The Perfect NCAA Bracket

Kevin Day, March 22nd, 2008

In grade school, my favorite game was “Starting Lineup Talking Baseball”, which simulated baseball games based on the players you had on your team. The success of each player depended on their real-life statistics, so it was important to memorize everyone’s stats.

The direct correlation of the players’ statistics to their success in that game may be why I think sports are deterministic and why I love trying to predict things about sports.

For a few years now, I’ve been thinking about the NCAA tournament. There are a few simple questions that I don’t think have been appropriately answered. This post answers the question, “What are the odds of picking the perfect bracket?”

Spoiler

Here’s the answer if you’re in a hurry and don’t care about the details. The chance that your bracket is perfect is:

1 in 924 trillion

… so you’re saying there’s a chance?

Well, put it this way. If 10 million people fill out brackets this year, there’s only a 1 in 100 million chance that anyone will have a perfect bracket.

Traditional Answer

The common way that I’ve seen this question answered (WSJ, Math Forum) is that the total number of possible brackets are calculated as 2^63, (neglecting the play-in game) which is 9.22 * 10^18. Then they go on to say that the chance of getting a perfect bracket is one in 9.22 * 10^18.

However, this assumes that every bracket is equally probable. Usually studies mention this caveat, but they don’t really take the next step and find out what the real probabilities are.

The Wall Street Journal article linked to above makes some simplifications about how accurate people guess winners and how many games can be considered a lock for the favorite, but in my opinion they’re still just guesses.

A better way to find the probability of the bracket being perfect is to first model the probability of each seed winning a game. Then, use that model to calculate the probabilities of different selection strategies people might use.

Modeling with Tournament Seeds

There have been a few studies that have modeled the relationship between a team’s tournament seed and their probability of winning a game (Carlin; Schwertman, et al). One of the simplest and most accurate models is

P(Team A) = Team B’s Seed / (Team A’s Seed + Team B’s Seed)

This says that if a #2 seed plays a #6 seed, the chance that the #2 seed will win is 6 / (2 + 6), which is 75%. The chance the #6 seed will win is 2 / (2 + 6), which is 25%.

This model does a good job representing historical win probabilities in the NCAA tournament, except for when #1 seeds play #2 seeds. Historically, #1 seeds only win 43% of the time instead of 67% predicted by the model (Schwertman, et al).

If you adjust for the #1 vs. #2 error, the model can be used to estimate the probability that any bracket will be perfect. All you have to do is go game by game through the bracket multiplying the win probabilities of each predicted winner. The result is the chance that every predicted winner in the entire bracket will actually win.

Below I calculate the probabilities of perfect brackets for five different strategies for selecting winners.

All Favorites

This strategy picks all higher-ranked seeds. At the Final Four, each team has an equal shot of winning since they’re all #1 seeds, so there’s actually 8 possible All Favorites brackets.

1 in 546,000,000,000

Most Favorites

This strategy is the one used by nearly everyone when they pick their bracket. Most of the higher seeds win, but there are several upsets as well.

I’m not making any distinction between strategies such as “teams hot at the end of the year are better” or “teams with big guys inside do better in the tournament.” Any bracket that choses favorites to win as frequently as they have in past NCAA tournaments fall within this strategy.

Since there can be many different Most Favorites brackets, I generated 8,000 brackets and took the average probability of all of them.

1 in 924,000,000,000,000

Random

This strategy assumes each team has an equal chance of winning. Again, I generated 8,000 brackets and took the average probability.

1 in 308,000,000,000,000,000,000

Most Upsets

This strategy is the opposite of the of the Most Favorites strategy. Underdogs are favored in most of the games, but higher-ranked teams win occasionally. Number 16 seeds are frequently chosen to win the championship with this strategy. This is the average probability of 8,000 brackets.

1 in 893,000,000,000,000,000,000,000,000,000

All Upsets

The opposite of the All Favorites strategy is picking 100% upsets. That’s right, your Final Four is all #16 seeds. What are the odds of that happening?

1 in 144,000,000,000,000,000,000,000,000,000,000

Conclusion

Not all brackets are created equal. A bracket with all upsets is 10 sextillion times less likely to occur than one with all favorites.

Unless you honestly have a legitimate way of predicting winners better than most people, the odds of your bracket being perfect are 1 in 924 trillion. Although not very good, these odds are 10,000 times better than the number usually reported. All right!

However, your odds could be a thousand times better if you just picked all of the higher seeds to win. Why don’t you?

First of all, it’s not fun. Most people like to take risks and then feel proud when they correctly predict upsets.

But I think there are other reasons as well. For instance, if everyone in your league picked all favorites, then someone would have the exact same bracket as you and you would have a lower chance of winning the league.

Or would you?

My next post will look into the trade-offs between these strategies when competing in a league. There’s some game-theory type questions that I’m not qualified to address but will try to answer anyways.

Stay tuned.

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