You
can have the best analytics on your side. You could even watch 100
hours of college basketball a week to prepare for filling out your
bracket. None of this matters for the wrong pool. And just about the
only wrong sort of pool is a big one. Let me repeat: DO NOT enter a big pool. Here's why.
Drunk people throwing darts at a dartboard
You and a
friend walk into a bar and find the employees of a start-up company
there. They have just secured their Series A funding and feel good
about their future. In anticipation of becoming millionaires, they
start throwing back flutes of Champagne. After a few too many drinks,
and against better judgment, they decide to play a game.
Each person gets one throw at a dartboard. Hit the bullseye, earn a free drink.
You laugh
at the spectacle, thinking that not a single person will hit the
bullseye. It even doesn't matter that the bar has installed a magnetic
field that directs all errant darts back towards the dartboard—great for
safety, but the bar still won't be serving up any drinks.
Your friend bets you that someone will hit the bullseye. Should you take the bet?
It depends
on the number of people lined up to throw a dart. Let's assume that a
dart from a drunk person has an equal chance of landing anywhere on the
dartboard thanks to the magnetic field. The odds that any one drunk
hits the bullseye is small, about 0.5 percent.
However,
to win your bet, you need every drunk person to miss. There is a 99.5
percent probability that the first drunk misses, but you must multiply
0.995 by 0.995 to get the likelihood that both the first and second
drunk miss. If the company has 20 drunks that step up to fling a dart,
there's a 90.5 percent chance that all of them miss. This implies a 9.5
percent chance that at least one drunk hits the bullseye.
For an
increasing number of drunks, the probability that at least one hits the
bullseye increases rapidly. At 100, there's a 39.4 percent chance for
someone to hit the bullseye, and this probability increases to 86.5
percent for 400 people.
The same
principle applies to your March Madness pool. Suppose you're filling
out a pool in 2010. Kansas has just capped an amazing regular season
and heads into the tourney as a 1 seed. Analytics agrees with this
assessment, as the Jayhawks top
The Power Rank heading into the tourney.
However,
everyone in your pool has also picked Kansas. According to data from
ESPN, 41.8 percent of brackets filled out on their site had Kansas as
champion. If Kansas wins, you and many others get those 32 points.
However,
just like the drunk people throwing darts, someone else in your pool
will hit the bullseye in the earlier rounds. They will get lucky and
pick two surprise Sweet 16 teams or a shocking Elite 8 team.
Only one
person has to get lucky on some weird pick or another to topple you from
the top of your pool. This gets more likely with more people in your
pool.
How your chance to win depends on pool size
Let's put some numbers behind how your chance to win a pool depends on its size.
Suppose you
fill out the bracket with all favorites. Since 2002, the higher ranked
team in The Power Rank has won 71.3 percent of 844 tournament games.
This suggest picking the higher ranked team for each game.
Most years,
the bracket will look boring, and you might stab yourself in the eye
from having to repeatedly cheer for teams like Duke and Kentucky. But
you want to win a pool, right?
To
determine how often this bracket of favorites wins a pool, I developed a
simulation method that accounts for two types of randomness in your
pool. Researchers use these types of "Monte Carlo" simulations to study
phenomena ranging from polymeric materials to the stock market.
Randomness in basketball games
First, the
simulation must account for the inherent randomness in playing the game
of basketball. In real life, the tournament only happens one time. In
2010, Kansas fell in the Round of 32 to Northern Iowa, and Duke went on
to win the tourney.
However,
if the same tourney happened again, the results would be different.
Northern Iowa's Ali Faroukmanesh misses that three point shot, and
Kansas survives and advances. Then maybe they beat Duke in the Final
Four and win the tourney.
For each game, a coin is flipped according to this win probability.
For
example, if Kansas has a 97 percent chance to win their first game, this
coin comes up heads on 97 percent of flips. Kansas advances on each
heads. We simply repeat this procedure for each game in each round to
simulate the results of the tourney.
Randomness in a pool
Second,
there is variability among the brackets in your pool. A person in your
pool might not know anything about college basketball and pick games
depending on which mascot he likes.
Another
person went to Connecticut as an undergrad and always picks the Huskies
as champion.
After Connecticut won 2 of 4 tourneys from 2011 through
2014 as a long shot, this person might not get invited back into the
pool.
For the variability in brackets in your pool, we consult the
data on brackets submitted to ESPN. To simulate a 50 person pool, the computer pulls 50 brackets at random from the millions submitted on ESPN.
This is
clearly an approximation to the actual brackets that might appear in
your pool. For example, if you live in North Carolina, your pool will
probably have more brackets with Duke and North Carolina as champions
than the national average.
How your chances depend on pool size
Then I
performed simulations at a number of different pool sizes and tracked
the fraction of simulations in which a bracket of favorites won. The
visual shows how your chance to win a pool decreases rapidly with pool
size for the 2010 tourney.
By picking
all favorites, you have a 43 percent chance to win a 5 person pool.
You'll win about every other year, which is pretty good. If winning is
your only goal, enter a small pool.
For a 30
person pool, your chance of winning the pool drops to 10.3 percent.
With your choice of Kansas as champion, there are enough others in your
pool with this choice that someone else will win based on luck in
picking earlier games.
In terms
of investment potential, a win probability of 10.3 percent for a 30
person pool is pretty good. If you took a random bracket from ESPN and
submitted it, you would have same chance as anyone else, or 3.3 percent,
to win the pool. By submitting the favorites based on my numbers, your
average return on this investment is over 300 percent. The public is
like those drunk people throwing darts.
However,
the same tournament won't happen again next month. Even if you could
get the same 10.3 percent win probability every year, you're waiting
about 10 years on average between years in which you win the pool. For
a 100 person pool, your probability to win a pool drops to 2.1. A 400
person pool? I'm not even going to do the calculation.
The
thing is, maybe you don't need to win your pool to have a good time. But
it's hard to deny that you have a little more fun in pools in which
you've got a shot at winning in the later rounds. That's less likely in a
larger pool, too. So, if you're entering a pool with the idea of
winning any money, or even if you're just looking to have fun, don't
enter a big pool. Even with the best analytics, you're better off
lighting your entry fee on fire.
