NCAA March Madness: why entering more than one bracket is questionable

Before I get into the math, a brief intro for international readers who might not be familiar with this event. The NCAA men’s college basketball tournament is a wild event that is often called March Madness. It features a tournament bracket of 64 teams competing in single-elimination games until a single champion is crowned.

The basketball is fun because early rounds have many games played during a single day and there is a potential for upsets. But the other appeal of the tournament is the related gambling in office pools. (I should mention these betting pools are usually illegal though the chance of prosecution is almost nil).

The way these office pools work is people fill out a bracket–a list of predictions of every matchup from the first round until the champion. Each submission costs money and is used to fund the pool of winnings. The brackets are scored based on accuracy (point systems may vary), often weighted towards predicting games deeper in the tournament like the champion and semi-finalists. The top bracket or perhaps top two get a prize.

There are many strategies for picking brackets. One of the more general issues is whether it makes sense to enter more than one bracket. The plus side is an extra bracket increases the number of picks and the chances of winning. The down side is an extra bracket costs money and makes losing more expensive.

Does it make sense to enter more than one bracket? It turns out it does not pay off from a strictly mathematical perspective. Here is why.

Expected values

The logic is best illustrated in an example.

Imagine a pool with 10 people (including you) and a cost of $5 per bracket. The pool stipulates a $5 fee for the organizer’s efforts. Also, the pool is a simple winner-take-all–the person with the best bracket wins the prize.

What is your expected payoff to this game, if everyone buys one bracket?

The monetary payoffs are $40 if you win (the other nine put in $45 less the $5 organizer fee) and -$5 if you lose.

As a starting point, suppose that luck is a big factor and each bracket is equally likely to win. The mathematical expectation is therefore:

Notice the expectation to this game is less than zero, a loss. That is because of the $5 fee given to the organizer.

What happens if you choose to buy one more bracket? You are still chasing after the same gain so you still get $40 if you win, but you have to pay $10 if you lose. But another entry affects your odds of winning. Now you have 2 out of 11 chances of winning and 9 out of 11 of losing.

Here is how the expectation turns out:

As you can see, the expectation is lower than before. This suggests that entering another bracket is a less favorable gamble because the increased odds of winning do not outweigh the extra cost of entry.

A similar argument can show that three brackets is even worse:

It should be clear that entering even more brackets will lower the expected value further.

The implication of all of this? If luck plays a big role in your office pool, consider taking your best guess and go for it.