# FiveThirtyEight

## Sports

Before any games have tipped off, the NCAA tournament already has an almost surefire winner: Warren Buffett.

As you’ve likely heard, the billionaire investor is backing a prize of $1 billion to anyone who correctly picks the winner of every game in the men’s college basketball tournament. Announced on Jan. 21, the contest grabbed far more attention than any perfect bracket competition ever had before. No forecast, even FiveThirtyEight’s, is a sure thing. But here’s one prediction we can make with almost complete certainty: Buffett won’t have to pay out. Mathematicians disagree on precisely how difficult it is to make a perfect bracket, but they agree the chance is so small that it’s not worth quibbling over. Big money has been used to spark innovation before. That’s the idea behind the X Prize for advances in energy, exploration and so on. The difference there is that inventors actually have a real shot at developing clean, renewable fuels. Even a$1 trillion come-on can’t inspire an unblemished NCAA bracket. No sum of money can beat the math.

“The bottom line,” said DePaul University mathematician Jeffrey Bergen, “is that Mr. Buffett has very little to worry about.”

To our knowledge, no one has ever produced a perfect bracket, the three-minute mile of office pools. Perhaps one is buried in the boneyard of photocopied, hand-filled brackets of yesteryear. Internet-era contests are easier to track. Against the standard of perfection, more than 30 million ESPN brackets have failed, one by one, in the past 16 years, according to a spokeswoman. Over the last two years, no CBSSports.com bracket remained perfect through the second day of the tournament, a spokeswoman said. Yahoo’s entrant who came closest to perfection got 58 of 63 games right in 2007, according to a spokesman; last year no one picked more than 50 games correctly at either CBS or Yahoo. (Yahoo is partnering with Buffett’s perfect bracket contest.)

Correctly predicting any one game in the tournament may be only a bit easier than calling a coin flip — not very good odds. Predicting all 63 games is nearly impossible. (Quicken Loans, the sponsor of the Buffett-insured offer, is omitting the four play-in games for its bracket contest, conforming to office-pool norms.)

Several mathematicians offered explanations for this difficulty, ranging from the abstract to the concrete. Bergen, in a 2012 YouTube video that resurfaced after Buffett offered his billion, pointed out that predicting 63 coin flips correctly was a 1-in-9.2-quintillion proposition.1 That’s because the probability of getting 63 out of 63 right is the product of the probability of getting each one right, which for a coin flip is 50 percent.

Most games aren’t coin flips, though. A No. 1 seed has never lost to a No. 16, which makes those round-of-64 games about as close to sure things as big sporting events get. Make a few assumptions about your chances of calling each game right and you can get estimates. They’ll be very sensitive to the assumptions, though, because they’re being multiplied together 63 times.2

The challenge is so enormous that some scientists consulted for this article shrank from providing detailed calculations in their estimates of the probability of choosing a perfect bracket. Their guesses ranged from about 1 in 5 billion to 1 in 135 billion. In response to a journalist’s query, University of Minnesota biostatistician Bradley P. Carlin used 1 in 128 billion, which is “just the opinion of one expert I found online,” he said.

That expert is Bergen, who cites the estimate in his video but doesn’t explain it. At my request, he provided his assumptions: that the probability of calling a first-round game correctly ranges from 51 percent for the No. 8 vs. No. 9 game to 100 percent for the No. 1 vs. No. 16; and that second-round games can be called with 65 percent accuracy. The figures are 60 percent for Sweet Sixteen games and 50 percent for every game from the Elite Eight through the final.

“This is based on some simplifying assumptions and could be tweaked based on historical data, but is probably not unreasonable,” Bergen said. (He has since posted a new video explaining his calculation.)

Bergen’s model, and his peers’ calculations, don’t incorporate any information about the actual players and teams that give March Madness its flesh and blood. A more empirical way to fill out a bracket is to use some sort of team power rating to calculate the probability of each game’s outcome, then choose the most likely set of 63 outcomes. A 2006 attempt using Ken Pomeroy’s ratings found the most likely bracket had a 1-in-722 billion chance of being right, and in 2011 the figure was 1 in 83 billion.

Suppose, though, that the $1-billion offer lets a thousand Ken Pomeroys bloom. Perhaps some previously unknown analyst solves college basketball’s unsolvable riddle. That’s how the Buffett offer looks through the X Prize lens. The trouble is, sports predictions can’t be solved the way space exploration or fuel economy can. When athletes, officials and physics collide, no one can be sure what will happen beyond some upper limit of probability. And while there’s room in all sports to improve on our current understanding, college hoops is a particularly tricky sport to forecast. Players have just a few years of track record, at most. Most pairs of tournament teams haven’t played each other, or played at all in the neutral host venues. “It’s probably not the best data set to use to make real improvements in statistical modeling,” said Mark Glickman, a research professor of health policy and management at Boston University who holds a doctorate in statistics. And even if someone makes a major breakthrough in predicting college hoops, there’s more reliable money to be made in nightly Las Vegas wagers than in lifting your expected yield from a perfect bracket contest to, say,$5 from 50 cents.

March Madness sits in a sweet spot of prediction contests: It’s hard enough to yield no winner yet popular and tempting enough to produce lots of entrants and publicity. The World Cup has greater global popularity and roughly the same number of games but lacks the simple, single-elimination bracket structure. Tennis Grand Slams share the brackets but lack the NCAA Tournament’s U.S. popularity. Besides, they’re too tough: With fields twice as large (and with about three times as many near-toss-up matches as in an NCAA tournament’s round of 64), choosing the first round is about as tough as winning the Buffett contest. NFL survivor pools are too easy: Roughly one out of every 2,600 entries in ESPN’s Eliminator Challenge, 166 in all, made it through to the end of last year’s regular season.

No matter how you slice it, any one March Madness entrant in a perfect bracket contest has next to no chance of winning. But if you open the gates to up to 15 million entries, with an option to allow more in — the terms of the Quicken offer — the risk may start to seem real. If all 15 million entries are different, the probability is roughly the number of entries multiplied by the probability of any one entry winning.3

But that last “if” is a big one. It’s fun to mess around with wild upset picks in a private contest for bragging rights with your friends. When entering a competition for $1 billion, though, why bother choosing a bracket with anything less than the best possible chance of winning? Mathematicians advise doing whatever it takes to stand out from the chalk crowd — those office-pool pests who rank highly simply by choosing higher seeds to win every game — while maintaining the same order of magnitude of probability of winning. (Duplicating other entrants’ brackets — or FiveThirtyEight’s, or any other widely published one — could be costly, since the$1 billion will be split among winners.) The obvious thing is to exploit questionable seeding decisions: Last year, Pomeroy rated eighth-seed Pitt above many higher seeds. More subtly, you could choose an unlikely outcome that opens up the draw and makes subsequent games easier to call. “Depending on how you assess prediction odds, the ‘top-ranked always wins’ bracket is not necessarily the most likely,” said John Pike, a mathematician at Cornell University.

But there aren’t 15 million especially likely variations on the theme of choosing a bracket with an eye to perfection. Statistician Scott Berry, president of Scott Berry Consultants, figures that a bracket picking all favorites has a 1-in-85-billion chance of being correct. Depending on how the 15 million entrants to Buffett’s contest choose, balancing the desire for a unique bracket with one for reasonable picks, Berry figures the collective probability of winning the jackpot ranges from one in 42,370 to one in 3.3 million.

(At Grantland, Ed Feng calculated an average probability for a perfect bracket in each of the last 10 years — if Buffett’s contest had run each year — of 1 in 590,000, which Feng calls “incredibly optimistic.”)

Damien Sutevski is trying to boost those collective odds. Sutevski, a graduate student at UCLA who studies fusion engineering, created the website Take Buffett’s Billion to coordinate entries. People who sign up are assigned brackets that are likely without ever being duplicative. They agree to donate any winnings to charities (the Immunity Project and Habitat for Humanity).

“I thought it’d be fun to game Buffett’s challenge by pooling people together,” Sutevski said. Based on simulations by Michael Beuoy, a sports analyst, he figures that the average probability of the 15 million most likely brackets is a little under half the probability of the single most likely one. So if they can enlist all 15 million entrants in the contest to join forces, they can drive the odds down to about one in 10,000, and boost their odds of winning some of the \$2 million set aside for the best imperfect brackets. As of Monday afternoon, 9,061 people had signed up.

The more duplicate ballots Buffett and Quicken get, the safer they can feel. In fact, their biggest risk isn’t from an NCAA Nostradamus but from a modern-day Kevin Mitnick or Cornelious Kelleher. A hacker or a match-fixer has the surest, though least legal, way to riches in a bracket contest. “The odds of someone winning the prize through fraud,” Pomeroy said, “are better than the odds of someone winning the prize honestly.”

That doesn’t mean that if somehow a forecaster defies the odds and wins the contest he or she is definitely a cheat — only that it’s the most likely explanation when simply predicting all 63 games is so very, very unlikely.

“There are two big risks in this,” Buffett told ESPN’s Rick Reilly. “One, somebody does it. Two, somebody tries to screw us.”

### Footnotes

1. A quintillion is a billion billion, or one followed by 18 zeros. ^
2. Suppose you’re building a model to enter the Buffett bonanza. If your chance of being right was 70 percent, on average, for every game, then you’d have a 1-in-5.7-billion chance of a perfect bracket.

But your chance of being right isn’t the same for every game. Some games will be more or less sure — for instance, some first-round games pit favorites against the last teams in, while Final Four games should be more competitive. So suppose you know the winner of 21 games with 90 percent certainty, another 21 at 70 percent and the last 21 at 50 percent — coin flips. The average probability remains 70 percent, but now your chance at Buffett’s prize has plummeted to 1 in 34.3 billion.

This follows logically from some basic algebra. Call your average probability of predicting a game correctly x. Suppose you’re predicting two games. In one scenario, you have an x chance of predicting each game, and an x^2 chance of calling both right. In an alternative scenario, you have an (x+y) chance of calling the first game right, where y is some number less than x and an (x-y) chance of calling the second. Remember the product of (x+y) and (x-y) is x^2-y^2. That is your chance of getting both games right, and it is less than x^2 — even though your average probability is the same for each game.

This has implications for your March Madness model. However refined it is, its predictions will contain some uncertainty. It’s based on finite data about teams full of college students who might get overwhelmed by the pressure or, for that matter, get injured. So even if your fancy model somehow spits out probabilities of 70 percent for each game, some might be 68 percent, or 72, or even 50 or 90. And so the overall probability you calculate of getting the whole thing right will probably overstate it, by the same (x+y)*(x-y) logic above, even if your confidence in getting any one game right is, on average, well founded. ^

3. More precisely, if each entry has the same chance of perfection, p, and there are x different entries, then the chance that there is a successful entry is one minus the chance that there is no successful entry. The amount to be subtracted from one is, in turn, the product of the probability that each entry is wrong, which is (1-p) multiplied together x times. Put it all together, and you get 1 – (1-p)^x. (1-p)^x is a polynomial with many terms. When p is very low, the first two are far greater than all the rest together, so (1-p)^x simplifies to 1-p*x, and so 1 – (1-p)^x is roughly p*x. Or, going back to our March Madness case, roughly 15 million times whatever we decide p is. If it’s 1 in 135 billion, this calculation implies there’s a 1-in-9,000 chance of a perfect bracket. ^

All Carl Bialik

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