In my last article, I posted this year’s iteration of my annual model for predicting the outcome of the NCAA Division I Men’s Basketball Tournament. I provided the modeled win probability of all 2278 possible matchups along with an “expected” bracket representing the outcome if the modeled favorite wins every game. Check that out here.

That bracket isn’t terribly practical, though. Just because the model gives Team A a 55% chance of winning doesn’t mean they’re going to win. That’s not much different from a coinflip! Furthermore, it doesn’t really reflect the other matchups that a team may have to go through. If Team A has a win probability of 20% against Team B in the second round, that doesn’t look too good for them. However, what if Team B’s win probability in the first round against Team C is only 51% and Team A’s win probability against Team C would be 80%? The “expected” bracket doesn’t show this, but that suddenly makes Team A’s outlook much better.

That’s where simulations come in. We can use the modeled win probabilities to simulate the entire tournament. We can run this simulation any number of times and compile the results to see how likely each time is to reach each round of the tournament assuming that the modeled win probabilities are accurate. I ran 10,000 simulations of the tournament – here are the results.

South Region

The top overall seed Alabama Crimson Tide are unsurprisingly the favorites to make it to the Final Four and win the region. Furthermore, their 12.5% probability of winning the tournament tops the region with No. 2 Arizona being the next best at 5.2%. Arizona looks to be a formidable team in their own right with a solid 20% likelihood of making the Final Four.

Afterwards, there’s a drop-off before the next tier of teams: No. 3 Baylor and No. 5 San Diego State. Their odds of making the Final Four, National Championship, and winning it all are all virtually the same. Everyone else is a long way down and not really worth a mention.

Midwest Region

Well, this one looks like a two man show.

There’s a lot to say about the Houston Cougars. They’re the favorites in Vegas to win the tournament and they boast one of the most stifling defenses in the country. With star Marcus Sasser set to return from injury shortly, the Cougars are poised to show off their best team in years. That’s saying a lot, given that they’re coming off of a Final Four appearance in 2021 and an Elite Eight loss in 2022.

Then we have the No. 2 Texas Longhorns. Texas is ranked 18th in offensive efficiency in 11th in defensive efficiency according to KenPom.com – with a 26-8 record and a recent Big 12 title, this balanced roster is coming into the tournament with some positive momentum.

The Cougars have an astonishing 44.6% likelihood of making the Final Four according to our model, with Texas following them up at 23.0%. The two favorites have 20.8% and 5.5% respective probabilities to win the entire tournament. No one else in the region is over one percent. It’s clear that one of these teams should be the ones to make it out of the Midwest.

East Region

No. 1 Purdue leads the East with a 5.7% probability of winning the tournament, followed closely behind by No. 2 Marquette at 4.9%. Spoiler alert, but this is the lowest mark for a region’s top team. Of all four regions, our model has the East as the least likely to field the tournament’s eventual champion.

Purdue and Marquette and virtually neck-and-neck for Final Four odds. Marquette is viewed as having a more favorable path to the Elite Eight, while Purdue is favored in a matchup between the two teams.

Afterwards, we have No. 4 Tennessee and No. 5 Duke – sleepers to steal the region and sneak into the Final Four, but not really viewed as massive title contenders (2.0% and 1.2% championship odds respectively).

West Region

Finally, we’ve got the West. Here’s a fun one – take a look at how much this model hates the No. 1 Kansas Jayhawks. It has the remaining top five seeds as more likely to make it to the Final Four – even No. 5 Saint Mary’s! If you want a one seed to fade, the Jayhawks may be the one for you.

At the top, we have the No. 2 UCLA Bruins with a fantastic 36.2% probability to win the region and a very nice 13.5% mark of winning the tournament. That’s behind Houston but slightly ahead of Alabama for the 2nd best overall. In order, they’re followed up by No. 3 Gonzaga, No. 4 Connecticut, No. 5 Saint Mary’s, and No. 1 Kansas.

Commentary

I think simulation analysis has a lot of value in revealing details in how the bracket shaped paths for the top teams. Last year, my simulations revealed that Kansas had the highest probability of making it to the Final Four. They weren’t viewed as the best team in the tournament and were not favored to win it, but the simulations demonstrated that they had the easiest path to the Final Four. Picking the Final Four teams correctly is a huge boost, so knowing which teams have easy/hard paths is immensely important.

This time around, Kansas isn’t so fortunate. They look like a fade candidate rather than a team poised to win their region. On the other hand, Houston has the highest Final Four probability in the tournament – if you want a safe Final Four pick, they might be your best bet!

Unfortunately, these margins are very small. We’re just playing the numbers here – anything can happen once these teams step on the court. There’s a reason no human has selected a perfect bracket yet, and there’s a reason it’s called March Madness. Don’t put too much stock on the statistics – there’s a lot that they can’t quantify. Most pre-tournament analyses like this one did not foresee the No. 8 North Carolina Tar Heels embarking on a miraculous Final Four run last season. We’re guaranteed to see more unexpected results this March, so don’t expect perfection. If you want to win your bracket pool, all you can do is play the numbers and hope for the best.

The post March Madness 2023: Tournament Simulation Results appeared first on The Spax.

For the fourth time in this website’s history, I’ll be training a model to predict the outcome of March Madness, the annual postseason tournament to crown a champion of men’s Division I college basketball.

Over the past four years, The Spax’s modeled bracket has varied in performance. Since a hot debut in 2019 where our model correctly predicted three of the Final Four teams (including five-seed Auburn) and the national champion Virginia Cavaliers, it has certainly cooled down.

Of course, scoring a lot of points is generally going to be difficult if you don’t predict the champion correctly. In two straight years of picking Gonzaga to win it all, the Bulldogs were not able to finish the job. Granted, most models in both years had Gonzaga winning it – they were the national favorites for a reason. That’s why an optimal bracket strategy would not be to simply pick the team with the highest probability to win each game like I do for the purpose of this analysis – rather, you should consider each team’s win probability relative to the public’s assessment of their win probability. Maybe some one-seed is modeled as the favorite to win their region, but not by that much, so you choose to pick the five-seed that has a solid 40% modeled probability to beat them head-to-head.

Anyway, let’s get to this year’s picks:

First Four

I slacked off this year and didn’t get this article out before the play-in games occurred, so all four of these games have actually ended by now. The forecast winner ended up going 3-1, the sole error being in Pittsburgh’s 60-59 win over Mississippi State.

South Region

Top Team: The top overall seed in the country is the University of Alabama, and they’re unsurprisingly the clear top dogs in their region as well. It would be a shock to see them fall prior to the Elite Eight, and they would also be favored to beat the two-seed Arizona Wildcats in that matchup. With top NBA Draft prospect Brandon Miller leading the way, Alabama has a chance to make a transformational run.

Potential Disappointment: There’s not a whole lot to say here – both No. 2 Arizona and No. 3 Baylor are forecast to make it to the Sweet Sixteen, and losing to one another wouldn’t really be a huge disappointment. However, it’s certainly possible that one of them falls in the Round of 32 – failing to make it to Sweet Sixteen because they suffered a loss to No. 6 Creighton or No. 10 Utah State would certainly be a slap in the face for either program.

Sleeper Pick: This is probably the least interesting region, to be frank. I don’t think there’s a great sleeper pick, but if I had to pick one I would probably go with No. 6 Creighton. I think there’s a chance that they snag a couple of early wins to find themselves in a Sweet Sixteen matchup against No. 2 Arizona.

Most Probable First-Round Upsets: No. 10 Utah State over No. 7 Missouri (70.70 percent chance)

Midwest Region

Top Team: The Alabama Crimson Tide are the top overall seed in the tournament, but according to sportsbooks, the Houston Cougars are the favorites to win it all. The Cougars have had a great run over the past five or so years, with coach Kelvin Sampson leading them to the Elite Eight in 2022, the Final Four in 2021, and the Sweet Sixteen in 2019. That’s a fantastic run, especially considering the injuries they endured (missing their most talented player in Marcus Sasser in 2022). They’ve already had postseason success, but the 2023 Cougars are even better than past iterations of the team and now all eyes are on a title. Sasser is entering the tournament with an injury and is questionable to play in the first round, but with a healthy lineup, the Cougars should be the safest Final Four pick in the tournament.

Potential Disappointment: The model has a Round of 32 match up between No. 3 Xavier and No. 6 Iowa State as a virtual coinflip. That early of a loss would be a let down for the Musketeers, and with a tough potential outing against Texas in the next round anyway, expectations shouldn’t be too high for Xavier.

Sleeper Pick: I’ll go with another six-seed here, this time the Iowa State Cyclones. Like I said, they’ve a shot to knock off Xavier if they can get past the first round. However, something interesting I noticed is that Mississippi State would’ve been forecast to beat Iowa State in the Round of 64. As you know, they lost to Pittsburgh in the First Four. Pittsburgh, on the other hand, are not at all favored to beat Iowa State. These sort of discrepancies are unsurprisingly common because the transitive property doesn’t apply to sports, but I wouldn’t be shocked to see Pittsburgh upset Iowa State.

Most Probable First-Round Upsets: No. 10 Penn State over No. 7 Texas A&M (37.73 percent chance), No. 13 Kent State over No. 4 Indiana (38.82 percent chance), No. 14 Kennesaw State over No. 3 Xavier (31.85 percent chance)

East Region

Best Team: The No. 1 Purdue Boilermakers enter the tournament with high expectations after 7’4 superstar Zach Edey led them to a 29-5 record and a Big Ten title. They’re tied with the Jayhawks for the 3rd best odds (Houston, Alabama) of winning the tournament per Bovada, and both our model and sportsbooks view them as the favorite to win the East.

Potential Disappointment: … but it’s not necessarily going to be a walk in the park. After the first round, the “easiest” predicted matchup for Purdue is in the Sweet Sixteen against Duke. In that affair, the Blue Devils are given a 34.87% win probability. Memphis has a modeled win probability of 37.74% in the Round of 32, and Marquette pose a threat of their own in the Elite Eight. It’s clear that the two candidates for disappointing one-seed are Purdue and Kansas this year.

Sleeper Pick: Watch out for the No. 8 Memphis Tigers. They’re coming off of a win over the Houston Cougars (albeit sans Sasser) to win the AAC Tournament as they enter the Big Dance with some good momentum. The same is true for Purdue, of course, who will undoubtedly be favored to win a potential matchup between the two teams in the Round of 32, but I wouldn’t write off the Tigers completely.

Most Probable First-Round Upsets: No. 10 USC over No. 7 Michigan State (40.47 percent chance), No. 11 Providence over No. 6 Kentucky (49.33 percent chance), No. 14 Montana State over No. 3 Kansas State (31.37 percent chance)

West Region

Top Team: The forecast winners of this region are the two-seed UCLA Bruins. While it’s always sticks out when we don’t pick a one-seed to advance, this result isn’t actually an upset – the Bruins are currently the favorites in Vegas to win the West region at +275, followed by Kansas at +350. The Bruins’ program has had a modern resurgence, with a Final Four appearance in 2021 and a Sweet Sixteen run in 2022. They’ll almost certainly make that three straight Sweet Sixteen appearances this year, with a key potential matchup against Gonzaga testing their ability to go the distance.

Potential Disappointment: Whenever a one-seed doesn’t get out of their region, it can be considered a disappointing tournament for them. The defending champion Kansas Jayhawks enter the West as the top seed, but as I mentioned before, they are not favored to come out on top. Furthermore, they’re actually forecast to fall to four-seed Connecticut in the Sweet Sixteen. It is clear that the Jayhawks will not have an easy road to the Final Four.

Sleeper Pick: Well, Connecticut’s a four-seed predicted to make the Elite Eight, so it suffices to say that they’re a solid sleeper pick to throw into your bracket. There’s not much else I’d buy into – I think picking Arkansas or Arizona State to get into the Sweet Sixteen would be a little too risky.

Most Probable First-Round Upsets: No. 11 Arizona State over No. 6 TCU (52.59 percent chance), No. 10 Boise State over No. 7 Northwestern (55.78 percent chance)

Final Four

The model forecasts the Final Four matchups to beat Houston v. UCLA and Alabama v. Purdue, with the favorites winning and moving onto the National Championship Game. The Cougars and the Crimson Tide are the clear two favorites to win the tournament, so it’s only fitting that they’d meet in the title game. If they do, we predict Houston to have the edge.

Needless to say, this is a pretty chalky Final Four. Just because this is viewed as the most likely outcome does not mean that it’s going to happen – it almost certainly won’t! Maybe Kansas will make it in over UCLA. Maybe Connecticut will get in over either. Perhaps Purdue falls in the Sweet Sixteen and Marquette takes their place. Or maybe some random double-digit seed comes in and takes a spot. Who knows? A good bracket probably shouldn’t pick these four teams in the end.

I’ve only shown a fraction of the possible matchups that can occur in March Madness. I’ll provide the win probabilities for every single one of them, though! If you’d like to see the model’s projections for matchups not listed above, you can search for them in table below which consists of 2,278 rows, one for every possible matchup. Just type the name of the two teams separated by a space. For instance, if you want to search for the Houston-Kansas matchup, type “Houston Kansas” into the search bar to get the win probability (69.34% for Houston).

In the next article, we’ll take a look at simulation results using these win probabilities.

The post March Madness 2023: Forecasting the NCAA Division I Men’s Basketball Tournament appeared first on The Spax.

In yesterday’s article, I described our annual logistic regression model for predicting the outcome of the NCAA Division I Men’s Basketball Tournament. I applied it to this year’s tournament to get the model’s “expected” bracket and the win probability for each team in every possible matchup. You can click the link here to see all of that good stuff.

Another good way to show the results is through a simulation – we can use the modeled win probabilities to simulate the entire tournament in less than a second. For example, I just ran a simulation of the tournament and in this hypothetical scenario, the Final Four teams are Gonzaga, Villanova, Kentucky, and Kansas. The Jayhawks went on to top the Gonzaga in the championship game. Not at all difficult to imagine that playing out.

We can run this simulation any number of times and compile the results to see how likely each time is to reach each round of the tournament assuming that the modeled win probabilities are accurate. We’ll go through each region to see the results.

West Region

Gonzaga won the West in 34.0% of our simulations, ahead of Duke at 26.1% and Texas Tech at 16.9%. There’s a drop-off afterwards before we get No. 4 Arkansas at 5.2% and No. 5 Connecticut at 6.2%. In terms of who can actually win the tournament, we see the same pattern of a “Big 3” with Gonzaga at 14.3%, Duke at 7.8%, and Texas Tech at 3.5%.

South Region

Despite the previous article having Houston in the Final Four in the situation in which the model is perfectly accurate, Arizona does have a substantially higher chance of reaching the Final Four in our simulated runs. The one-seed Wildcats reach the Final Four in 33.9% of simulations versus 20.7% for Houston followed by 15.7% for No. 3 Tennessee and 15.0% for No. 2 Villanova. These are also the only four teams who won the tournament in more than 1% of simulations.

Houston is a super interesting case. Most metrics love them, but there is plenty reason for concern – a weak schedule, two key injuries, and the fact that they’ll probably run into Arizona in the Sweet Sixteen. I would personally wager that the analytics are overrating the Cougars’ chances, so I’d be weary of picking them over Arizona to make the Final Four. Their performance in the tournament is probably the thing I’m most interested in keeping an eye on.

East Region

Now we have the East, the most tightly contested region. Baylor is favored to win it at 27.1% followed by Kentucky, Purdue, and UCLA at 24.8%, 18.7%, and 16.3% respectively. That’s an incredibly tight race between the top four seeds.

It’s worth noting that the Baylor Bears are dealing with injuries right now (not explicitly accounted for in the model) which should give even more reason for you to not pick them in your Final Four. I also do not think any teams outside of the top four seeds in the region are worth picking – I would choose between Kentucky, Purdue, and UCLA.

Midwest Region

Finally, the Kansas Jayhawks won the Midwest in 37.8% of simulations. Auburn represented the region in 21.0% of the iterations followed by the five-seed Iowa Hawkeyes at 15.3%. Impressively, the 11-seed Iowa State Cyclones managed to break into the Final Four in 4.5% of the simulations.

Based on this data, it also seems that the model views the Jayhawks’ path as being the easiest. It’s not easy seeing any other team coming out of the region and winning the entire tournament.

Commentary

The two most important factors in picking a successful March Madness bracket is to have an accurate Final Four and accurately pick the champion. And part of having a strong FInal Four is to be smart with the seeds that you select. On average, 1.7 one seeds make it to the Final Four, so you don’t want to go all “chalk.” Baylor is probably not a one-seed that should be selected, leaving you to pick between Gonzaga/Kansas/Arizona.

It is also fairly common for a seven-seed or lower to make it to the Final Four. A few interesting picks to make it to the Final Four are No. 8 San Diego State (3.0%), No. 9 Memphis (2.5%), and No. 11 Iowa State (4.5%). Or maybe you’d go with slightly higher seeded sleepers like No. 5 Iowa (15.3%), No. 4 UCLA (16.3%), and No. 5 Houston (20.7%). Based on ESPN bracket data, all six of these “sleepers” are picked to advance to the Final Four at a lower rate than they actually do in simulations – perhaps they’re strong values pick in larger pools.

As for who’s going to win… well, that’s a tough question. Gonzaga (14.3%), Kansas (14.0%), and Arizona (12.8%) won the tournament most often during these 10,000 simulations, followed by a drop-off before we get teams like Houston, Duke Baylor, and Kentucky. Gonzaga is actually given a significantly higher chance of winning by sportsbooks (25% implied odds) and the public (27.8% of ESPN brackets picking them to win). Whether it’s rightful or not, the model does not view Gonzaga’s chances as highly as others. In this case, Kansas is an interesting contrarian pick as just 8.5% of brackets have them winning it all.

Unfortunately, these margins are very small. We’re just playing the numbers here – anything can happen once these teams step on the court. There’s a reason no human has selected a perfect bracket yet, and there’s a reason it’s called March Madness. Don’t put too much stock on the statistics – there’s a lot that they can’t quantify. Most pre-tournament analyses like this one did not foresee the No. 11 UCLA Bruins embarking on a miraculous Final Four run last season. We’re guaranteed to see more unexpected results this March, so don’t expect perfection. If you want to win your bracket pool, all you can do is play the numbers and hope for the best.

The post March Madness 2022: Tournament Simulation Results appeared first on The Spax.

Since creating this website in November 2018, I’ve created yearly statistical models for the purpose of predicting the annual NCAA Division I Men’s Basketball Tournament.

The first time was in 2019 when we successfully forecasted the national champion Virginia Cavaliers and three of the four Final Four teams (including the five-seed Auburn Tigers). The predicted bracket finished in the 99th percentile of ESPN’s Tournament Challenge.

The tournament returned in 2021 after the COVID-19 pandemic forced the cancellation of the 2020 edition. We weren’t quite as successful this time around – while the model picked three of the four Final Four teams again (including the two-seed Houston Cougars), the Baylor Bears unexpectedly bested the Gonzaga Bulldogs in the National Championship Game. Not picking the correct champion (which is worth by far the most points) meant the model’s corresponding bracket finished in just the 79th percentile of all brackets submitted to ESPN. Although I did submit a variation of it that had Baylor correctly besting Gonzaga in the national championship and finished in the 99.6th percentile, most bracket pools don’t allow multiple entries so that isn’t particularly relevant.

In any case, I believe the first two iterations of The Spax’s March Madness model have been reasonably successful and I’m glad to bring it back again this year with some changes to its methodology, including a bonus for games previously won in the tournament. For example, if we were predicting the probability of Georgia State (16) beating Boise State (8) in the Round of 32, we’d automatically factor in the fact that Georgia State must have beaten Gonzaga (1) in the first round in order for that matchup to even take place. A few other variables have been added as well, including one for a team’s tournament experience.

With that said, let’s get started by taking a look at the First Four matchups.

First Four

Notre Dame v. Rutgers is viewed as essentially a coinflip while Texas Southern and Indiana winning their respective games seems to be a “safer” bet. Meanwhile, Bryant is predicted to have a ~58% probability of besting Wright State to earn a R64 matchup against Arizona.

Most of these odds roughly line up with sportsbooks’ odds with the exception of Bryant vs. Wright State – Bryant will be entering that game as 3.5 point underdogs.

West Region

Top Team: Gonzaga will be playing their third consecutive tournament as the top seed in their region. While they definitely aren’t as viewed as strongly as they were last year (NCG loss vs. Baylor), they are still considered the favorite to win the tournament and they have the best modeled probability of reaching the Final Four. An upset before then would not be the craziest thing to happen, though – instead of an 88% win probability in the Round of 32 like last season, the Bulldogs have “just” a 75% win probability of beating Memphis in the second round this year. While they should make it out of the region, their path is not the easiest it could be and it’ll definitely be something to watch.

Potential Disappointment: The Duke Blue Devils will be seeking a magical farewell run for the legendary Coach K, but it will not be an easy path for them at all. While their potential Round of 32 matchup against Michigan State isn’t considered as worrisome by the model as some fans & analysts view it, Texas Tech is given a slight edge over them in the Sweet Sixteen matchup. Interestingly, though, if Duke and Gonzaga do matchup in the Elite Eight, the Blue Devils have a respectable modeled win probability of 42.44%. I could realistically see Duke losing at any point after the first round so I’m not sure exactly what to expect.

Sleeper Pick: It’s not obviously looking at the bracket, but the nine-seed Memphis Tigers are an interesting team to watch this year. Given how good Gonzaga is and what Memphis’ seeding is, a 25% win probability is relatively interesting. In a potential Sweet Sixteen matchup against five-seed Connecticut, Memphis has a modeled 43.15% win probability and a 29.05% modeled win probability in an Elite Eight bout against Texas Tech. None of these are even close to guaranteed and obviously even their first round matchup is viewed as a tight one, but I do think Memphis is the most interesting mid-low seed in the region.

Most Probable First-Round Upsets: No. 10 Davidson over No. 7 Michigan State (36.43 percent chance)

South Region

Top Team: According to our model, the five-seed Houston Cougars would represent the South region in the Final Four if the modeled favorite won every game. This may come off as a surprise, but analytics do seem to like the Cougars – they rank 4th in KenPom’s rankings and 2nd in BartTorvik’s. It’s not at all difficult to make the case that Houston may be underseeded. On the other hand, two of their four most important players are likely out for the season (Marcus Sasser and Tramon Mark). The model does not explicitly account for injuries (it does place extra weight on more recent games where Mark and Sasser did not play, though) so this should definitely be something to keep in mind when filling out a bracket. You don’t want to just put one seeds in the Final Four so naturally Houston is a pretty good Final Four pick to go against the grain, but giving them a 51.01% win probability over Arizona is probably unrealistic. They will also have to go through Illinois and Tennessee to reach the Final Four, so there’s no reason for excessive optimism – it’ll be quite the tough path. Hell, UAB is even a popular 12 seed upset pick in the first round.

Potential Disappointment: Anytime a one-seed doesn’t make it out of the Sweet Sixteen, it can be considered a disappointment. And in a hypothetical third round matchup against Houston, the top seed Arizona Wildcats have a modeled win probability of 48.99%. While that’s basically a 50/50, it is an interesting possibility for an early exit for the team with the second-best odds of winning the tournament according to sportsbooks. It’s worth noting that Arizona would have modeled win probabilities of 67.96% and 76.19% in hypothetical Elite Eight matchups against No. 2 Villanvoa and No. 3 Tennessee respectively. If they manage to get past Houston, or escape a matchup against them, they have as good of a chance of making the Final Four as the other one seeds.

Sleeper Pick: This region has the potential to be a bit wild – a couple first round teams to watch are No. 13 Chattanooga and and No. 11 Michigan. The model gives Michigan a 60% probability of pulling off the first round upset while the Chattanooga has a relatively high 33% probability for the massive upset over Illinois.

Most Probable First-Round Upsets: No. 10 Loyola Chicago over No. 7 Ohio State (41.92 percent chance), No. 11 Michigan over No. 6 Colorado State (60.00 percent chance), No. 13 Chattanooga over No. 4 Illinois (32.89 percent chance)

East Region

Top Team: The one-seed Baylor Bears are the modeled favorites to come out of the East. The Bears won the 2021 NCAA Division I Men’s Basketball Tournament (also as the one-seed) with a blowout win over the undefeated Gonzaga Bulldogs in the National Championship Game. While going for back-to-back titles will be an incredibly difficult task, it would not at all be a surprise to see them in the Final Four again …

Potential Disappointment: … but it also wouldn’t be surprising to see an earlier exit for the Baylor Bears. They’re not expected to have an easy time in the East – the very real threat of potential matchups against No. 2 Kentucky, No. 3 Purdue, and No. 4 UCLA pose true danger to the Bears. Baylor would have modeled win probabilities of 62.65%, 60.40%, and 62.49% in these matchups respectively. While that makes them consistent favorites, they’re far from a sure thing. Any of those top four teams could realistically be seen in the Final Four. In particular, the two-seed Kentucky Wildcats are a very formidable threat to win the East and disappoint the Baylor Bears.

Sleeper Pick: In my opinion, the four-seed UCLA Bruins are the most interesting team in the East. I expect them to advance to a Sweet Sixteen matchup against No. 1 Baylor where they have a solid 37.51% modeled win probability. And in hypothetical Elite Eight matchups against No. 2 Kentucky and No. 3 Purdue, the Bruins have modeled win probabilities of 35.77% and 53.65% respectively. In 2020, the No. 11 Bruins unexpectedly made a miraculous run all the way to the Final Four (and went toe-to-toe with Gonzaga) and they look even better in 2021.

Most Probable First-Round Upsets: No. 10 San Francisco over No. 7 Murray State (64.36 percent chance), No. 11 Virginia Tech over No. 6 Texas (36.43 percent chance)

Midwest Region

Top Team: Bill Self has coached the Kansas Jayhawks to the one-seed for the ninth time and after dominating the Big 12 Tournament, they look like a team ready for the big dance. The one-seed Jayhawks should probably not have much of a problem advancing to the Sweet Sixteen (although they’d “only” have a modeled win probability of 71.26% in a R32 matchup against SDSU). Iowa and Auburn may not be walks in the park but Kansas could have a harder path.

Potential Disappointment: The No. 11 Iowa State Cyclones actually have an above 50% modeled win probability in the first round against No. 6 LSU – despite the Tigers being 3.5 point favorites in Vegas, the model really doesn’t like their chances against the Cyclones. And in a potential second round matchup against the No. 3 Wisconsin Badgers, the Cyclones have a modeled win probability of 43.02% – pretty remarkable for an 3 vs. 11 matchup. As such, the three-seed Badgers are a glaring candidate for an early exit according to the model.

Sleeper Pick: The model clearly considers the aforementioned Iowa State Cyclones as a possibility for a surprising run. Looking elsewhere for another sleeper, their in-state rivals serve as an interesting candidate. The Iowa Hawkeyes are expected to get past Richmond and South Dakota State in the first two rounds to meet the one-seed Jayhawks in the Sweet Sixteen. And a 30.58% modeled win probability against Kansas is not at all bad. 42.68% modeled win probability versus No. 2 Auburn in the Elite Eight and a whopping 68.21% win probability against No. 3 Wisconsin. And just for fun, Iowa would have the 60.81% edge in an in-state Elite Eight matchup.

Most Probable First-Round Upsets: No. 11 Iowa State over No. 6 LSU (55.78 percent chance)

Final Four

And we’ve reached the Final Four! We have three one-seeds (Gonzaga, Baylor, Kansas) and a five-seed in the Houston Cougars. Funny enough, it reminds me of last season where the model forecasted Houston beating Baylor in the Final Four to advance to the NCG where they were predicted to lose to Gonzaga.

This time around, Houston is barely favored over the Jayhawks while Gonzaga has a decent edge (as far as you can expect from a 1 vs. 1 matchup) over Baylor in a rematch of last season’s National Championship Game. And once again, the model foresees a Gonzaga vs. Houston matchup in the championship, with Gonzaga coming out on top again.

If we predict the same championship matchup every year, it has to happen eventually. Right?

Jokes aside, it would be odd if Gonzaga wasn’t favored. They are viewed by the favorites by most statistical models, Vegas’ odds, and public opinion. But we play the games for a reason – anything can happen. If Kentucky reaches the Final Four instead of Baylor, Gonzaga’s modeled win probability would drop to 57.65% – a much closer matchup.

Or maybe we get No. 2 Duke vs. No. 1 Baylor (50.86% win probability for Baylor). Or No. 2 Duke vs No. 2 Kentucky (56.76% win probability for Kentucky). Or No. 1 Arizona vs. No. 1 Kansas (58.28% win probability for Kansas).

Needless to say, there’s a lot of possible matchups in March Madness. We’ve only presented a subset of the possibilities so far. If you’d like to see the model’s projections for matchups not listed above, you can search for them in table below which consists of 2,278 rows, one for every possible matchup. Just type the name of the two teams separated by a space. For instance, if you want to search for the Gonzaga-Houston matchup, type “Gonzaga Houston” into the search bar to get the win probability.

In the next article, we’ll take a look at simulation results using these win probabilities.

The post March Madness 2022: Modeling the NCAA Division I Men’s Basketball Tournament appeared first on The Spax.

Introduction

Counter-Strike: Global Offensive (CS:GO) is a first-person shooter that was released in 2012 and has been played competitively as an e-sport ever since. In CS:GO, a single game is played between two teams of five players and consists of two halves, each containing a maximum of fifteen rounds. One team plays the first half as the terrorist (T) side while the other team plays as the counter terrorist (C) side. The teams switch sides at the end of the first half and the game stops if one team reaches 16 rounds before the other team wins 15 rounds (if the game ends 15-15, an overtime period begins which was not included in this analysis).

The professional CS:GO scene has grown dramatically over the past five years. The last major tournament finished on November 7, 2021 in Stockholm, Sweden and consisted of 24 teams competing for $2 million in prize money. The grand final was won by Ukrainian side Natus Vincere with a peak of 2.74 million international viewers. Despite its apparent popularity worldwide, statistical analysis in e-sports has lagged compared to traditional professional sports like football and basketball, where analytics are used to inform decisions and improve evaluation.

In this analysis, we will leverage public match data on CS:GO professional matches to train a new system of player evaluation based on regularized regression. We hope to expand on the traditional rating system based off of simple statistics such as a player’s eliminations, assists, and deaths. The objective of our new player evaluation framework is to isolate and quantify an individual’s contribution to winning.

Methods

Data Collection and Preparation

We scraped hltv.org, a website dedicated to CS:GO coverage and statistics, to obtain data for 19034 professional games since December 3rd, 2015. HLTV includes a star rating from zero to five for each game where no stars indicate that neither of the teams competing were ranked in the world top 20, while five stars indicate a map played between two top three teams. We limited our analysis to games with at least one star – including games without any top 20 teams would introduce data for over 50000 more matches, including many semi-professional teams in lower tier leagues. Most of these less successful teams & players would never actually compete against the best players and teams, thus limiting the model’s ability to estimate their value due to the lack of interactions.

The data set was split into two rows for each game, each representing one half of play. The score of each match was recorded and the percentage of rounds won by the T team was calculated to account for varying half lengths. For example, if the T side won three rounds in a half versus five for the C side, the response variable tWinPct would be .

The objective was to have a column representing each player who played in any of the 19034 games over the past six years. If a player played for the T side in the game represented by any row, the value for their respective column would be where represents the player rating for a T player in that half. If a player played for the C side, the value for that column would be for a C player . If a player did not play in that game, the value would be . A player’s rating was obtained from hltv.org and is included in the analysis to serve as a way to approximate how much credit one player should get for their team’s performance. The rating is calculated using statistics such as kills, assists, deaths, survival rating, damage rating, etc (Milovanovic, 2017).

The reciprocal of is taken for each player because higher ratings are better and we want higher model coefficients to correspond with better players. Greater input values would correlate with greater model coefficients, so the reciprocal is calculated. Recall that the response variable tWinPct measures the performance of the T side. Thus, the column value is for players because the negative sign indicates that a greater performance from would be expected to have a negative impact on the performance of the T side.

Ridge Regression

The data set consists of 2263 players. Each row will consist of 10 nonzero values (five positive, five negative) for these 2263 players because there are five players on each side. In addition, dummy variables based on the setting (or “map”) of the game are added to account for any bias. Certain maps are more favorable to the T side than the CT side, so this adjustment introduces eleven dummy variables for the eleven different maps played in the given time frame.

Thus, the input matrix consists of 2274 variables and 38068 observations while the output vector (or tWinPct) has 38068 values. We also have a weight vector representing the number of rounds played in each observation. Then we are looking to find the estimates for the vector where .

For example, suppose a game is played between a team with players and a team with players . Team begins the game on the T side and ends the first half up nine rounds to six for team . Team then plays as the T team in the second half, winning six rounds again while team wins the necessary seven rounds needed to win the game with an overall score of 16-12. The game was played on map in . Then would be equivalent to the matrix below where represents the player rating of the corresponding player.

All other player and map variables in would have a corresponding value of for these two rows. The weight vector would contain values and , representing the total number of rounds played in each half. Finally, the vector representing the response variable would have values and , representing the percentage of rounds won by the T side.

The traditional ordinary least squares (OLS) solution would be to compute the estimates for as . To handle collinearity in the data (as teammates will be playing with each other at the same time), we introduce a penalty term that reduces variance by shrinking the coefficients towards zero. Thus, the ridge regression solution is denoted as . The ridge parameter is a constant that represents the degree of regularization; when , the ridge solution is equivalent to the OLS solution. If , then all of the coefficient estimates would be zero.

We ran k-fold cross validation to find an optimal using the “one-standard-error” suggested by the authors of the glmnet package used for modeling (Friedman et al., 2010). The cross validation plot of mean squared error and is shown in Figure 1.

Using this value of , we were then able to use the ridge regression solution to compute the model estimates corresponding to each variable in . The effect of on the model coefficients can be seen in Figure 2, where a vertical line denotes our chosen ridge parameter. The reduction in variance and the shrinkage towards zero of the model coefficients as increases can be seen in this graph.

Results

The ridge regression model’s coefficients for map are shown in Table 1 along with the average T side win percentage on that map.

The coefficients for the remaining variables represent the estimated impact of each player in the model. The players with the ten highest coefficient estimates are shown in Table 2. We included their HLTV ratings and the corresponding percentile based on the rating as a way to compare the model results.

We explored the relationship between the coefficient estimates and the HLTV player ratings that were used as prior information in the model. Figure 3 shows a plot of a player’s average HLTV rating versus their coefficient estimate among players with at least 300 games played.

The correlation coefficient between rating and coefficient estimate was found to be 0.684 for these points, and the weighted correlation coefficient with games played as the weight for all points was found to be 0.658.

We also examined the relationship between the aforementioned HLTV rating and coefficient estimate with a player’s overall win rate in the time frame of interest. A weighted correlation matrix (weighted on games played) between these three variables is shown in Table 3.

Discussion

The map coefficient estimates in Table 1 are clearly related with the proportion of rounds won by the T side in the data set. If the T side has a win proportion of greater than 0.5 on any given map, the coefficient estimate is positive. Otherwise if the T side has a win proportion below 0.5, the corresponding estimate for that map is negative. The absolute value of the coefficient estimate depends on the difference between the proportion and 0.5, along with the proportion’s confidence interval. The 95% confidence interval was computed for the proportion of rounds won by the T side on each map (tWinPct) and the size of the interval varies based on how high the sample size is for each map. The map Tuscan has a massive 95% confidence interval for tWinPct from 0.335 to 0.541 because the map was only played in eight games (a total of 89 rounds), so it also has the coefficient estimate closest to zero.

The remaining model coefficient estimates represent the estimated impact of each player. As shown in Table 2, we’ve found that seven of the players with a top ten coefficient estimate also had an HLTV rating in at least the 95th percentile. The players with the five highest HLTV ratings (minimum 100 maps played) are all in the top six for coefficient estimates: s1mple, ZywOo, sh1ro, device, and NiKo. Oleksandr “s1mple” Kostyliev, Nicolai “device” Reedtz, and Nikola “NiKo” Kovač are all regarded as some of the greatest CS:GO players of all-time, while Mathieu “ZywOo” Herbaut and Dmitriy “sh1ro” Sokolov are young stars who are considered top five players in the world today. These claims seem to be supported by the players’ statistical dominance.

The weighted correlation matrix of player rating, win rate, and coefficient estimate (Table 3) suggests that the model estimates blend together both contribution to winning (0.609) and individual performance (0.658) in a way that neither metric can do on their own. By being able to combine both intertwined aspects of the game, teams can identify players that put up “empty stats” (racking up a high number of eliminations, but not contributing as much to their team’s winning chances) or players that quietly impact the game more than their basic statistics suggest.

Regularization has previously been used in a similar way for player evaluation in the National Basketball Association (Sill, 2010) and the same methodology appears to be applicable to an e-sports context. Future research should begin to explore the predictive capabilities of a regularization framework and continue to expand the use of advanced statistical methods in e-sports. Our regularization methodology can also be altered to better handle players with fluctuating skill levels throughout the time frame covered in the data set. For example, a player who peaked at a high skill level and then regressed later on would be treated as a single variable despite having varying impact throughout the data set. Also, the map dummy variables do not account for updates to each map that have taken place over the past years – these updates can affect the impact each map has on the terrorist team’s win rate. Despite the room for improvement, we believe that our research is a strong starting point for more advanced methods of player evaluation in e-sports.

References

1. Friedman, J. H., T. Hastie, and R. Tibshirani. “Regularization Paths for Generalized Linear Models via Coordinate Descent”. Journal of Statistical Software, vol. 33, no. 1, Feb. 2010, pp. 1-22, doi:10.18637/jss.v033.i01.
2. Milovanovic, P. (2017, June 14). Introducing rating 2.0. HLTV.org. Retrieved December 6, 2021, from https://www.hltv.org/news/20695/introducing-rating-20.
3. Sill, Joseph. “Improved NBA adjusted +/- using regularization and out-of-sample testing.” Proceedings of the 2010 MIT sloan sports analytics conference. 2010.

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Last month, I released my modeled predictions for the NCAA Division I men’s basketball tournament. It was the second version of the machine learning model after it performed exceptionally well in 2019. Unfortunately, the bracket had a lot less green this time around.

Despite Gonzaga obviously being favored to win the tournament by every predictive model, they collapsed with an embarrassing performance in the national title game at the hands of the Baylor Bears. While the model correctly picked three of the Final Four teams, it did not pick Baylor to advance to the national title game, let alone win it all. As one-third of the maximum number of points for an ESPN bracket come from the final three games, Baylor’s unexpected success obviously had quite a toll on the model’s final performance. However, we can still evaluate its performance as we did two years ago.

Hits & Misses

Hits

Our model started off with a 3-1 record in the First Four after picking all four of the underdogs. Most of the win probabilities were quite close and all of the First Four games were pretty close too, so it doesn’t really mean much.

In the initial article, I said that “USC and Oregon both have relatively solid odds to make it to the Elite Eight.” The six-seed Trojans had a modeled probability of 45% to beat three-seed Kansas (they did) and the seven-seed Oregon Ducks had a modeled probability of 45% to beat the two-seed Iowa Hawkeyes (they did). The model also favored USC to beat Oregon in their matchup and for Gonzaga to annihilate USC (although that wasn’t a hot take). Not a bad job in the West region.

Houston advancing to the national title game was the surprising pick from the model. While they didn’t, they at least made it to the Final Four instead of the one-seed Illinois Fighting Illini. But they also had a historically easy route to the Final Four, so who knows how good they really were?

Misses

The East region was a bit of a mess. UCLA obviously shook everything up — the model didn’t pick them to advance past the first round and they ended up winning the entire region. Alabama lost early, Texas lost early, Connecticut was a sleeper pick and they lost in the first round, etc.

Alright, I don’t think anyone saw Oral Roberts winning two games. The model’s performance in the South region also wasn’t superb, but it’s at least a bit more understandable. I don’t think anyone saw Oral Roberts winning two games. The model also did not like Arkansas’ chances in the region, and while Arkansas did advance to the Elite Eight, they didn’t make it convincing.

Baylor annihilating Houston in the Final Four cost our bracket 160 points, so it was obviously an impactful miss. It would’ve been more understandable if the game was close (after all, Houston was given a 63% win probability, nowhere near a guarantee), but the Cougars didn’t look like they belonged on the same court as the eventual champion Baylor Bears.

Comparison to Other Models

Compared to the three most prominent March Madness models, The Spax comes in at third ahead of FiveThirtyEight and behind Model284 and SportsLine.

The Spax’s model was the highest performing by a slim margin through the Elite Eight, as it had a good start in the Round of 64 while also correctly picking three of the Final Four teams (unlike FiveThirtyEight, who incorrectly had Illinois in the Final Four instead of Houston).

Model284 and SportsLine pulled ahead in the Final Four where their model forecasted Baylor to move on to the national title game. That one game, being worth 160 points, ended up making the difference. All four models incorrectly predicted that Gonzaga would win the championship.

Conclusion

Every year, I submit a bracket to the ESPN Tournament Challenge which simply follows the model’s pick. Last year’s bracket finished with 1360 points (ranked 138,776th in the world, 99.2 percentile). I created 24 variations of that bracket, none of which performed any better. This time around, the model’s bracket finished with 920 points (ranked 3,115,289th in the world, 78.8 percentile). Not so good. I did create 15 variations, though. Unfortunately, I was overconfident in Houston beating Baylor, so I only made one bracket in which Baylor won the championship. This bracket was fantastic, accumulating 1400 points and ranking 64,554th in the world (99.6 percentile). Of course, no bracket pool lets you submit 25 entries, so that doesn’t mean much.

How can our model be improved for next year? There are plenty of ideas I’d like to implement into the model for future tournaments, such as accounting for injuries, a superstar ¹ factor, tournament experience, etc.

While the model was unable to live up to the crazy standards set by its performance in 2019, it wasn’t too far off from how other models performed. Maybe it’ll have more success next year.

The post March Madness 2021: Assessing Our Statistical Model’s Performance appeared first on The Spax.

Two years ago, I created a model to predict the 2019 March Madness tournament. It performed pretty well, most notably picking Auburn to advance to the Final Four and correctly predicting that Virginia would win the National Championship. On the whole, the model’s forecasted bracket finished in the 99th percentile of ESPN’s Tournament Challenge.

Back then, I used a massive Excel spreadsheet to train a linear regression model that violated many fundamental rules of statistics. While the result may have been solid, the actual creation of the model was awful and it almost certainly would’ve performed poorly in the long run. Come on.

Of course, I was a beginner at the time, so it’s hard to blame myself too much. Since then, I’ve learned a lot about data science, and this time around I started from scratch and created the model in Python. I conducted exhaustive feature selection and hyperparameter tuning in order to optimize its predictive ability, which was not the case in 2019.

While the model is different, the idea at its core remains the same. I want to holistically assess a team’s chance of winning each game based on a variety of different variables, including their recent performance leading up to the tournament, their performances against opponents representative of a tournament field, playstyle, etc.

Anyway, let’s get into the predictions.

First Four

Wichita State beating Drake is the closest thing to a safe bet here, but even that’s not exactly a sure thing. Oddly enough, the projected losers in all four of these games are actually the odds-on favorites to win. Not by much, of course.

West Region

Top Team: Gonzaga is entering the tournament as the clear favorites — Vegas gives them implied odds of around 32% to win the tournament, with Illinois at a distant second at just 16%. There’s a very real chance that they will finish their season undefeated. And the model doesn’t really think they’re facing any stout competition in their region. Anything short of a Final Four berth would be a surprise.

Potential Disappointment: Neither the No. 2 Iowa Hawkeyes or the No. 3 Kansas Jayhawks are given a super optimistic shot at making it to the Sweet Sixteen. Both of them have their hands full with No. 6 USC and No. 7 Oregon respectively in the second round.

Sleeper Pick: Along the same lines, USC and Oregon both have relatively solid odds to make it to the Elite Eight. If USC were to upset Kansas and move on to play Iowa in the Sweet Sixteen, they’re projected to have a 48% win probability in that game. And if Oregon moves on to play Kansas in the Sweet Sixteen, the model actually slightly favors them to advance with a 53% win probability. Both teams would still be massive underdogs against Gonzaga in the Elite Eight, but still. Knocking off the No. 2 and No. 3 teams in the region would be a superb rub.

Most Probable First-Round Upsets: No. 11 Wichita State over No. 6 USC (40.58 percent chance)

East Region

Top Team: Michigan isn’t considered a juggernaut to the degree of Gonzaga, but they’re certainly still the favorites to come out of the East. None of Michigan’s projected opponents have a win probability of 33% or higher against them.

Potential Disappointment: Alabama’s 57% chance of defeating No. 7 Connecticut in the second round is not as high as you’d expect for a 2v7 matchup. It would be a massive disappointment if Alabama was knocked off that early, although it would be great for the Huskies, who would have a projected 55.28% chance of beating Texas in the Sweet Sixteen.

Sleeper Pick:  The team in this region that has the best shot at knocking off Michigan if they played is actually the No. 4 Florida State Seminoles. In this hypothetical matchup (which the model does not think will occur), the Seminoles would have a projected 43 percent chance of coming out on top, in which case they would advance to the Elite Eight to most likely play Texas or Alabama. The model projects that Florida State would have a 46% chance of winning both of those matchups — not an awful shot at a Final Four berth. Oddly enough, the No. 7 Huskies are actually projected to have a better chance (56%) at beating Florida State in an Elite Eight matchup. That would be an even more exciting run.

Most Probable First-Round Upsets: No. 12 Georgetown over No. 5 Colorado (35.09 percent chance)

South Region

Top Team: Baylor’s projected path is quite similar to Michigan’s in the East. Neither team is expected to dominate like Gonzaga, but they’re also not expected to have a particularly tough time. Unlike Michigan though, there’s no hypothetical interregionional matchup that poses a serious threat for an upset. Baylor maintains their high win probability against teams like Purdue, Arkansas, Texas Tech, etc.

Potential Disappointment: The model really doesn’t seem to love Arkansas. For whatever reason, the 15 seed Colgate Raiders are forecasted to have a 37.5 percent chance of defeating No. 3 Arkansas in the first round. Wow. And while Arkansas is still projected to advance to the second round, their run is expected to stop then with a loss to Texas Tech. A second round exit (and potential first round loss!) for a three seed would be a huge disappointment.

Sleeper Pick: The No. 6 Texas Tech Red Raiders are the obvious pick here. The model gives them the edge over No. 3 Arkansas and projects that they have a solid 43% chance at knocking off No. 2 Ohio State. Their chances at advancing to the Elite Eight aren’t bad at all. The bigger sleeper, though, is in the first round. Texas Tech actually only has a 53% chance of advancing to the second round, as the No. 11 Utah State Aggies are projected to have a favorable 47% win probability. That’s almost a coin flip. If they were to win, the Aggies would have a solid 45% chance of topping Arkansas in the second round to advance to the Sweet Sixteen. Not bad.

Most Probable First-Round Upsets:  No. 11 Utah State over No. 3 Texas Tech (47.27 percent chance), No. 14 Colgate over No. 3 Arkansas (37.50 percent chance)¹

Midwest Region

Top Team: The No. 2 Houston Cougars are the only two seed favored by the model to win their region and it’s not even that close. While Vegas gives the No. 1 Illinois Fighting Illini the second best odds to win the tournament, our model doesn’t even think their chances of beating Houston in the Elite Eight is above 35%. Houston would also be expected to crush teams like No. 7 Clemson, No. 5 Tennessee, No. 4 Oklahoma State, and No. 3 West Virginia if those matchups occurred.

Potential Disappointment: The No. 3 West Virginia Mountaineers are projected to lose in the second round to the No. 6 San Diego State Aztecs, and it’s not exactly a close call. The Mountaineers are only given a 32.6 percent shot at advancing to the Sweet Sixteen, and if they do, don’t expect them to go any further. Houston would have a 76% edge in such a matchup if it were to occur.

Sleeper Pick: The San Diego State Aztecs are clearly projected to make an impressive Sweet Sixteen run, although the model’s confident that they won’t go any further than that. Liberty and Syracuse are given decent chances to pull off first round exits, but they would both have bad to advance to the Sweet Sixteen after that. Tennessee would have a slightly better chance (64%) at beating Illinois than Oklahoma State, but not by much. All in all, don’t expect any massive surprises. The model likes the chances of a one seed versus two seed matchup in the Elite Eight.

Most Probable First-Round Upsets: No. 13 Liberty over No. 4 Oklahoma State (36.52 percent chance), No. 11 Syracuse over No. 6 San Diego State (39.35 percent chance), No. 10 Rutgers over No. 7 Clemson (51.94 percent chance)

Final Four

Michigan isn’t expected to have a great shot at defeating Gonzaga in the Final Four, and while the Baylor-Houston matchup is a tighter call, Houston still has a decisive 63% win probability. In the projected National Championship, Gonzaga has their toughest matchup of the tournament against the two seed Cougars, but they’re predicted to come out on top nonetheless. And with that, the Gonzaga Bulldogs are the projected 2021 champions of the college basketball world.

Afterword

It’s funny that this model is much improved since the 2019 version, yet it will almost certainly perform worse. Predicting the tournament is a very delicate thing — New Mexico State was two missed free throws away from making the Auburn Final Four prediction look very stupid. There are a few features of the model that were probably quite obvious to you already. For instance, it’s shy. The one seeds certainly have a greater than 90% chance of winning in the first round, but the model refuses to be so optimistic. There are also plenty of variables that are unaccounted for, such as injuries, the whole pandemic thing impacting the tournament, and other tournament results.²

In any case, I’m hopeful that I’ll be able to continue improving the model throughout the years just like I did this time around. Maybe it won’t finish in the 99th percentile of ESPN brackets, though.

If you’d like to see the model’s projections for matchups not listed above, you can search for them in table below which consists of 2,278 rows, one for every possible matchup. Just type the name of the two teams separated by a space. For instance, if you want to search for the Gonzaga-Baylor matchup, type “Gonzaga Baylor” into the search bar.

The post Forecasting the 2021 NCAA Division I Men’s Basketball Tournament appeared first on The Spax.

The most popular game show in America has had many great players compete in its long history. Just like most professional competitions, there has never been a consensus greatest player, though. Fans have spent much time debating the age-old question, but most of these discussions were based on conjecture. Until now, that is. Last week, the three greatest minds in the history of Jeopardy! competed in a seven-game series to try to settle this debate.

First, some background on the three contestants. In 2004, Ken Jennings won 74 consecutive Jeopardy! regular season games. The second-longest win streak ever is 32 games, which pales in comparison. This record may never be broken. Jennings won $2,520,700 in this streak, which is also a regular season record.

Let’s go back to the length of that streak, though: 74 consecutive wins. It’s hard to conceptualize just how incredible this accomplishment is. According to FiveThirtyEight, Jennings’ stats during his run suggested that he had a 97.9% chance of winning any given game.¹ That’s incredibly high — yet the expected streak length for that win probability is “just” 47 games.²

Luck was undoubtedly a significant factor, which is why the record is unlikely to ever be broken. The stars have to align perfectly. However, this doesn’t diminish Ken’s achievements. You have to perform at an astounding level to even have a shot at 74 straight wins, and Ken’s expected win streak of 47 games would still be an all-time record. But winning 74 games just adds to his lore.

However, when it came to tournaments, Jennings underperformed relative to the high expectations established by his regular-season play.

Brad Rutter competed on the show in 2000. At the time, a player could not win more than five games. After a player’s fifth victory, they were automatically retired, a limitation that was obviously lifted before Ken Jennings appeared on the show. Brad won these five games and earned $55,102 (and two Chevrolet Camaros). Of course, these regular season achievements aren’t quite as jaw-dropping as Jennings’. He wasn’t the only five-day champion at the time, after all. Rutter’s claim to fame was his performance in Jeopardy!‘s tournaments.

A month after the end of Ken’s streak, the Jeopardy! Ultimate Tournament of Champions was announced. It was a 15-week, 75-show competition between 145 players with a prize of $2,000,000 for the winner. It would provide another chance for five-time champions (from before the rule change) to shine. Rutter took that chance. He decisively defeated Jennings in the final round of the tournament and took home $2,000,000. Rutter also won the Million Dollar Masters, the Battle of the Decades, the All-Star Games, and a Tournament of Champions.. Going into the GOAT tournament, Rutter had never lost to a human on the show. His only losses came against the IBM Watson robot.

In total, Rutter has accumulated $4,688,436 in all-time winnings from regular-season play and tournaments, the most ever. Jennings comes in at second with $3,370,700.

And then there’s the new kid: James Holzhauer.

James won 32 straight games in 2019 in which he won a total of $2,462,216 in regular-season winnings. That’s $74,612 per game, far greater than Ken’s average of $33,609 during his record-breaking streak.

In 2010, Roger Craig set the record for the highest single-game total on the show with $77,000. James topped this 16 times in his streak, peaking at $131,127. James also won the recent Tournament of Champions to put his all-time winnings at $2,712,216.

All things considered, these are by far the three highest earning players in the history of the show.

A matchup between the three juggernauts quickly became the most anticipated event in game show history.

The format was simple: every day of the tournament, there would be two matches. The points scored in the two matches will be combined for each player and the player with the most points in a day is given the win for that day. The first player to reach three wins takes home $1,000,000 and, most importantly, the title of the greatest Jeopardy! player of all-time.

Jennings won Day One and Day Three while James Holzhauer won Day Two. Ken took care of business on Tuesday, winning Day Four and ending the tournament with a 3-1-0 lead over Holzhauer and Rutter respectively. Fifteen years after his legendary 74 game win streak, Ken solidified his status as Jeopardy!’s greatest player of all-time.

Let’s take a look at each contestant’s Coryat score³ in each day of the tournament. By using Coryat score, we can eliminate some of the luck involved in finding Daily Doubles.

One of the first things that pops out to me is Jennings’ remarkable consistency. The range of his Coryat scores is just 1,800 points. Ken’s combined Coryat score over all four days was 146,000, higher than James’ 141,800 and significantly higher than Brad’s 61,400. Furthermore, his average Coryat score of 36,500 is higher than every single day Coryat scores of his opponents save for James’ Day Four performance.

James was really good on the fourth and final day of the tournament. James answered 23 questions correctly and zero incorrectly in this first match, while Ken went 22/24 and Brad went 4/4. Unfortunately, James couldn’t find any Daily Doubles so he finished the first match down by 31,419 points to Ken.

In the second match of the fourth day, James kept up his performance by answering 30 questions correctly and just one incorrectly. More importantly, James won a wager of 20,200 points on a Daily Double which otherwise would’ve effectively clinched the tournament for Ken. Instead, James went into Final Jeopardy! with a chance to win Day Four with a correct response. He had only missed one FJ question during his 32-game regular season win streak, so the odds were looking pretty good. As we all know, though, James answered “Who is Horatio?” instead of “Who is Iago?” on the Shakespeare-related question, and Ken was crowned the Jeopardy! GOAT.

One of the storylines leading up to the game was how fast Jennings and Rutter would be with the buzzer compared to the younger Holzhauer. Here are the day-by-day percentages of clues in which a player was the first to buzz in on.

James appeared to be the fastest with the buzzer, although Ken wasn’t too far behind. The way Brad holds the buzzer makes it difficult to tell when he’s actually trying to buzz in. Was he first to buzz so rarely because he didn’t know as much as the other contestants, or was he just slow? Or both? Historically, Ken was the first to buzz in on almost 59% of the clues in games he played in, versus 56% for James and just 41% for Brad. Maybe Brad was never quite as good with the buzzer as his two opponents.

Side note: Brad’s extremely lower percentage on the fourth day can be partially explained by strategy. He was out of contention in the second match, so it was in his best interest to put the buzzer down and hope James would come out with the win and prolong the tournament (obviously giving Brad another chance to stage a comeback).

Finally, let’s wrap things up by taking a look at the total correct and incorrect answers for each contestant.

As the stats on buzzing implied, James answered more clues than Ken. Ken was slightly more accurate (but the difference is marginal), but his higher Coryat score suggests that he was answering questions of greater value (and presumably greater difficulty).

And then there’s Brad. The highest earning player in Jeopardy! history struggled immensely throughout the tournament. While 93.2% of Ken’s answers and 93.1% of James’ answers were correct, Brad hit on just 85.3% of his buzz-ins. Despite buzzing in on 107 less clues than James, Brad provided the same number of incorrect answers.

Despite Brad’s disappointing performance, the GOAT Tournament was excellent. If this is the last hurrah for legendary host Alex Trebek and the final appearance for Ken Jennings, it’s the perfect farewell. It’s obviously not an ideal way to see Brad’s career potentially come to an end, so we can only hope that he’ll get a chance at redemption. All three contestants have left a lasting legacy on the show and they will hopefully not be the last dominant players the game has to offer.

The post ‘Jeopardy: The Greatest of All-Time’ Recap appeared first on The Spax.

Before the first week of the 2019 college football season, Clemson quarterback Trevor Lawrence and Alabama quarterback Tua Tagovailoa were the early favorites to win the most prestigious award in college football: the Heisman Trophy. It’s no coincidence that these two quarterbacks faced off in the 2018 College Football Playoff National Championship. Clemson and Alabama traded national championships from the 2015 season through the 2018 season. It was a safe bet for one of their quarterbacks, arguably the two best college quarterbacks in the country, to win the award for the most outstanding player in the NCAA.

As we all know, LSU quarterback Joe Burrow eventually won the 2019 Heisman Trophy via the most lopsided vote in the long history of the award.

During the course of this legendary season, Burrow has completed 77.6% of his passes for 5208 passing yards, 55 passing touchdowns, and just six interceptions. He even ran for an additional 311 yards and four touchdowns.

Burrow’s passing efficiency rating of 204.6 is the highest in a single-season in college football history, far higher than Tua Tagovailoa’s record-setting mark of 199.4 last season. Of course, this stat on its own does not mean much because league-wide PER has obviously increased dramatically over time:

So, I used Pro Football Reference’s methodology of creating era-adjusted passing metrics in order to see how Burrow’s 2019 PER stacks up once you adjust for the inflation of the passing game. Turns out he has the highest single-season passer efficiency rating since 1980 with or without an adjustment for era.

Don’t forget that LSU’s schedule has been far from a cakewalk. It’s been a gauntlet for the undefeated Fighting Tigers. They beat #2 Alabama, #7 Florida, #9 Auburn, and #9 Texas in the regular season. That’s four teams ranked in the top-ten in the country at the time of their game against LSU. They went on to rout #4 Georgia by 27 points in the SEC Championship Game before advancing to the College Football Playoff for the first time in school history. They stomped #4 Oklahoma in the semifinal by a whopping 35 points and will now face off against #3 Clemson (who beat #2 Ohio State) in the national championship. The same Clemson Tigers that have won 29 straight games. The fact that Burrow is putting up these stats against this schedule is simply incredible.

If they finish the job against Clemson tomorrow, the team will go down as one of the greatest of the modern era. Furthermore, Burrow’s season will have an even stronger argument for the greatest single-season performance in college football history.

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