## Sunday, January 26, 2014

### The \$1 Billion Bracket - Part One

This is the first of a two part blog post on the probability of picking a perfect March Madness bracket. This first post will focus on what we can learn from the betting markets. The second post will take a more robust look at the problem using my college basketball rankings.

### One......Billion Dollars

If you're looking for bigger upside in your March Madness bracket this year, Quicken Loans is offering a \$1 Billion payout to anybody that can submit a "perfect" bracket. Quicken Loans founder and chairman Dan Gilbert is a wealthy man (he's also majority owner of the Cleveland Cavaliers), but a billion dollars is still a lot of money. So he has partnered with Warren Buffett (through Berkshire Hathaway) to provide the financial backing for the contest.

Buffett did not disclose how much they are receiving from Quicken in exchange for insuring this contest, saying only that the amount was less than what he would have wanted, but more than what Gilbert would have liked to pay.

### The (im)probability of a perfect bracket

So, what would be a fair price for Berkshire Hathaway to charge for this insurance contract? Being an actuary by day and stats-blogging man-of-mystery by night, one could say that such a question falls right in my "wheelhouse" (one could, but not me, because I hate that word).

In order to figure out the fair price, you need to know the probability of choosing a perfect bracket. This is not a new question, and several before me have taken a stab at it.

• There's the "simple" estimate, which sets a lower bound on the probability. If you assume each game (all 63 of them) is a 50/50 proposition, a coin flip, the probability of getting all 63 coin flips correct is 1 in 9,223,372,036,854,775,808.
• DePaul mathematics professor Jeff Bergen claims in this video that the true odds of a perfect bracket are much better than that, more on the order of 1 in 128 Billion. The frustrating thing about the video is that he spends most of the video deriving the simple "coin flip" estimate above, and then goes all Pierre de Fermat on us, baldly asserting that 128 Billion is the number you get if you "know basketball" (I imagine his derivation was "truly marvelous").
• Carl Bialick of the Wall Street Journal (aka "The Numbers Guy") took a pretty in depth look at this question back in 2006. To be honest, my post here covers similar ground (I had completed the bulk of the analysis before I discovered Carl's article). Like any good numbers guy, Carl calculated the number in various ways. His "best" estimate, in my opinion, used the Ken Pomeroy rankings, and resulted in a 1 in 728 billion chance of picking a perfect bracket.

### Evidence from Betting Markets

One way to price an insurance contract (or anything really), is to see what kind of price one could get on the open market for a similar product. So let's say it's March 20, 2013, the eve of the 2013 tournament, and you know (somehow) that you have the perfect bracket. How could you have maximized the financial value of your prescience? One way would be to bet the moneyline on each game, and roll over each game's winnings into a moneyline bet on the next game. A moneyline bet, in contrast to the more popular point spread bet, pays out in proportion to the odds of your selected team winning.

In reality, this betting strategy may be hard to put into practice since March Madness games are played concurrently, especially during the first two rounds. But there are probably parlay options one could cobble together to get a reasonable approximation of this sequential betting strategy. But set that aside for now. The wagering practicalities aren't the point here, we're just using the market information as a means to an end.

So, let's say you start with a single dollar, and start placing sequential moneyline bets based on your perfect bracket. What would be your ultimate payout? The gambling website sbrforum.com, archives betting information for multiple years and multiple sports, including moneyline bets. I pulled the moneyline payouts for all 63 games of the 2013 tournament (I'm ignoring the play in games). Starting with only a single dollar, your bankroll would have ballooned to nearly eight hundred trillion dollars (\$798,844,225,777,125 to be exact), or about 10 times the gross world product. Of course, no sports book in the galaxy is going to back that kind of action, so let's express this result in a different way.

Instead of starting with a dollar, what would your initial investment have to be in order to result in a \$1 billion bankroll by the end of the tournament? The math is simple, it's just \$1 billion divided by the world-economy-crashing \$800 trillion amount. That works out to \$0.0000013, or 1 ten thousandth of a cent.

We now have a price tag we can place on the "actuarial value" of that bracket: 1 ten thousandth of a cent. This could be considered a fair value for Warren Buffett to charge for insuring a billion dollar payout for a single entry. According to the short form rules, the contest is limited to a maximum of 10 million entries. Multiplying 10 million by \$0.0000013 results in a premium of twelve dollars.

I think I lowballed it. And there are good reasons to believe this is a lowball estimate. But if you can hold on to your objections for now, here is the same analysis for the past four years of the tournament:

Moneyline Payouts for "Perfect" Brackets - 2010 to 2013
SeasonPayout for \$1 Bet\$1 Billion Payout Price10 Million Entries
2009-2010\$4,661,532,755,065,560\$0.00000021\$2.15
2010-2011\$20,033,918,992,775,900\$0.00000005\$0.50
2011-2012\$571,114,367,106,784\$0.00000175\$17.51
2012-2013\$798,844,225,777,125\$0.00000125\$12.52

The moneyline approach results in fair premiums ranging from \$0.50 to \$17.50. The implied probabilities range from 1 in 571 trillion to 1 in 20,034 trillion. We've shaved several orders of magnitude off of the coin flip estimate, but we're still many orders of magnitude away from the estimates from Professor Bergen and Carl Bialick.

Part of the problem with the above estimates is that they are based on the value of one particular bracket, and a sub-optimal one at that (just because it turned out to be right doesn't mean it was the most likely outcome, a priori). For Buffett to price this right, he needs to account for a range of bracket submissions, from the optimal (picking favorites straight through), to the sub-optimal (picking lots of upsets).

In part two of this post, I'll calculate the expected value of the optimal bracket, with an assist from my college basketball rankings.

1. > ...I imagine his derivation was "truly marvelous"...

At least Fermat's last theorem is true. Perhaps that 1 in 128 Billion is a floor. I calculated the Shannon Entropy of the odds I found for the last two NCAA tourneys, and it suggests the odds of getting a perfect bracket are around 1:10^15-1:10^16. (I didn't find the referenced pages at SBRforum, so YMMV if you do the calculations yourself. Anyone good enough to actually have a 1:10^12 chance of getting a perfect bracket should probably go to Vegas and wager 'till the casinos hire them or kick them out.)

Conveniently enough, if the chance of hitting a perfect bracket is no more than 10^(-15) and there are 10^7 entries for a payout of 10^9 dollars, then the total EV is 10 dollars. Buffet probably charges more for the administrative costs...

As you point out, your odds based estimate is a lowball - it seems like the house edge sucked out an order of magnitude there.

1. I'm a bit skeptical of the 128 billion number myself (I'll address it in part two). And his final point regarding the odds if everybody in the US submitted a bracket is misleading at best.

You are correct that the vig biases the numbers here, but I think a vig-free estimate would lead to an even lower lowball estimate. My approach for part two will be based on pure probabilities though, so it shouldn't be an issue.

Here is a direct link to the moneyline odds: SBR Moneyline Archive

2. > ... I think a vig-free estimate would lead to an even lower lowball estimate.

Sure, the calculated EV would be a worse estimate of the insurance premium, but that's because the insurance premium is not a fair price. It's either negotiated based on a 'free market' or driven by administrative and opportunity costs. Just negotiation time with Buffet is worth thousands of dollars.

1. True. The actual price probably had very little do with the EV in the end, and was more a function of how much the publicity was worth to Quicken. If Buffett demanded too high a price, they would just forego the contest.