## Bracketology

March is a little bit madder than usual this year, ever since investment mogul and noted suit-wearer Warren Buffett put up (along with Quicken Loans) a cool \$1 billion for anyone who can perfectly predict the results of  the NCAA men's basketball tournament.  No purchase necessary.

While that's certainly an astounding offer, it comes with a bit of a mathematical asterisk.  It's almost impossible to win.  That might not be totally surprising, given the magnitude of the prize.  Or maybe you followed last year's bracket-busting: none of the more than 8 million people in ESPN's online contest even got through the first round unscathed.  But just how slim are your chances of buying that island?

For the uninitiated, there are 64 teams in the tournament (Buffett is ignoring the play-in games, so we will too), which means there are 63 games: the first round has 32 match-ups; the 32 winners play 16 more games in the second round; all the way until the last two surviving teams play in the national championship game.  If you keep a running total, you'll see that there must be 32 + 16 + 8 + 4 + 2 + 1 = 63 games.  Or, more beautifully, notice that the tournament must eliminate every team but one, with the other 63 teams getting ousted along the way.  That ousting happens at a rate of exactly once per game, so there must be 63 total games.

Each game has two possible outcomes: one team or the other must win (no ties!), which means there are 263 possible tournament results.  In other words, there are 263 possible brackets.  (Just to give you some context, that's conservatively 20 million brackets for every star in our galaxy.)  So if you were hoping to pick your winners by throwing darts, your chances (1/263) are indistinguishable from zero.

Of course most people don't fill out their brackets completely at random.  They try to use information about teams' regular season performance to make educated guesses about who will win each match-up.  So let's say someone can pick winners with 75% accuracy.1  That's pretty great.  Gambler Hall of Fame great.  But 0.7563 is still only something like 3 in 200 million...not so amazing.2

But all hope is not lost!  Mr. Buffett is keenly aware that no one is going to win the \$1 billion grand prize, but he's still giving out \$100,000 to each of the 20 best brackets.  That's not island money, but it's not bad for sitting around watching basketball.

So what does it mean to have the "best" bracket?  Should you get a lot of credit for picking the tournament winner?  Should the person who correctly picks the most games score the highest?  Should games in later rounds be weighted more heavily than those in earlier rounds?

There are several different ways that online and office pools throughout the country score their brackets, but two common ones are the Default (or Traditional) and Progressive systems.  The Default system is a geometric progression: correct predictions are worth one point each in the first round, two points in the second round, four points in the third round, and continue doubling each round until the championship, which is worth 32 points.  The Progressive system is an arithmetic progression: first-round games are still worth one point each, but then the point value simply increases by one per round, so that the final game is worth six points.

There is (perhaps unsurprisingly) much debate about which system is better.  Default scoring places a huge amount of emphasis on picking the winner of the tournament (as much as picking the whole first round perfectly), which many people think is the most important factor in a winning bracket.  The Progressive system places more emphasis on choosing lots of winners, rather than just late winners, which many people think shows more skill and knowledge of the teams.

For what it's worth, Buffett's challenge uses Default scoring.  So who do you think will take it all this year?  It could be worth a lot of money.  Well, not really.  It's still fun, though.

Teachers, want to have this conversation in class?  Check out our lesson materials.

1.  Even though which teams win in each round is obviously highly dependent on which teams won in the last round, let's assume for simplicity that this 75% accuracy is independent.  Based on an incredible track record of gambling, this person picks winners 3/4 of the time.

2.  Our estimate is highly generous, because we're technically giving the gambler a chance to pick a winner in a game (s)he may have already blown with bad picks in a previous round.  The real probability is even smaller than we're calculating.