---

I was briefly interviewed for a Wall Street Journal article on Warren Buffett's insuring of a $\$1$ billion dollar prize for any contestant who correctly picks all 63 games in the 2013 NCAA basketball tournament (see also here). Don't ask how this happened; it involves consorting with a bunch of degenerates. As stated in the article, "Mr. Buffett's Berkshire Hathaway would take on the risk, and earn a fee for doing so". My position as stated in the article is that this paid premium to Berkshire Hathaway was unnecessary, since the odds of picking the outcome of these 63 games correctly is so unlikely. A great deal for Mr. Buffett, but that is indeed his reputation. I expand on these thoughts below.

Naively, the odds of picking 63 games correctly are as follows. If the outcome of a single game was random, the odds of either team winning (or of correctly picking the winner) would be 1 in 2, or 50%. The odds of anyone picking the correct outcome of 2 games (team A vs team B; and team C vs team D) are 1 in 4 (either A and C win; A and D win; B and C win; or B and D win), or 25%. A pattern quickly emerges: the chance of picking $N$ games correctly goes like $1/2^N$. This generalizes even further: assuming odds $O$ of any particular choice being correct (which is 50%, or 0.5, above), the likelihood of picking $N$ games correct is $O^N$. Naively assuming odds of 0.5 for each of 63 games in the tournament, the chances of getting them all correct is $0.5^{63} = 1.1 \times 10^{-19}$, roughly 1-to-$9.2 \times 10^{18}$, or 9.2 billion billion. This is also approximately the number of grains of sand on the entire planet Earth. Such is the power of exponential growth, that you can double something, double it again, and by doing this 63 times you have an entire planet's worth of material. This process is not unlike the growth in technology we currently find ourselves in the midst of, where computing power is basically doubling every 18-24 months. A quick computation suggests that we are around 20 doublings into this process.

So in this ideal world, $\$9.2$ billion billion should be the monetary payout for correctly calling all 63 games of the NCAA tournament, not a mere $\$1$ billion. To think about this another way, each dollar in that $\$1$ billion payout should itself be worth $\$9.2$ billion. The vig on this bet would be enormous, perhaps unprecedented.

Now, the above assumes that the outcome of each game is basically a coin flip. But we know this to not be the case. Teams are seeded based on (among other considerations) their perceived strength, from 1 to 16, with #1 seeds expected to proceed farther into the tournament than #16 seeds. In addition, the first and sixteen seeds play each other in the first round of the tournament, so this is not nearly a coin flip. In the WSJ article, Mr, Buffett is quoted as saying the odds can't be calculated, and while this might be technically true (no-one can predict the future), they can be estimated.

Estimating The Odds

To understand the true odds of calculating all 63 games of a NCAA tournament correctly, one would ideally like many realizations of all possible scenarios, to understand what fraction of the time the one particular scenario of interest played out. This is of course impossible without the assistance of a Level-III multiverse, let alone that there are 9.2 billion billion ways a single tournament in our Universe could unfold. So we take one step in a data-driven look at this problem, which is to calculate the average odds $O$ that the favored team beats the underdog. This is somewhat of a "best case scenario", in the sense that one could pick the next round of a bracket after each previous one, picking only the higher seed of the two competitors for any given game.

As a caveat, this is a very coarse parameterization of our ignorance of just how the NCAA seeding rules create matchups that are able to be predicted. An additional complication is that the seeding rules yield different teams in a given seed each year. And each of these teams will have in general different players and perhaps even coaches than its previous entry. This is the system complexity that Mr. Buffett alludes to in the WSJ article.

To start this investigation, I obtained here a list of all the matchups in the NCAA basketball tournaments going back to 1985. From here it was a trivial computation to determine what fraction of higher-seeded teams typically won. Since there are 4 teams in each tournament with a given numerical seed, I have ignored all competitions where equal-numbered seeds went against each other.

Results

I first split the data up into the 6 rounds: "First Round", "Second Round", "Sweet 16", "Elite Eight", "Final Four", "National Championship". The fraction of top seeded teams that won in each round are listed below:

As expected, the higher ranked seed performed exceptionally well in the first round, winning approximately 75% of their games. From the second round on, this advantage is somewhat lessened, but there are also fewer games in the sample. Note, this analysis does not take into account e.g. a particularly outstanding lower seed that left a trail of correlated destruction in their wake (e.g. George Mason, 2006). For the overall numbers, the higher seeds win 72% of their games.

So how does this wrap into the bracket contest above? We proceed by looking at a "best case" scenario, where we can pick the results of the "Second Round" after seeing the results of the "First Round", of the "Sweet 16" after seeing the results of the "Second Round", etc. First, use a likelihood of 0.75 for guessing all 32 "First Round" games correctly, 0.70 for all 16 "Second Round" games, 0.72 for 8 "Sweet 16" games, 0.57 for 4 "Elite Eight" games, 0.65 for 2 "Final Four" games, and 0.74 for the 1 Championship Game. This yields a 1-in-1.4 billion chance that the higher ranked team wins every game in the NCAA tournament, assuming you know the exact teams that go into each and every game (in truth this is not how the tournament plays out, as you have to fill out your bracket all at once, before any of the 63 games are played). However, the above numbers potentially suffer from the process of "overfitting" where we have subdivided the data too finely (e.g. per round) and end up multiplying together a bunch of noisy, uncertain values. If we instead use the overall average odds of 0.72 for the higher seed to win, we find a more favorable scenario, a slightly worse than 1-in-1.1 billion chance of the desired outcome.

No matter how you slice it, given the data on-hand, the likelihood of picking a tournament correctly is exceedingly miniscule. Only if you were able to interactively pick your bracket after each round, and only chose the favorites in each and every game, would you have a 1-in-a billion chance to win $\$1$ billion dollars. Which is not too shabby. And the $\$11$ million premium being paid to insure the money? This is about 1% of the total prize; if the odds of winning the prize were 1 in 100, this might seem like the appropriate level to insure the kitty. However, given the rules of the tournament, the true likelihood of correctly guessing all 63 games is far less than our best case scenario, which is still only a 1-in-1.1 billion chance. Given this absolute best likelihood is worse than 1-in-1 billion, and the value of the prize is (perhaps not coincidentally?) $\$1$ billion, the true premium on insuring this bet should likely be less than $\$1$.

Bottom line, this is great bet for Berkshire Hathaway to make, and unnecessary insurance for the sponsor of the competition. It is, however, good PR for all involved.

• Update 1 (2012-02-10): A friend has pointed out that the contest is being capped at 10 million entrants, whereas my analysis above is for a single entry. This detail actually provides an order-of-magnitude risk of 1% of this contest being won, assuming the best case scenario. This then calls into question what is the real risk of the contest, as designed, being won by any of 10 million entries. An estimate using the geometric mean (which is useful in order of magnitude calculations) of the best case and worst case scenarios suggests that the risk per entry would be around 100,000 billion-to-one ($\sqrt(9.2 \times 10^{18} * 1.1 \times 10^9)$). So the chance of being won by any of the 10 million entrants is about 1-in-10 million. Meaning I would insure that billion for a cool $\$100$.