# Perfect Billion-Dollar Madness

Every year, the United States goes mad in March over basketball. The NCAA’s single-elimination men’s basketball tournament, known as “March Madness,” is one of the United States’ most-watched sporting events. It starts with 68 teams and narrows to 64 after two play-in games. The madness begins with the following 32 games, played over 2 days, in the first round of the tournament.

As buzzers mark the end of games, millions of March Madness fans compare the outcomes with their brackets. To create a bracket, you pick a winner for each first round match up. Then, you choose a second round winner from your two predicted first round winners. You continue predicting winners round by round until you have your pick for national champion. The trick is (and for this audience the math lies in) who to pick!

This year, the madness may hit rabid levels as Warren Buffett’s company, Berkshire Hathaway Inc., has insured a billion dollar prize for creating a perfect bracket. His company’s pay-off? The prize is actually offered by Detroit-based Quicken Loans Inc., the nation’s fourth-largest mortgage lender. Quicken Loans pays Berkshire Hathaway Inc. a fee to insure the prize money. If a perfect bracket happens, Buffett’s group writes the check.

So, Buffett is essentially betting that no one will create a perfect bracket and announcing it this week means that no one, including you and me, will derive new methods that can!

Keep in mind, there are 2^{63} ways to fill out a 64-team bracket and 2^{63} nanoseconds is just under 300 years. Note, ESPN and CBS Sports online tournaments have never contained a perfect bracket. So, it looks like Buffett’s got a sure thing!

Note though, it may not be necessary to create 2^{63} brackets. The brackets are divided into groups – South, West, East and Midwest regions. In each, the teams are ranked from 1 to 16 with the number 1 seed playing the number 16 in the first round. If you chose the number 1 team in each region to win the first round, you would have been right in every tournament ever played.

Assuming equally likely outcomes results in a 0.5^{63} or 1.1 x 10^{-19} chance of creating a perfect bracket. This is about the same chance as being dealt 14 straight blackjacks in a row. Assuming all brackets are equally likely essentially assumes a 50% chance of either team winning. Clearly, that isn’t the case. Suppose you look at historical data and decide that on any given game there is a 70% chance of knowing who will win the game. Then the probability jumps to 1.7 x 10^{-10}. If we have 10 million different brackets, suddenly the odds of having a perfect bracket are one in a thousand. Depending on how you model things, Buffett has a sure thing or is being quite risky.

Notice further that significant digits become very significant in this context. Suppose rather than assuming 70% predictability for a game, 71% is more accurate. Now, the chance of a perfect bracket climbs to 4.2586894e-10, which is close to 2 and a half times more likely than assuming 70%. Comparing this to our earlier value, this is as likely as being dealt 7 straight blackjacks.

Berkshire Hathaway studied these types of mathematical issues before insuring the billion-dollar prize. This isn’t the first billion they’ve insured either. In 2003 and 2004, they insured the Pepsi contest “Play for a Billion.” Studying probabilities inherent in such contests is essential in order to insure such colossal prizes.

Mike Lawler, Vice President in the Berkshire Hathaway Reinsurance Group, has been involved in the March Madness and Pepsi contests. Discussing the probability of a perfect bracket, Lawler commented, “There isn’t one such probability. It depends on the tournament.”

For example, in 2006, no number 1 seeds reached the Final Four but 11th seed George Mason did. In 2008, Kansas’ win over, my very own Davidson College, placed every number 1 seed in the Final Four. These two years would be quite different in how likely anyone would be to create a perfect bracket.

In fact, the variability of the tournament can be seen in the first round alone. Last year, it took only 24 games to whittle the field from 8.15 million to 1 on ESPN. Only 1 bracket perfectly predicted those games. In 2012, the best bracket out of 6.45 million predicted 30 of the 32 first round games. And again, ESPN’s tournament, which has run for 16 years with over 30 million submissions, has never seen a perfect bracket – ever.

For Berkshire Hathaway, here lie two math problems. First, how do you compute or quantify such a probability enough to insure a billion dollar prize? Second, how do you ensure everyone plays fair? Lawler noted, “Remember, it’s really easy to make a perfect bracket after the games are played.”

So, underneath insuring such a prize lies another important issue at the intersection of math and computer science – security. There is a billion dollars now on the line. Remember, Target’s credit card information was recently hacked. Lawler noted, “When you have a prize this big, some people can have incentives to spend lots of money to rig it.”

That’s exactly why Berkshire Hathaway has developed various checks and balances along the way and at completion to secure fair play. For example, how many perfect brackets do you expect after the first round, second round? What verification systems are in place to ensure no new brackets slip into the pool? Lawler noted that they are “insuring the luck of picking a perfect bracket and not the tampering of one.”

And so, Warren Buffett’s making a bet that no one can create a perfect bracket. The company is putting a billion dollars on the line. Behind that bet is math and computer science to ensure the game is fair and, like a casino, that the house (Berkshire Hathaway) has an advantage – in this case a heavy one. Want to take the bet? Make a bracket – better yet, come up with mathematical techniques that help you choose your picks. And then, watch the games. Be forewarned, however. With so much money on the line, it’s definitely going to be maddening – one way or the other.