As the COVID-19 related pause continues for the Michigan State University basketball team, I have continued to think about and look forward to March Madness. Last week, I shared the results of an analysis that demonstrated that the total number of teams does not make a big impact on the odds for the top teams to win the tournament.
However, it also occurred to me that there other ways in which the tournament could be modified which might impact the results. While we continue to wait for MSU basketball to resume, I decided to explore some of these other formats in more detail. As a refresher, I am using Kenpom efficiency margin data from Jan. 1, 2021 to both select and seed teams. I use the same data to generate odds for individual games which act as an input to a Monte Carlo simulation of the full tournament.
In my previous post, I looked at the impact of changing the size of the tournament from 16 teams up to 357 teams (all of Division I). In order to complete that story, I also simulated the results for even small tournaments of only eight, four, and two teams. The results of these simulations are shown below:
Figure 1: Impact of the size of the tournament on the odds that the highest ranked team wins
As expected, if the tournament is small enough, the odds that the top ranked team cuts down the nets does increase. If the “tournament” is simply a single game played by the two top rated teams at the end of the season (in this example, Gonzaga and Baylor) the result of that game is likely a near toss-up on a neutral floor and thus the highest ranked team’s odds are naturally slightly over 50 percent.
As the number of teams in the field expand, those odds fall to 36 percent (for a four-team field), 28 percent (for an eight-team field), and finally to around 23 percent for any tournament with 16 teams or more. As explained last time, the reason that these odds seem to level off is that while the top rated teams have to win more games, so do all of the other top teams. A bigger tournament tends to generate more upsets which makes the average path for any given team slightly easier.
Another interesting result that follows from this analysis is that it implies that the current 68-team format for the NCAA Tournament is close to ideal. In order to include all of the conference champions, the field must be larger than 32 teams. If the field were to expand to greater than 68 teams, it is clear that the odds for each individual team do not really change.
Furthermore, my math suggests that the odds for any at-large team ranked outside of the top-45 or so in Kenpom to win the tournament is only about one percent. It is very unlikely that any team that does not make the field in the current format would go all the way. So, there seems to be no compelling reason to expand the tournament any more than it is today, unless there are extenuating circumstances...such as a global pandemic. More on that later.
Other Tournament Formats
The impact of the size of a standard single elimination tournament is now clear. But, in the leagues such as the NBA, the “tournament” consists of several rounds where the teams face off in best-of-seven game series.
It is natural to assume that in a multi-game format, the favored team is more likely to win, as the result would mostly likely “regress to the mean” over a large sample size (number of games). In other words, the underdog is less likely to win a majority of the games due to random chance (such a cold or hot shooting night) as the total number of games increase.
I was able to use simple probabilities (the binomial expansion) to estimate the odds that the favored team wins a multiple game series based on the odds of the favored teaming winning an individual game. To keep things simple, I assumed that all games would be played on a neutral court. The results are shown below in Figure 2.
Figure 2: Change in the overall odds when moving from a single game to a multi-game format
The shape of this graph is essentially what one might expect to see. In all cases, the more games are played, the more likely that better team is projected to win the series. For near toss-ups, the effect is small, but it appears to increase as the gap between the teams grow.
Another way to visualize this data is to converter the odds to win both a single game and the series into an effective equivalent point spread. That data is shown below in Figure 3.
Figure 3: Effective single-game point spread equivalent for various multi-game series relative to the spread for a single game
As the figure shows, the relationship is essentially linear and there is a simple correlation between the point spreads for the single game and the equivalent point spread for a multi-game series. For example if the projected point spread for two teams on a neutral court is 10 points in a single game, the odds that the better team would win in a three-game series is the equivalent of winning a single game with a point spread than is 1.48 times bigger, or roughly 15 points.
In a five-game series, the difference is a factor of 1.86. So, it is like moving from a 10-point spread to an 18.5 point spread. In a seven-game series, the effect is just a bit larger than the odds of doubling the point spread.
By itself, this data is interesting, but perhaps not that useful. However, it does allow us to answer another interesting question: what would happen to the odds in the NCAA tournament if somehow there was enough time to make each round a multi-game series of three, five, or seven games? Of course, this is completely impractical, but what if it were possible?
In addition, we could envision a double elimination format for the NCAA tournament. For the sake of comparison, I simulated a 16-team tournament using both a double elimination structure as well as a best-of-three game structure and compared it to the standard single elimination format base case. That result is shown below in Figure 4.
Figure 4: Comparison of the odds of each team winning a 16-team tournaments with different formats
In this case, we can finally see a significant difference in the odds, and the primary beneficiaries are the top two teams. Moving from a single-elimination format to double elimination boosts the odds of the top two teams by about four percentage points.
If each round were instead a best-of-three series, the odds would be even better for the top teams. In this case, the No. 1 overall seem gets a boost of almost 10 percentage points while the No. 2 seed’s odds improve by around seven percentage points.
These boosts come at the expense of the weaker teams. For this data set, the No. 3 overall seed has essentially the same odds in all three scenarios, while the odds for the rest of the field go down in both the best-of-three and the double elimination scenario.
Though not shown in Figure 4, an additional simulation suggests that moving to a seven-game series in each round would push the odds for the No. 1 team up by an additional 10 percentage points to around 45 percent. In other words, an NBA-style best-of-seven tournament would very likely identify “the best” college team in the tournament. But, it would also likely temper much or all of the madness that we have come to expect and love.
The Full Monty
In the previous piece on this topic, I mentioned that the idea had been floated to expand the 2021 NCAA Tournament to include all eligible Division I programs. At first, I thought that this idea was a bit silly, but as COVID-19 related cancellations continue to mount, it seems less and less likely that the full 140-game Big Ten schedule is going to be completed. Other teams and conferences will have the same problem.
With unbalanced and abbreviated schedules, it is going to be much more difficult than usual to select the best 40 to 45 at-large teams this year. So, I am starting to see the logic in just letting everyone play. Perhaps for selfish reasons, this would also ensure that MSU’s current tournament streak remains intact.
But, a full roughly 350-team tournament would last a full nine rounds and the logistics are likely too difficult. But, with the release of the schedule of the NCAA tournament dates and venues this week, it occurred to me that there is a way to possibly run a full national tournament in 2021 with limited impact to the current schedule. This is how I would set it up:
Seeding of the tournament might get messy, but the NCAA could even decide to use a model similar to the NHL where teams are reseeded after the play-in round. In other word, the No. 1 overall seed would automatically be paired with the weakest champion of the regional pod round in the round of 128. If this round is also all played in the state of Indiana, it would be pretty simple to accomplish this.
As I showed in my previous analysis, there would likely be a higher number of upsets overall in the NCAA Tournament due to its larger size, but the top teams would have no worse odds to cut down the nets in early April. That seems like a win for fans, players, and coaches. I would certainly not want to move to this type of model on a regular basis, but for one year, after everything that we have gone through as a society, I think that it would be fun.
That does it for this time. We still have another week to wait for MSU to take the court once again. Hopefully this will be the final “pause” of the season for the Green and White. My fingers remain firmly crossed.
However, it also occurred to me that there other ways in which the tournament could be modified which might impact the results. While we continue to wait for MSU basketball to resume, I decided to explore some of these other formats in more detail. As a refresher, I am using Kenpom efficiency margin data from Jan. 1, 2021 to both select and seed teams. I use the same data to generate odds for individual games which act as an input to a Monte Carlo simulation of the full tournament.
In my previous post, I looked at the impact of changing the size of the tournament from 16 teams up to 357 teams (all of Division I). In order to complete that story, I also simulated the results for even small tournaments of only eight, four, and two teams. The results of these simulations are shown below:
Figure 1: Impact of the size of the tournament on the odds that the highest ranked team wins
As expected, if the tournament is small enough, the odds that the top ranked team cuts down the nets does increase. If the “tournament” is simply a single game played by the two top rated teams at the end of the season (in this example, Gonzaga and Baylor) the result of that game is likely a near toss-up on a neutral floor and thus the highest ranked team’s odds are naturally slightly over 50 percent.
As the number of teams in the field expand, those odds fall to 36 percent (for a four-team field), 28 percent (for an eight-team field), and finally to around 23 percent for any tournament with 16 teams or more. As explained last time, the reason that these odds seem to level off is that while the top rated teams have to win more games, so do all of the other top teams. A bigger tournament tends to generate more upsets which makes the average path for any given team slightly easier.
Another interesting result that follows from this analysis is that it implies that the current 68-team format for the NCAA Tournament is close to ideal. In order to include all of the conference champions, the field must be larger than 32 teams. If the field were to expand to greater than 68 teams, it is clear that the odds for each individual team do not really change.
Furthermore, my math suggests that the odds for any at-large team ranked outside of the top-45 or so in Kenpom to win the tournament is only about one percent. It is very unlikely that any team that does not make the field in the current format would go all the way. So, there seems to be no compelling reason to expand the tournament any more than it is today, unless there are extenuating circumstances...such as a global pandemic. More on that later.
Other Tournament Formats
The impact of the size of a standard single elimination tournament is now clear. But, in the leagues such as the NBA, the “tournament” consists of several rounds where the teams face off in best-of-seven game series.
It is natural to assume that in a multi-game format, the favored team is more likely to win, as the result would mostly likely “regress to the mean” over a large sample size (number of games). In other words, the underdog is less likely to win a majority of the games due to random chance (such a cold or hot shooting night) as the total number of games increase.
I was able to use simple probabilities (the binomial expansion) to estimate the odds that the favored team wins a multiple game series based on the odds of the favored teaming winning an individual game. To keep things simple, I assumed that all games would be played on a neutral court. The results are shown below in Figure 2.
Figure 2: Change in the overall odds when moving from a single game to a multi-game format
The shape of this graph is essentially what one might expect to see. In all cases, the more games are played, the more likely that better team is projected to win the series. For near toss-ups, the effect is small, but it appears to increase as the gap between the teams grow.
Another way to visualize this data is to converter the odds to win both a single game and the series into an effective equivalent point spread. That data is shown below in Figure 3.
Figure 3: Effective single-game point spread equivalent for various multi-game series relative to the spread for a single game
As the figure shows, the relationship is essentially linear and there is a simple correlation between the point spreads for the single game and the equivalent point spread for a multi-game series. For example if the projected point spread for two teams on a neutral court is 10 points in a single game, the odds that the better team would win in a three-game series is the equivalent of winning a single game with a point spread than is 1.48 times bigger, or roughly 15 points.
In a five-game series, the difference is a factor of 1.86. So, it is like moving from a 10-point spread to an 18.5 point spread. In a seven-game series, the effect is just a bit larger than the odds of doubling the point spread.
By itself, this data is interesting, but perhaps not that useful. However, it does allow us to answer another interesting question: what would happen to the odds in the NCAA tournament if somehow there was enough time to make each round a multi-game series of three, five, or seven games? Of course, this is completely impractical, but what if it were possible?
In addition, we could envision a double elimination format for the NCAA tournament. For the sake of comparison, I simulated a 16-team tournament using both a double elimination structure as well as a best-of-three game structure and compared it to the standard single elimination format base case. That result is shown below in Figure 4.
Figure 4: Comparison of the odds of each team winning a 16-team tournaments with different formats
In this case, we can finally see a significant difference in the odds, and the primary beneficiaries are the top two teams. Moving from a single-elimination format to double elimination boosts the odds of the top two teams by about four percentage points.
If each round were instead a best-of-three series, the odds would be even better for the top teams. In this case, the No. 1 overall seem gets a boost of almost 10 percentage points while the No. 2 seed’s odds improve by around seven percentage points.
These boosts come at the expense of the weaker teams. For this data set, the No. 3 overall seed has essentially the same odds in all three scenarios, while the odds for the rest of the field go down in both the best-of-three and the double elimination scenario.
Though not shown in Figure 4, an additional simulation suggests that moving to a seven-game series in each round would push the odds for the No. 1 team up by an additional 10 percentage points to around 45 percent. In other words, an NBA-style best-of-seven tournament would very likely identify “the best” college team in the tournament. But, it would also likely temper much or all of the madness that we have come to expect and love.
The Full Monty
In the previous piece on this topic, I mentioned that the idea had been floated to expand the 2021 NCAA Tournament to include all eligible Division I programs. At first, I thought that this idea was a bit silly, but as COVID-19 related cancellations continue to mount, it seems less and less likely that the full 140-game Big Ten schedule is going to be completed. Other teams and conferences will have the same problem.
With unbalanced and abbreviated schedules, it is going to be much more difficult than usual to select the best 40 to 45 at-large teams this year. So, I am starting to see the logic in just letting everyone play. Perhaps for selfish reasons, this would also ensure that MSU’s current tournament streak remains intact.
But, a full roughly 350-team tournament would last a full nine rounds and the logistics are likely too difficult. But, with the release of the schedule of the NCAA tournament dates and venues this week, it occurred to me that there is a way to possibly run a full national tournament in 2021 with limited impact to the current schedule. This is how I would set it up:
- Cancel all conference tournaments. As all teams are new eligible, there is no reason to stage them.
- Give all of teams from the six power conferences (75 teams in the ACC, Big Ten, Big 12, Big East, SEC, and Pac-12) a first round bye as well as the top 21 non-power five teams, as selected by the committee. The total number of teams with a first round bye is 96.
- This leaves a total of about 250 teams from mid- and low-major conferences. These teams would play in a set of 32 regional tournament (like mid-major conference tournaments) of seven or eight teams each during the week of March 1 (when most mid-major conference tournaments take place). Each tournament is only three rounds and the 32 total winners would advance to the full tournament of 128 teams. This would require just one additional round compared to the current tournament.
- Now, perhaps the tournament can move to Indiana with the one additional round of 128 to be played during the time when the Big Ten tournament was scheduled to be played (March 11 to 14). This total of 64 games can be split over four days and the six or so venues already planed for use in the city of Indianapolis and the state of Indiana. Each team only plays once, so logistics are fairly simple for each team
- Now, we have 64 teams. Eliminate the First Four and continue with the existing schedule.
Seeding of the tournament might get messy, but the NCAA could even decide to use a model similar to the NHL where teams are reseeded after the play-in round. In other word, the No. 1 overall seed would automatically be paired with the weakest champion of the regional pod round in the round of 128. If this round is also all played in the state of Indiana, it would be pretty simple to accomplish this.
As I showed in my previous analysis, there would likely be a higher number of upsets overall in the NCAA Tournament due to its larger size, but the top teams would have no worse odds to cut down the nets in early April. That seems like a win for fans, players, and coaches. I would certainly not want to move to this type of model on a regular basis, but for one year, after everything that we have gone through as a society, I think that it would be fun.
That does it for this time. We still have another week to wait for MSU to take the court once again. Hopefully this will be the final “pause” of the season for the Green and White. My fingers remain firmly crossed.
