Bracketology!

I just checked and my March Madness brackets are officially losers.  I will finish either second or third in my league of 6.  And I know this even though he scores are pretty close together and the only team anyone has left that’s still alive is Louisville.  (By the way, congratulations to MikeD on the victory.  They were definitely inspired in the second half.  I hope that kid will be OK.)  Which led me to think that bracket math is kind of odd, and deserves to be investigated.

March Madness is a single-elimination tournament of 64 games.  The first round is 32 games, and the 32 winners go into the second round.  That’s 16 games, and the 16 winners go into the third round (they’re nicknames the Sweet 16).  You see how this works now.  The sixth round is a single game between the two remaining teams whose winner is the national champion.  That’s a total of 63 games (obvious, since each game eliminates one team, and it needs to reduce a field of 64 to one winner.) A bracket is like the picture above, but with all 64 teams filled in at the far left and far right. It’s up to you to predict the winner of each game in the first round, and based on those choices, predict winners for rounds 2 through six. You get points for each winner you predict correctly. Note that the first round is the only one where you know for sure which teams are playing. If you pick every first-round game wrong, you can’t earn any points in later rounds, because the teams you picked won’t even be playing.

There are different ways to score brackets. All of them make late-round games worth more than early-round games, but by how much varies. Here, I’ll use ESPN’s scoring, which makes each round worth a possible 320 points:

• Round 1: 10 points per game
• Round 2: 20 points per game
• Round 3: 40 points per game
• Round 4: 80 points per game
• Round 5: 160 points per game
• Championship Game: 320 points

That’s a total of 1920 possible points.

Very naively, you might think that since each game is a 50-50 shot, picking randomly would give you an average 31.5 games correct, and an expectation of 1920/2 = 960 points. Quite incorrect, though. Recall that bad picks in earlier rounds will result in your not having picked either of the teams playing a game in a later round. So, while it is indeed true that random picks will give you an average of 16 winners in round 1 for 160 points, that will (on the average) result in a round 2 where:

• In four games, you have predicted neither of the teams playing to advance that far. No chance to win.
• In eight games, you have predicted only one of the teams that is this round. One quarter of a chance to win, since you will have picked that team to win this round only half the time, and it will actually win only half of those times.
• in four games you have picked both teams that are in the round. That gives you a half-chance of winning (you win if you’ve picked the right one to win.)

So your expected number of wins is (4 * 0) + (8 * .25) + (4 * .5) or 4, for a point total of 80. This makes sense, because of the 16 winners sent into round 2, overall you’ll pick half of them to win, and half of those will win. So each winner in round N becomes, on the average, a quarter of a winner in round N+1.
Continuing this logic, your total expectation is:

• Round 1: 16 winners for 160
• Round 2: 4 winners for 80
• Round 3: 1 winner for 40
• Round 4: 1/4 winner for 20
• Round 5: 1/16 winner for 10
• Championship Game: 1/64 winner for 5

Which makes your total expectation 21 5/16 winners for 315 points.
Now, clearly between seedings, expert predictions, and your own knowledge of the game, you can do better than 50%. Let’s call probability of arriving at the right answer p. We can generalize what’s above to

• Round 1: 32p winners for 320p
• Round 2: 16(p**2) winners for 320(p**2)
• Round 3: 8(p**3) winners for 320(p**3)
• Round 4: 4(p**4) winners for 320(p**4)
• Round 5: 2(p**5) winners for 320(p**5)
• Championship Game: p**6 winners for 320(p**6)

Here are some results for different values of p:

In other words, because of the way losses add up, you could pick games correctly at a 90% rate, and still only get 70% of the possible score.