Anti-Buzz: Bracketology

newface-620x461So in American Sports Fandom there is this thing called March Madness, famous for its high variance and its invitation to fill out predictive tournament brackets. ESPN has coined the phrase “Bracketology” and when our President participates, they call it “Baracketology.” Some of my Canadian colleagues have admitted that they quietly have the stereotype of Americans “loving brackets.” If somehow you didn’t know, March Madness, or the NCAA Men’s Basketball tournament, is going on right now, and this year’s tournament came with a little bit more buzz due to the Quicken Loans Billion Dollar Bracket Challenge, backed by Warren Buffet. That is, a billion of Warren Buffet’s dollars to anyone who can perfectly guess the entire tournament. So once again journalism and mathematics meet, and things get a little confused.

In some ways, this contest is a modern rendition of Scientific American’s Largest Number game. The contest offered a million dollars in a raffle, and contestants could submit a number indicating the number of entries they wished to submit. The catch? The total prize money would be divided by the number of entries. You could submit a million entries yourself, but the total (total), prize would drop down to one dollar, and that’s if you were the only one playing. The similarity with the Billion Dollar Bracket Challenge is that nobody running either contest expects to have to pay out any cash. (To be fair to Quicken Loans, they are offering much-less-than-a-billion-dollar prizes to those who do the best).

Many outlets have warned contestants that they only have a 1 in 9.2 Quintillion chance of winning, odds that make the Powerball look like a safe bet. If you need perspective, A quintillion is a billion billion. Those odds aren’t actually accurate, but to understand what a monumental task predicting the entire NCAA tournament is, we can begin with where that number came from. As I recently expounded on powers of 2, it might not take you long to devise that there are about 9.2 quintillion different ways the tournament could play out. You can think of the bracket like a tree as I have described in weeks past, but if you want an even easier explanation, there are 64 teams, each game played eliminates one team, (you don’t even have to care what round it is in), and so it takes 63 games to finish the tournament. Each game has two possible outcomes, which means in total there are 2-to-the-63rd-power possible combinations – roughly 9.2 billion billion.

If you tried to guess the exact sequence of 63 coin flips, you’d be facing the 1-in-9.2 quintillion odds, but we all know that basketball games are not coin flips. This is why sports journalism has so much heft. In general, sports outcomes are much too complex to model reliably, and so everybody’s intuition about the narrative that best explains a team’s odds are or best explains the outcome of an event is fair game. They were tired, they had the momentum, they came to play, that key injury really cost them, this team is better but matches up poorly with their opponent. All of that is armchair statistics in a void where real statistics does not have much traction.

But anyway, what all the giddy sports fans know is that their sports knowledge gives them an edge over those mighty nine quintillion. And it does. A person who filled out a bracket at random, would have the 1-in-9-quints shot of winning, (because their random bracket is just as likely to be “good” as it is “bad”), so injecting any amount of knowledge into your bracket improves your shot of winning. But the odds are still so long that Warren Buffet’s billion dollars should be safe.

Imagine I had computed the most likely bracket. In fact, somebody who entered the contest probably has. The odds of this bracket winning are still very very small. A popular misconception about probability is that the most likely outcome is likely. I know that sounds weird. Let me explain:

Consider that I flipped four coins and asked you guess how many times heads will come up. You’re intuition is correct: the most likely outcome is 2. However, the odds of this outcome happening are outweighed by the odds of it not happening; there is a 3/8 chance exactly two heads come up – the most likely outcome – but there is a 5/8 chance that something else happens.

So let’s say your expert sports knowledge led you to the best bracket and that this bracket improved your odds by a billion. A billion! Improving your odds to … 1 in 9.2 billion. Still a long shot. If you don’t like that, consider that, provided seeding and rankings are fair, the the most likely outcome for every NCAA tournament would always be for the top seeded teams to always win, something that famously refuses to happen during March Madness. It should be clear by now that the odds of the most likely thing happening are slim, which by the way is at the crux of the NCAA tourney’s hype: Unlikely happens!

As for the details of the contest, there was a cap of 15 million entrants. With 9.2 quintillion possible realities, even 15 million guesses is paltry, and this assumed no duplicate submissions. With 15 million distinct random brackets, the odds of Warren Buffet losing his billion are still about 1 in 600 billion. Even if the world colluded against him and all the entries combined to be the 15 million best guesses, I don’t see things getting dangerous.

Still things might be hairier than anybody expected; With the first round of the tournament already over, 32 of 63 games have been played, and the 15 million entrants have been reduced to 16, a survival rate of about one in a million. With the other half of games yet to be played, a similar survival rate means the odds of one of those 16 winning is about 1 in a million, still a very long shot, but those are now way-better-than-powerball odds.

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