Breaking the Tyranny of Chalk Brackets in March Madness

March Madness is the greatest sporting event in recorded human history.¹ The entire sports world joins together to make outlandish predictions, grow emotionally attached to teams they’ve never heard of, and hate Duke.

What makes this all possible are brackets. A surprising number of fans and non-fans alike fill out one or more every year to compete in bracket pools of varying legality. It brings families, friends, offices, and mortal enemies together, creating shared bonds and experiences that will last a lifetime.

There’s just one flaw, and it’s a major one: Bracket pools incentivize picking no upsets!

There, I said it. Pick a chalk bracket (i.e., one with no upsets), and you’ll give yourself the best chance to win your pool.

Most people know this fact. Despite that—or perhaps because of that—unwritten rules govern how many upsets a bracket should include. Picking none isn’t just seen as boring but as a kind of cheating. If that bracket wins, it’s dismissed as luck: “They just picked the better seeds! Anyone could’ve done that. Getting a couple of upset picks right—that takes real skill.”

Some have tried to combat this by adding in an upset bonus—a bonus that rewards correct upset picks. Correct upset picks mean your selected winner advances over a higher ranked team, regardless of whether you picked the higher ranked team to make it that far. There are two types of upset bonuses we’ll discuss here:

• Flat bonus: The same bonus is rewarded regardless of the seeds involved in the upset. A No. 16 seed over a No. 1 seed gets the same bonus as a No. 9 seed over a No. 8 seed.
• Seed difference bonus: The bonus is proportional to the difference in seeds. Correctly picking a No. 16 seed over a No. 1 seed is worth 15 times as much as a No. 9 seed over a No. 8 seed.

For both bonus types, the scale can change from one round to the next. For example, we could reward 2 times the seed difference in round 1 and 4 times the seed difference in round 2.

The major flaw with the typical upset bonus is this: it’s either not large enough to matter or so large that it’s senseless to pick anything but an upset.

For example, let’s take the 5­–12 matchup in the first round. Historically, the No. 5 seed wins 65% of the time (see Figure 1). If there’s no upset bonus, you should generally pick the No. 5 seed. This is boring, so you introduce an upset bonus of the difference between the seeds, which is 7. Let’s say a correct pick gets 1 point in the first round. If you pick the No. 5 seed, you get 0.65 * 1 point = 0.65 expected points. But if you pick the 12, you get 0.35 * (1 + 7) = 2.8 expected points. With this type of bonus, picking the No. 12 seed to win is a far better strategy.

How do we fix it? Calibration. Instead of rewarding 7 bonus points, let’s calibrate it so that the expected value is the same for the two. We can turn this into an equation:

In that example 𝑥 is the factor we multiply the upset bonus by. A little algebra later, x is solved:

This means if we reward an upset bonus of 0.12 times the difference in the seeds, then picking a No. 5 or 12 seed will have the same expected value.

Unfortunately, this upset bonus factor is different for all seeds. The figure below shows the optimal factor for each first round higher ranked seed. The No. 1 seed is by far the largest because, even though 15 points is a huge bonus, it’s so unlikely that we’d need to incentivize by much more to get people to pick it. The No. 8 seed is negative because No. 9 seeds have won more than No. 8 seeds in the first round historically.

How would we pick a single number out of this? We’d probably choose 0.12, since that makes the 5­–12 and 6–11 equal in expected value and the 7–10 close to equal.

Another approach we could take is a simple flat upset bonus. We reward the same bonus whether it’s a No. 16 seed or a No. 9 seed who won. In that case, the optimal multipliers are a bit higher:

There’s not that much practical difference between this and the seed-difference example. Here, we would set the bonus to 0.6, making the 6–11 and 7–10 matchups equal in expected value and the 5–12 close. The 8–9 matchup is a sunk cost—any upset-rewarding system is going to incentivize picking all the No. 9 seeds to win in the first round. And there’s no way around this.

So far, we’ve shown basic upset bonuses leave a bit to be desired, but there might be a way to optimize them with the right tweaking.

Now is a good time to answer the question: what does the optimal March Madness scoring system look like? What do we want it to do?

This is a subjective question, but here are the things we’re looking for:

1. 1. Picking no upsets should not be the only optimal approach for both early and late rounds.

1. Certain upsets should not be so incentivized that picking all of them gives you a clear advantage. Better teams retaining higher expected value is better because it’s more intuitive and doesn’t lead to people gaming the system.
   • 8–9 first round matchups are an unavoidable exception to this rule.
2. It should be a simple and explainable system. To be simple and explainable:
   • Upset bonuses should be integers and either a simple flat bonus or a multiple of seed differences. Flat bonus is preferred for simplicity.
   • This multiple can change from round to round, but it must be the same within a round. For example, we can’t use a different multiple for the 5–12 and 1–16 matchups. It’s better if it follows a defined pattern from one round to the next.
   • It must be possible to implement this system on Yahoo’s March Madness site.

Since flat bonuses and seed difference bonuses are practically similar for the first round, we decide to use flat bonuses for simplicity. Another benefit for the flat bonus is that upsets are more likely to have smaller seed differences in later rounds, where No. 1 seeds are comparatively dominant.

Figure 1 shows how often each seed reaches each round. No. 1 seeds are twice as likely to reach the Final Four as any other seed, 3 times as likely to reach the national championship, and 5 times as likely to win the national championship. Some hefty bonuses are needed to prop No. 2 and 3 seeds up to similar expected values.

Figure 1: How often each team reaches each round historically

We can then repeat this process for every subsequent round. This brings an inherent challenge though. Let’s say you want to calculate the expected value of picking a No. 5 seed to reach the Sweet 16. The upset bonus will always be rewarded for winning the first-round game, but in the second round the No. 5 seed could either be facing a No. 4 or 13 seed. Beating the No. 4 seed would reward an upset bonus but beating the No. 13 seed would not.

To solve this problem, I first estimate the likelihood of facing a better-seeded team by averaging across all seeds each team could face in each round. In this example, there’s a 78.9% chance the No. 4 seed wins in the first round, so 78.9% is the likelihood of a No. 5 seed facing a higher seed in the second round.

To estimate the probability each team has of reaching each round, I use the historical probabilities shown in Figure 1.²

An important aside: up to this point, we’ve been looking at expected values. It’s important to note that maximizing your expected bracket score is not the right objective. The right objective is winning your bracket pool.³ The good news is that these are usually aligned, but there are exceptions.

The best way to illustrate this is an example. No. 1 seeds have about a 16% chance of winning the championship, and No. 16 seeds (let’s say) have a 0.0001% chance. We could set up an upset bonus to bring these to the same expected value by having the cumulative bonus sum to 16,000 for the No. 16 seed. But this would not actually incentivize picking a No. 16 seed to win it all. It’s so unlikely to happen you’re almost certainly not going to get the reward, regardless of what it is.

The goal is winning, not scoring the most points. It doesn’t matter if you win by 1 or 10,000 points—a win is a win. Since bracket pools often have between 10 and 100 members, any event that occurs rarely is going to be suboptimal to pick, regardless of how much bonus we add to it. Therefore, for this analysis, I chose to remove seed-round combinations below 1.5%.

We will use the geometric baseline scoring system, which rewards double the points in each round for correct picks. For example, if a correct pick is worth 1 point in the first round, then it would be worth 2 points in the second round, 4 points in the third round, and so on until the championship, where a correct pick is worth 32 points.

Finally, we optimize the upset bonuses to meet these criteria:

1. Reward having as many matchups as possible have similar expected value.
2. The underdog having a greater expected value is bad, so this is penalized 5 times as much as the favorite having a greater expected value.
3. All upset bonuses should be integers.
4. There should be some pattern in how the upset bonuses grow over time.

This results in the scoring values shown in the table below. The “Optimal Upset Bonus” is what comes out of the optimizer, and it’s a good starting point. The upset bonuses mostly increase with later rounds, but they do not follow a defined pattern like the baseline scoring does. This is unfortunate since it adds complexity. However, we noticed that if you spread out the bonuses, you could make them exactly half of the baseline scoring values and still have a decent result.

How do we know how good our results are? Figure 2 displays the expected cumulative score for picking each seed to reach each round. We black out seed rounds that are rare (less than 1.5%). Additionally, we place a yellow box around the seeds that are competing with one another for each open bracket slot.

Figure 2: Expected cumulative score for picking each seed to reach each round. Rare seed rounds are blacked out (less than 1.5%). Yellow boxes are around seeds competing for the same bracket slots.

There are a few great outcomes from this:

1. Only twice does a lower seed have a higher expected cumulative score for a bracket slot:
   • The 8–9 first round matchup, which is unavoidable.
   • No. 3 seeds over No. 2 seeds for winning the championship.
2. The expected values have been brought much closer for several matchups:
   • First round: No. 5–12, 6–11, and 7–10 seeds
   • Second round: No. 4–5 seeds and 3–6 seeds
   • Third round and on: No. 2 and 3 seeds
   • Championship: No. 1, 2, and 3 seeds
3. The bonus is simple and easy to explain (always half the point value for picking the correct winner).

In conclusion, the tyranny of the chalk bracket has been adequately challenged by our custom scoring system without opening the door to gaming the upset bonus. Is this the perfect scoring system? Are we geniuses? Will a new Virginia basketball dynasty start next year? I can only hope the answer will be yes to all the above, but only time will tell.

Happy bracketing!

Footnotes

¹ Source: me, a sports fan

² These probabilities are normalized to account for low sample size. For example, No. 7 seeds are 1 for 1 in winning national championships. This is adjusted downward using the average winning percentage for lower seeds in the championship game.

³ https://www.elderresearch.com/blog/how-to-pick-a-winning-march-madness-bracket/