Your Upset Picks). This year I'll update it with information about the 2012-13 season.

Before I get to that, however, I want to emphasize the nature of the game again. For a deeper explanation, refer to the earlier pieces.

The NCAA men's basketball tournament starts with 68 teams and uses a single-elimination format (one loss and you're out) to get down to a single national champion. Since each game eliminates one team, that means we need 67 games. Some bracket games use all 67 games, some use only the 63 after the “first four” games are used to select the “play-in” teams.

If our goal were to maximize the number of games we picked correctly, we would pick all favorites. However that's a foolish strategy in any reasonably large pool, because someone will almost certainly pick more upset winners than upset losers, and will beat us. Getting a lot of games right isn't the point; getting more games right than anyone else in your pool is. Your best chance of that is to make sure your selections are different from everyone else's.

There is an analogy with the Powerball lottery. Suppose the grand prize were given out to the person who got the most balls right, rather than being reserved for someone who gets all six balls right. If everyone else picks at random, there is nothing you can do to improve your odds. You have something like one chance in 200 million of winning the prize outright, and about two chances in a million of sharing the prize (likely with about 20 other people).

But what if no one else were allowed to pick your numbers? Now your chance of winning outright goes up to about 1 in 250, and your chance of sharing the prize to about 1 in 20 (but you'll have to share it with over half a million other winners). The fact that no one shares any of your numbers does nothing to increase the probability of you getting matches, but it makes a dramatic difference to your chance of having the most correct picks.

In Powerball, all numbers are equally likely to be chosen. In the NCAA tournament, in order to make your bracket sheet different from everyone else's, you'll have to pick upsets.

Here's another way to think about it. Suppose for each slot there were only two outcomes: favorite wins or upset wins (that's true only in the first round, but the deviations don't make much difference). We could write your total wins as the sum of the number of times the favorite occupies a slot, plus the upsets you picked correctly, minus the upsets you picked incorrectly. Since the first term is the same for everyone, the pool will be won by the person with the most correct minus incorrect upset picks. Only the upset picks matter.

Someone will almost certainly get at least zero, perhaps by picking all favorites, or perhaps by picking an even number of upsets and winning half. So you need to win more than half your upset picks to have much chance. Therefore you cannot pick too many upsets as that would require picking some low probability upsets, and the law of aveages would catch up with you. You would be very unlikely to win more than half. But you also don't want to pick too few upsets, or you won't have much chance to build up the net advantage you need to win.

Emily Dickinson expressed this poetically when her brother criticized her for making too many upset picks in her pool, and for failing to pick local favorites Amherst and Mount Holyoke. She wrote him:

March Madness is divinest Sense -
To a discerning Eye -
March Sense - the starkest Madness -
’Tis the Majority
In this, as all, prevail -
Assent - and you are sane -
Demur - you’re straightway dangerous -
And handled with a Chain -

In an average tournament, about two-thirds of the bracket slots are occupied by the lowest seeded team that can get there (or, for first four and final four games where teams of equal seed can play, the team with the higher RPI). About one-fourth are occupied by the second-lowest seeded team that can be there. Slots occupied by teams with higher seeds than the second-lowest seed possible are rare, five or six in a typical tournament.

An interesting regularity is second-lowest seeds occupying slots are roughly equally likely in all rounds. In the first round, for example, upsets are relatively rare because most of the games are unequal. But if there is an upset, it has to be by the second-lowest seeded team, since only two teams can win. In later rounds the games get more even, so upsets are more common, but it's more likely that someone other than the second-lowest seed is doing the upsetting.

This year is much less predictable than most. There is no dominant good team, in fact no team that seems to be able to win consistently versus even marginal opponents. Some of the non-elite teams have shown flashes of brilliance, or have gotten stronger or weaker as the season went on. Seeds are based primarily on overall season performance, so a team on the upswing will be under-rated and a team in decline will be overrated.

Basketball analysts usually exaggerate the extent to which this is true. Every season has surprises and ups and downs. A lot of the impression is overinterpretation of random results. But there is some basis of fact in the observation as well.

Many people will use this unpredictability as an excuse to make more upset picks. There is little mathematical basis for that. What it does argue is for making more of your upset picks in the first round, because that is the round in which you are confident who the winner will be if there is an upset.

Generally, you want to select the most probable upsets, subject to the caveat that you don't want the upsets everyone else is picking. I've put together my personal chart below of all upset possibilities from  most likely to least likely.

My advice is to stay away from upset candidates that are strong basketball schools in general, or that were strong last year, or that are favorites in your office. Look for the lesser-known schools without local partisans, and happen to be strong this year.

To get the full advice, read last year's pieces. The basic idea is to pick upsets where the higher-seeded team (or the lower RPI team where seeds are the same) has a relatively good chance to win. I rank the games from 1 to 67 for upset attractiveness. The table below shows my rank starting from the most attractive (Florida instead of Georgetown in the fourth or semi-final round of the South regional, I think Florida is significantly more likely than Georgetown to occupy the winner's spot of that game). The slot is the regional (East, Midwest, South or West) and round (the first round is the “first four” play in round, then there are four rounds in each regional to get down from 16 teams to 1 team, then the national semi-finals and national finals). The favorite is not necessarily the betting favorite, it's just the lowest seed team that can win the slot; similarly the second favorite is the second-lowest seed team (when seeds are equal I go with the higher RPI).

In the first six slots, I think the second favorite is more likely to win than the favorite, so I would pick these in any bracket (you may disagree, of course). The next three are close to toss-ups, although I think the favorite has a slight advantage. As you continue down the list, you give up more and more by picking the second favorite (it rarely makes sense to pick anything lower than the second favorite). Remember that the higher seeds are not only usually better teams, but they have easier paths to the game. Still, you have to pick upsets to have a chance, and it's usually best to select the likely ones.

As always, good luck.
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