Men’s Lead

Women’s Boulder

Women’s Lead

This also provides a unique challenge in predicting a climbing competition. Most models for predicting sporting outcomes involve assigning some numerical strength to each competitor and then somehow comparing these strengths to predict performance in a competition.

In predicting a competition in bouldering or lead climbing, such a model would probably work fine. If climber A climbs higher than climber B on about 70% of lead routes, then any model that gives climber A a 70% chance of beating climber B seems pretty fair. The final rankings in the competition don’t care about how much higher A can climb or where climber B ends up on the wall, so a predictive model need not care either.

With the current combined format used for the Olympics, however, the effect of the route becomes impossible to ignore. Every top, zone, and hold on a lead route has a measurable impact on the final results, and there is a fixed relationship between boulder and lead scores. Predicting a boulder and lead competition relies not only on predicting what the climbers will do, but what the routesetters will do.

Background/Scoring System

The 2024 Olympics feature a combined boulder and lead climbing event. While the previous Olympics featured a combined event with scoring based on climbers’ ranks in each event, the points system for this year’s event assigns points based on performance on each boulder and lead route.

Each boulder is worth 25 points. A climber gets 5 points for reaching the first of two intermediate zone holds, a total of 10 for reaching the second zone hold, and a total of 25 for reaching the top of the boulder. The climber loses 0.1 points for each failed attempt before reaching their highest-scoring part of the boulder.

The lead route consists of at least 40 hand holds, the last 40 of which are worth points. Of those 40, the first 10 are worth 1 point each, holds 11-20 are worth 2 points each, holds 21-30 are worth 3 points each, and holds 31-40 are worth 4 points each. An additional 0.1 point is awarded if the climber successfully makes progress off of their final hold.

Thus, boulder and lead are each worth a maximum of 100 points, for an overall maximum of 200. At the Olympics, the top 8 athletes after the semifinals advance to the final, where points are reset and climbers have another 4 boulders and lead route to try to get another chance at up to 200 points.

The goal of the model is to determine each climber’s ability on both boulder and lead, and then to simulate a large number of boulders and lead routes to estimate the chances of each climber performing well at a competition.

The Model

The model is divided into 4 parts: rating climbers’ boulder ability, rating climbers’ lead ability, simulating boulders, and simulating lead routes.

Each climber is given an Elo rating in both boulder and lead. Elo ratings are a system invented for chess in which competitors’ ratings go up after wins, especially against stronger opponents, and down after losses, especially against weaker opponents. In the spirit of “competing against the route,” ratings are also given to boulders and lead routes.

Boulder Ratings

Every climber in the first competition in my dataset (the 2008 World Cup in Hall, Austria) was given an initial boulder rating of 1500. Every subsequent climber was also given a 1500 rating prior to their first competition.

Each boulder is divided into 2 sections (or 3, if the route has 2 zones). A climber is credited with a “win” against each section that they complete, a “loss” against each section that they fail to complete, and no result against a section that they fail to reach. For example, if a climber reaches the low zone on a B&L combined route but not the high zone, they get a “win” against the low zone and a “loss” against the high zone, but they are not penalized again for failing to reach the top, since they did not get to the high zone. Attempts were not considered in assigning ratings, so a flash is considered the same as a top in 10 attempts for the purpose of these ratings.

At a boulder competition, each section of each boulder is given a “performance rating,” a concept also taken from chess. This corresponds roughly to what the section’s Elo rating would have to be in order for you to expect the climbers in the competition to have the success rate that they actually had against the boulder. A section’s performance rating is calculated by taking the average Elo rating of the climbers who attempt the section, and then adding rating difference term based on the percentage of climbers that successfully complete it. For example, if half of the climbers who attempt the section successfully complete it, the rating difference term is 0. If all of them fail, the rating difference term is 800, and if they all succeed, the term is -800. Rating difference terms were taken from the FIDE (world chess federation) handbook.

Once each section of each boulder is given a rating, climbers’ ratings are adjusted based on what they are able to climb in the competition. If a climber successfully completes a boulder, their Elo rating is increased by

If they fail, their rating is decreased by

Here, C is the climber’s rating and S is the section’s rating. K is a constant that determines how quickly climbers’ ratings change with each competition. A higher K means that climbers’ ratings will depend more heavily on recent competitions. The boulder model currently uses K=32.

These ratings adjustments were run on every World Cup, World Championships, and OQS from 2008 (as long as data were available) to today to produce the boulder ratings.

Lead Ratings

Lead ratings were computed in a similar way, but with some key changes. Rather than assigning a rating to each section of the lead wall, each hold was given a rating corresponding to the difficulty of getting to that hold from the start of the route. A hold is credited with “beating” all of the climbers who did not reach that hold, and with “losing” to all of the climbers who did not reach that hold in determining a performance rating. Extra 0.1 points are not considered in generating lead ratings.

Thus, a route with 40 holds would be given a sequence of 40 nondecreasing ratings going from the bottom to the top. The best performing climber(s) is (are) credited with a “win” against the last hold that they completed, and the worst performing climber(s) is (are) credited with a “loss” against the first hold. Each other climber is credited with a “win” against the last hold that they reached, and a “loss” against the following hold. Because each climber has at most 1 win and 1 loss per lead route, compared to as many as 12 wins and losses in a boulder round, a K value of 64 is used in updating lead ratings.

Boulder Simulation

Simulating a boulder round requires some idea of how difficult each top and zone tends to be in each qualifying, semifinal, and final round. Since the only recent competitions with multiple zones are Boulder and Lead combined competitions, these were the only competitions used in generating boulder simulations.

To simulate a semifinal round, all boulders from the semifinals of the 2022 Morioka World Cup, the 2023 Combined World Championships, and OQS events were given performance ratings. The average and standard deviation of the performance ratings were taken for all of the Z1s, all of the Z2s, and all of the tops. To simulate a single boulder, a Z1 rating is generated by sampling from a normal distribution with the mean and standard deviation corresponding to the mean and standard deviation of the Z1s from previous semifinal rounds. A Z2 rating and a top rating are sampled similarly to get the 3 difficulty ratings necessary to define a single boulder. This is repeated for every boulder in the semifinal round, and then again for the final round (as well as the qualifying round if the competition being simulated has a qualifying round).

To simulate a climber attempting to climb a boulder, we first randomly determine if they climb Z1. If they succeed, we randomly determine if the climb Z2, and if they succeed again, we randomly determine if they reach the top. Once a climber reaches a section, their probability of successfully climbing the section is

Where S is the section’s Elo rating, and C is the climber’s Elo rating. Again, attempts are not considered, so a climber is given 5 points for reaching zone 1, 10 for reaching zone 2, and 25 for reaching the top. The simulated scores are thus probably slightly higher than they should be, but this effect is hopefully small enough and similar enough for each climber that it doesn’t give any climber too much of an advantage or disadvantage.

There are many problems with a method like this, including the fact that these difficulties are not independent (a hard zone 1 often leads to an easier zone 2, and some rounds are systematically too hard or too easy across the board), but this approach seems like a good start.

Lead Simulation

Simulating a lead route was much trickier. The ratings for each hold can’t be simulated independently, because the ratings have to be nondecreasing. While hold 10 might have a rating of 1700 on one route, and hold 11 might have a rating of 1500 on a different route, those ratings can’t both occur on the same route.

The approach is as follows: to simulate a semifinal lead round, get the ratings for each of the last 40 holds for every World Cup, World Championship, and OQS semifinal since 2021 (if a route has fewer than 40 holds, just copy the rating of the first hold until there are 40 ratings). Then get the mean and standard deviation of the ratings for each hold number from 1 to 40. To simulate the rating of hold 1 on a lead route, do the same thing from the boulder simulations: simulate from a normal distribution with the mean and standard deviation corresponding to the mean and standard deviation of the hold 1 ratings from every route considered.

To simulate other holds, we can’t use normal distributions, since that could result in a decrease in rating from one hold to the next. Instead, we use a distribution that decays like an inverse cube (differences between successive hold ratings seemed to behave this way). Specifically, we sample the difference between hold i+1 and hold i from a distribution with

This distribution has mean m. m is set to be equal to the average rating of hold i+1 minus the average of the following two things:

• The average rating of hold i
• The rating of hold i on the route being simulated

This ensures that each hold has approximately the right difficulty on average, while preventing the possibility of having too many easy holds or too many hard holds in a row. Some more adjustments are made (m has to be at least 3, the change in hold rating has to be at most 500, and all ratings are adjusted at the end so that they have the right mean).

Once a route with 40 simulated hold ratings has been simulated, each climber must get a simulated result. To do this, the probability of the climber falling before each hold is computed (this will start close to 0, and then increase as the route goes on until it is close to 1). The climber is then assigned a random number between 0 and 1. Whichever two hold probabilities that number falls between is that climber’s simulated performance on the route. For example, if the climber’s random number is 0.6, and their probability of falling before hold 21 is 0.59, and their probability of falling before hold 22 is 0.62, then the climber’s simulated hold reached is 21.

Again, this is done repeatedly to simulate each semifinal and final round.

Finishing the Simulation

To simulate the Olympics and get probabilities, we simulate 100,000 boulder and lead semifinal rounds. In each, the top 8 climbers are put into a simulated final round to determine final placings. Probabilities are then determined based on the number of simulations in which each result occurred.

Final Thoughts

This writeup went on much longer than I expected. This model could definitely use some improvements (adding dependence between boulders and trying to handle easy or hard lead routes come to mind), but it hopefully gives a rough sense of likely outcomes for the Olympics and beyond. Simulating a competition like this has been a really fun technical challenge, and I have really appreciated the feedback and interesting discussion that I’ve gotten from this project. If you have any more feedback, you can reach me at [email protected].

Update 2024/08/03

I’ve made a few small updates to the model. I’ve added recent world cup events to the Elo ratings, as well as the European, Asian, and North American Olympic qualifying events (I previously only had world championship and world cup events). I also modified some parameters, and made some slight changes to simulations, in which climbers’ performances are slightly correlated, leading to favorites being given slightly higher probabilities of winning.

As I mentioned in my first post, the odds of picking a perfect bracket are somewhere between roughly 1 in a billion and 1 in 100 billion. One possible conclusion to draw from this is that it will take somewhere between 1 billion and 100 billion brackets in order to find the perfect one.

This intuition turns out not to be correct. These estimates, from FiveThirtyEight forecasts and from DePaul professor Jeff Bergen, are the probability of the most likely bracket being perfect. Specifically, those estimates are trying to measure the probability that the favorite will win every game. As we know from watching March Madness in previous years, there have always been plenty of upsets, and thus the all-favorites bracket has never come anywhere close to being perfect.

Here’s another incorrect conclusion to draw from this: because the NCAA tournament is much more likely to have, say, 12 upsets than 0, then it must be “better” to predict a bracket with 12 upsets than one with 0 if you’re trying to be perfect. It may initially seem impossible for the all-favorites bracket to be the most likely bracket, while also being much less likely than having 12 upsets in the tournament.

However, here is another way of looking at the problem that shows why these two ideas are still compatible. If we assume that every game has one team whose winning would be considered an “upset” (maybe not realistic in, say 8 vs. 9 seed games, but still a reasonable idea), then the number of ways for there to be 12 upsets is equal to the number of ways to choose 12 games out of a collection of 63. This number, known as 63 choose 12 and equal to about 2.7 trillion, is the number of ways for 12 upsets to occur in the tournament. It is far more likely for the tournament to have 0 upsets than for any one of these outcomes to occur, but it is not 2.7 trillion times more likely, so the “12 upsets” outcome, whose probability is equal to all of those 2.7 trillion probabilities put together, ends up being more likely than the single outcome with no upsets at all.

This has some implications for both individual people filling out brackets and my bracket challenge. Just because there’s usually, say, at least one 13-seed in the second round, doesn’t mean that you should pick one. You’ll look really cool if you get the prediction right, but this only happens because there are a lot of opportunities for it to happen. There are 4 ways for a 13 seed to win but only 1 way for all of them to lose, so you’re still usually better off picking 4 4-seeds than picking a 13.

For my challenge, it means that I need to be careful to pick the most likely brackets rather than ones that just look reasonable. Picking out all of the most likely brackets as quickly as possible will be key to the success of the challenge.

Probabilities for the Project

With that tangent about likely brackets out of the way, I’ll describe my first attempt at finding a perfect bracket. The challenge can be divided into two main parts: writing out as many brackets as possible as fast as possible, and being careful to choose the brackets that are the most likely to happen.

So far, I’ve been running my program on this year’s NCAA tournament. I told the program who is playing in the tournament and what the initial bracket looks like, but not the outcomes of the games, so I can simulate what it will be like next year, when the program will be trying to produce brackets between the tournament selection show and the start of the first round of games.

In order to ensure that I’m choosing the most likely brackets, I will need some way to compute probabilities of various outcomes. I may end up trying to produce my own model, but for now, I’m taking data from FiveThirtyEight. Every year, at the beginning of the tournament, they release their March Madness projections. In these projections, each team is given a rating, and the probability of either team winning a game is based on the difference between the teams’ ratings. In particular, they use this relatively simple formula for the probability of team 1 beating team 2:

where is team 2’s rating and is team 1’s rating. This is a slight modification of the probabilities that form the foundation for the Elo rating system used most notably in chess ratings.

There are a few problems with this system, most notably in how it handles upsets. At the start of the 2022 tournament, 2-seed Duke had a rating of 89.3, and 8-seed North Carolina had a rating of 84.0, giving Duke a 72% chance to win that matchup. By the time they played each other in the final four, their ratings were 92.7 and 90.0, giving Duke just a 62% chance. UNC won the game and advanced to the final.

Of course, I cherry-picked a game in which the underdog won, but this is a pattern that comes up often when a team has multiple upset wins during the tournament. When a team like 2022 Duke wins a few tournament games, their rating doesn’t change much, since they were expected to win (Duke was the better seed in every game they played in that tournament). When a team like 2022 UNC makes a run (they upset a 1 seed and a 4 seed in the early rounds), their rating goes up quite a bit more. The ratings of the remaining teams tend to converge toward each other as the tournament goes on for this reason, so using ratings from the beginning of the tournament to try to predict later rounds will tend to underestimate the chance of upsets later on.

The First Attempt

With those probabilities (and caveats on accuracy) in mind, I wrote a program to try to produce as many likely brackets as possible. What I would like to do is just write out the brackets in order of likelihood: write out the most likely bracket, then the second most likely, and so on. I just get the program to do this for three and a half days, and once the time runs out and the first round of games is about to start, I turn the program off after it’s written the n most likely brackets, for some (hopefully) large value of n.

The main problem with this is that it is very difficult to figure out the most likely brackets in order. Pretty much the only way to do it successfully is to write out all of the brackets, calculate their probabilities of occurring, and then sort that list in order of probability. If it were possible to just write out all of the brackets in three and a half days (let alone sort them), then this project would be too easy. Furthermore, the point of this project is to write brackets. If I’m writing out brackets, computing their probabilities, and then sorting them, I would be much better off just skipping those last two steps and writing more brackets.

Thus, my approach will try to combine the approach of carefully choosing the most likely brackets with the speed of just blindly writing brackets without checking. I will do this by dividing the bracket into 3 sections:

1. Left half of the bracket, first two rounds (24 games)
2. Right half of the bracket, first two rounds (24 games)
3. The last four rounds of the tournament (15 games)

While there are too many overall brackets to ever have any chance of writing them all out, each section is small enough to consider every possible outcome individually. Sections 1 and 2 each consist of 24 games, so there are 2^24=16,777,216 possible ways for each of those sections to go. Once the results of the first two rounds are determined, there are 2^15=32,768 possible outcomes for section 3. These are large numbers, but are very manageable for a computer. Dividing up the bracket this way gives the following approach to generating brackets:

1. Write out all 16,777,216 possible outcomes for section 1 of the bracket, and compute the probabilities of each of these outcomes occurring using FiveThirtyEight’s model.
2. Sort the possible outcomes of section 1 from most to least likely.
3. Repeat steps 1 and 2 for section 2 of the bracket.
4. Combine the most likely outcome of section 1 with the most likely outcome of section 2. Then combine these first 2 rounds of the tournament with the 32,768 possible outcomes for section 3, and use these to create 32,768 completed brackets. Then write these 32,768 brackets to a file. We have created our first completed brackets.
5. Repeat step 4, but with different combinations of results from sections 1 and 2. If b(i,j) is the partial bracket formed by combining the ith most likely outcome of section 1 with the jth most likely outcome of section 2, then step 4 gave us all 32,768 possible brackets with b(1,1) as the outcome of the first two rounds. We are going to repeat step 4 with b(1,2), b(2,1), b(1,3), b(2,2), b(3,1), b(1,4), etc. For every sum k>1, and every i and j such that i + j = k, we will produce all 32,768 brackets with first two rounds b(i,j). We start at k = 2 and keep increasing k until the program runs out of time.

After running this program for about a day and a half, I generated many billions of brackets but a perfect bracket probability of only a few tenths of a percent. While this is far from what I would need to have a decent shot at a perfect bracket, I found this to be a promising result for a few reasons. First, I was putting the program under worse conditions than it would be under during the actual bracket challenge next year. The main point of blindly including all 32,768 possible outcomes of the last 4 rounds of the tournament is for speed: it’s much faster to just write down 32,768 brackets than to actually calculate their probabilities. However, since I was testing the model, I actually went ahead and computed those probabilities to see how the program was doing. During the real thing, I won’t have to do this, which will speed the program up.

Second, the current form of the program leaves a lot to be improved. Blindly printing 32,768 brackets at a time is good for speed, but it results in a lot of extra brackets being written down that have very little chance of happening (even less of a chance than one would normally have of picking a perfect bracket). Historically, a tournament is less likely to have a champion seeded 4 or lower than it is to have a 13 seed in the sweet 16, but at least a quarter of my brackets will have such a team as its champion.

There’s also a more subtle issue in the way that I’m choosing which outcomes to consider. Which of the following is more likely?

• Section 1 has its most likely outcome, and section 2 has its 999th most likely outcome
• Section 1 and section 2 each have their 500th most likely outcomes

It’s far from obvious, but it turns out that the first event is far more likely than the second, and this difference becomes more exaggerated as we move further down the list of the most likely outcomes. However, my program is currently treating these as equally likely, and adding them to the list of brackets at around the same time. I could do a much better job by, say, including b(1,2000) before b(500,500). This is probably important enough for me to write a post on this phenomenon later on.

The third, and possibly most important effect (if the least interesting) is that my choice of programming language could have a major effect on the speed of the program. I’ve been using python so far, but while python is easier to use, it tends to run much more slowly than something like C++ or even Java, so I could probably speed things up quite a bit by switching. I’ll probably work on this once I start running out of improvements to make on the bracket generation process.

I’ve already made some improvements on my approach, but the same basic idea remains: divide up the bracket into sections, get every possible outcome for the early rounds, and then extend the most likely outcomes into the later rounds. The project is hopefully on its way to constructing a perfect bracket!

The original purpose of this blog was to provide a public record of the challenge, so that, if I succeed, it’s possible to verify that I didn’t cheat. I was initially reluctant to post anything about the challenge too far before next year’s tournament, out of fear that a better programmer might come along and beat me at my own challenge. I ultimately realized that (1) that would be really cool and (2) that would mean that at least one other person is reading my blog, which would be really cool.

With those ideas in mind, here are the rules to the challenge. These are the rules that I will follow, and can also be seen as the guidelines for anyone else who wants to try it. The general rule of thumb is that the program is trying to simulate a single fan who just happens to be absurdly fast at filling out brackets.

The Bracket

First, I’ll clarify what a bracket should look like. A bracket, as I’ll choose to define it, is a string of 63 1’s and 0’s, in which each digit represents the winner of a single game of the tournament*. The first 32 digits represent the outcomes of the first round games, the next 16 digits represent the second round, etc., with the last digit representing the outcome of the final. Within a round, the games on the left half of the bracket are listed from top to bottom, followed by the games on the right half of the bracket listed from top to bottom.

*As in most bracket challenges, I will not be including play-in games in my brackets. If one of my brackets predicts that a team from a play-in will win one or more games in the first round or later, those count as wins for either team that wins the corresponding play-in. This is the same as the rule used in most bracket challenges, where brackets can be filled out before the play-in games are played.

If ‘0’ is listed for a game, that means that the team listed higher in the bracket is predicted to win, and if a ‘1’ is listed, that means that the team listed lower in the bracket is predicted to win. In the final, a ‘0’ predicts a win by the team from the left half of the bracket, while a ‘1’ predicts a win by the team from the right side of the bracket. Since different sources sometimes depict the bracket in different ways, the “official bracket” used for the challenge will be the one that appears in the NCAA tournament selection show.

I couldn’t think of a reasonable way to word all of that, so here’s a diagram of a bracket, with the games numbered from 1 to 63, and the teams corresponding to 0 and 1 listed for each game:

And, as an example, here is the 2021 men’s bracket. We can verify that it corresponds to the following bracket string:

011010000000000101001000011010100111111100010100110111011010111

Brackets will be submitted in one or more text files, named bracket_file_0.txt, bracket_file_1.txt, etc., in which each line consists of a 63-character bracket string.

I realize that a string of 1’s and 0’s isn’t really the same thing as a bracket, but I think it maintains the important characteristics. It doesn’t have the same physical structure or the names of the teams, but it still has the property that one string corresponds to exactly one bracket, so anyone can look at one of these strings before the tournament and know exactly what has to happen for that “bracket” to be perfect. It’s also not that different from an actual bracket challenge, where users have to choose which boxes to check rather than having to enter team names. This will also allow the code to run faster, and thus generate more brackets, so I think the tradeoff is worth it.

Timing

As stated above, the code should be simulating a single fan who happens to be extremely fast at filling out brackets. The challenge will follow the timing of the ESPN bracket challenge. Even though I won’t be competing in the challenge (the billions of brackets would probably crash their servers), the code cannot be run until the challenge opens on selection Sunday, and the code must be halted before the challenge closes prior to the start of the first round on Thursday (Friday for the women’s tournament).

Since brackets will have 1’s and 0’s instead of the actual teams, that could provide some opportunities to “cheat” in generating brackets. In theory, it would be possible to create a bunch of brackets (or parts of brackets) ahead of time, using heuristics like advancing 1 seeds most or all of the time. This is not allowed. No brackets or parts of brackets may be generated prior to the opening of ESPN’s bracket challenge. This violates the “really fast fan” test, since it would correspond to a fan filling out part of their bracket before the challenge opened.

However, any basketball-related analysis may be done beforehand. Any external models (KenPom, FiveThirtyEight, etc.) may be used, and participants are welcome to do any modeling they want. If, for example, a participant wants to calculate a rating for each team, or compute a probability of any one team beating any other team, they are welcome to do these computations and store the results beforehand. Any single fan could do this, so it’s allowed in this challenge.

This should hopefully cover most of the things that people would want to do to complete the challenge. For anything else, the “really fast fan” test should serve as a good guideline to what is reasonable.

Computation Limits

Any code used should be able to run on a personal computer during the ~88 hours that the ESPN tournament challenge is open. I’m not an expert on processors, but hopefully that’s a reasonable enough guideline. I’ll be completing the challenge on my 2020 MacBook Pro with an M1 chip.

Furthermore, no parallelization may be used. Brackets must be filled out one at a time (although writing one bracket, modifying it slightly, and then writing the new bracket is allowed). Writing brackets in parallel would be a violation of the “really fast fan” test, although I suppose the personal computer requirement is just a limit on how fast the fan can be.

If there are any questions about the rules, or any loopholes I might be missing, feel free to submit a comment, and I can try to address it.

The tournaments also bring a surprising amount of attention to an interesting mathematical object: the bracket. Every year, millions of people fill out tens of millions of brackets in an attempt to show off their predictive skills (or simply to express their love for certain teams’ mascots or uniforms). Every bracket that’s filled out brings with it a slight hope of achieving the holy grail of March Madness: the perfect bracket, with correct predictions for every game, from the first round to the championship.

It is, however, hard to overstate just how slight that hope really is. Various attempts have been made to quantify the probability of a bracket being perfect, and all make the prospects look quite bleak. The first, most naive value that’s often quoted is the probability of picking a perfect bracket by guessing the outcome of every game uniformly at random: there are 63 games in most bracket challenges, so the odds of such a bracket being perfect are 1 in 2^63, or 1 in 9.2 quintillion.

Of course, there are far better ways to fill out brackets than flipping coins. DePaul math professor Jeff Bergen adjusted teams’ win probabilities based on the historical success of their respective seeds (1 seeds almost always beat 16 seeds, but 8 vs 9 matchups are more like a coin flip) and found that a more reasonable probability was 1 in 128 billion. This is of course much more manageable, but 128 billion is still orders of magnitude more than the number of brackets that have ever been filled out.

I think that probability is still a bit low (I’ll probably make a post about that in more detail). If we take FiveThirtyEight’s pre-tournament forecasts from this year, the probability of the favorite winning every game is about 1 in 26 billion. Thus, there is at least one way to fill out a bracket (namely, the one where we always pick the better team according to FiveThirtyEight’s model) that gives us odds of 1 in 26 billion. FiveThirtyEight put the odds as high as 1 in 1.6 billion back in 2015, which they said was a particularly top-heavy year, making upsets less likely than usual. I haven’t yet crunched the numbers on every year in which data are available, but it seems that we can’t hope for much better than about a 1 in a billion chance of picking a perfect bracket.

The Challenge

Now that we’ve established just how impossible it is to make a perfect bracket, I’ll outline my goal: I want to write a perfect bracket for the 2024 NCAA Division I men’s basketball tournament. After spending this much time going through how long the odds are, how am I still conceited enough to think that I can accomplish what hundreds of millions of brackets have not? Well, I’m going to need a lot more than hundreds of millions of brackets.

My goal, stated more precisely, is the following: next year, between the time that brackets are released and the time that the first games start, I want to make a perfect bracket. This will (hopefully) be accomplished by writing a computer program that prints out as many plausible brackets as it can. I will not get the bragging rights associated with actually predicting the games myself, and I won’t be able to submit all of my brackets to any official bracket challenge, but if all goes well, I will achieve the distinction of having technically produced the first ever perfect bracket.

As far as I can tell, a challenge like this has been attempted once before. In 2019, Chris Wilson from Time Magazine randomly generated 10 million brackets using a formula to give preference to better seeds. As we might expect, some of them did very well, but 10 million was not enough. Those 10 million brackets included some very good entries, but every bracket was busted by the end of the first round, and the best bracket did well enough to finish 8th in ESPN’s bracket challenge.

While 10 million brackets may have fallen short, it may still seem like computer assistance should turn this into a trivial problem. After all, if I can just print brackets at will, what is stopping me from just printing all of them? It turns out, however, that computer technology is just advanced enough to make this an interesting problem. Printing 1.6 billion brackets on a personal computer in 4 days is very doable. Printing 9.2 quintillion is not.

Furthermore, making a perfect bracket isn’t as simple as writing out 1.6 billion brackets, even if next year’s tournament turns out to be more like 2015 than 2022. While the most likely bracket might have odds of 1 in 1.6 billions, the billionth most likely bracket will have odds that are much lower, and filling out more brackets will quickly produce diminishing returns. Without filling out all 9.2 quintillion possible brackets, it is impossible to guarantee that one will be perfect, but there is a lot that we can do to improve our odds.

Thus, this is a challenge of trying to maximize the quantity of brackets produced without sacrificing quality of predictions made in each individual bracket. I’ll put out the full rules of the challenge in another post, and I’ll continue to post updates on my progress. Let the madness begin!