Analagous NCAA brackets; intense risk

References in these links.  Good luck!
https://twitter.com/salilstatistics/status/843103562825646080
http://fivethirtyeight.com/features/the-odds-youll-fill-out-a-perfect-bracket/
https://twitter.com/salilstatistics/status/843298735690043393
https://twitter.com/salilstatistics/status/843556160183635973


With NCAA’s March Madness later this month, tens of millions of people will be part of the speculation fever by populating playoff brackets.  The odds of haphazardly guessing and still completing a “perfect bracket” of 64 teams is 1 in 9.2 quintillion (18 digits to the right of the “9”).  Or 3 billion multiplied by another 3 billion!  And Quicken Loans has offered prizes such as a billion dollars if someone in fact performs this feat, underwritten by Warren Buffett.  Of course this is not the true odds for such a bracket challenge, since most do not randomly guess their way through the selections.  And it turns out that there is a great deal of interrelated strategies that people use when they complete their forms (very similar to the correlated strategies people use in predicting turns in the financial markets or who is the likely winner of a presidential referendum).  This results in many sports fans having the same poor chance of seeing the return of their office pool money, yet a higher chance than many sports promoters, ESPN, and insurers realize when they offer large prizes for what they falsely think is a far rarer event.  Logically and empirically we expect many more top seed teams going further into the tournament (and linear spreads based on seeds), and many more brackets are also completed in a similar fashion.  The resulting variations are typically in the matched seeds that are closer to one another through the bracket.  And therefore the true chance of a perfect bracket, conditional on this reality, is closer to 1 in a 1 trillion (12 digits to the right of the “1”).  With tens of millions of people playing, the probability an insurer will have to pay a billion dollars for a winner(s) of a perfect bracket is therefore 0.005%.  Certainly low for a given year, but imprudent to offer annually over one’s lifetime (and certainly in the rare case of a perfect bracket it is possible -not highly probable but possible- that we have multiple matching winners with a concentration of similar brackets causing bouts of success and busts around these probabilistic averages).  The Lottery as a comparison has odds that are nearly 1 in 0.3 billion, and nearly the same number of playing tickets, but with winners (including ties) multiple times a year.

So in order to make March Madness a little less mad, there are multiple sub-contests that allow one to take part in more realistic gambling pools, with a good chance that there are winners without too many years going by.  First we should identify that for the most part the probability of picking each round perfectly is multiplied as a product of one another, and therefore can be decomposed into small competitions.  Let’s discuss this in topics below.

 

First four

The first four games winnow the 68 college teams to the 64 who are part of the bracket.  These games are played mid-March.  There is no formal probability theory that works every year in how to refine the odds of selecting these four team winners beyond random luck, however lower seed teams often play against higher seed teams and we can assume that correcting predicting the “First four” (which is often excluded in bracket challenges starting with the 64 teams) as something on the order to (3/4)⁴, or ~1/3.  If one is predicting the brackets starting with the First four, then the probability is therefore about 1/3 those who start after the First four, and with the 64 remaining teams.

 

First round

The first round consists of 32 games among 64 teams.  Using the conditional logic described in the initial paragraph, the best odds of perfectly selecting the first round of (n=64 teams) is:

 

As one can imagine on Day One, where ½ the 64 teams play in 16 matches, the number of brackets that are no longer perfect will be at least 1-0.6%, or 99.4%.  Perhaps more when too many “upsets” occur.  By the end of Day Two, the number of brackets no longer perfect rising to over 99.995%, from 99.4%. 

Through second round and on through finals

And as we described in the initial paragraph, if we continue this process smartly, then a dozen or so every year will have a perfect bracket heading into the Sweet 16.  And 0.1 Berkshire Hathaway employees on average would therefore win this type of highly valuable competition.  This makes the risk to underwrite Berkshire nearly tens of thousands of times more likely (if just the Sweet 16 picks per below) to result in a payoff versus Quicken Loans’ competition and a smaller -though still higher- payout of a $1 million annually for life.  And again advancing all the way through The Finals, there will be only 1 in a trillion (far more than actually play) would have won with a perfect bracket.

The morale again is that while winning the entire bracket perfectly isn’t a manageable probability, in order to make it realistically more fun with frequent enough winners, often sub-contests are performed in selecting teams in the early rounds only or simply selecting who is in the Final rounds.  Or points are assigned imperfectly-linear based on the difficulty of selecting that rounds winners.

It is interesting to note that in the past few decades, the public has selected lower seeds (say seed 12 through 16) to be present in the Final Four roughly 1 percent of the time.  Perhaps a feeling one has towards their own alma mater.  Yet never in that period of time have these lower seeds advanced that far.

Who is in the Sweet 16 or the Semi-finals

Another method to compete on the brackets is to only look at selecting the participants at a certain stage of the tournament (e.g., who will be in the Sweet 16), or who will win beyond a certain stage of the tournament (e.g., the 4-team semi-finals bracket).  Let’s first look at the probability of selecting the Final Four teams.  Within each bracket quad of 16 teams, we generally expect most of the time the top few seeds only will advance to the Final Four (more specifically 41% seed 1, 20% seed 2, 11% seed 3, 10% seed 4, etc.)  The probability of selecting any one of the four teams is therefore roughly equal to how the public also chooses it:
41%²+20%²+11%²+10%²+…, or 24% per quarter of bracket. 

And the probability of a perfect Final Four is therefore 24%⁴, or 0.3%.  Let’s see the NCAA challenge probabilities of tens of millions of participants annually in recent years, and we see the empirical average below matches our 0.3% theoretical probability as well:

Year	Percent
2011	0.0%
2012	0.3%
2013	0.0%
2014	0.01%
2015	1.6%
2016	0.1%
Average	0.3%

The math is less tractable using calculus theory similar to above for just the Sweet 16 picks but it is obviously more likely than the 1:19k for the Round of 32!  The last probability is perhaps the most obvious, a welcome relief for those who have come this far.  And that is the probability of selecting a perfect bracket from a starting point of the semi-finals onwards.  And since it is generally 2 semi-finals games and then one 1 final (so three nearly equally paired matches), it is ½³, or 12%.  Or more than the chance comic Nate Silver and other MSM pollsters gave Donald Trump’s chance to win the Presidential election.