Big Ten Basketball Mathematical Analysis, Addendum: The Ideal Schedule – The Only Colors

A few weeks back, I presented a three-part series in which I performed a detailed analysis of the Big Ten schedule. In Part One, I looked at the Big Ten conference slate and did some math related to the strength of each teams schedule. The results suggested that MSU was given a schedule of average difficultly, while potential conference favorite Wisconsin (according to the Kenpom preseason projections) was given a remarkably easy schedule.

In Parts Two and Three, I made additional calculations based on Monte Carlo simulations to estimate the odds for each Big Ten team to win both the regular season title and the Big Ten Tournament Title. In both cases, Wisconsin also had the best preseason odds.

In the days that followed, the idea that the schedule was biased in favor of the Badgers weighed on my mind. I wondered if it would be possible to construct a schedule that was more fair. It soon became clear to me that I could use the same tools that I used to analyze the strength of each teams Big Ten schedule to address this issue. Once I saw the entrance to the rabbit hole, I had to try to find the bottom.

The first step was to craft an algorithm that would generate a random Big Ten conference schedule based on its current form. The schedules have each team playing a total of 20 games. Seven of the opponents are played twice (once at home and once away). The remaining six opponents are either played only at hone or only on the road. With a bit of work, I was able to write some simple code that allowed me to generate random schedules for the entire conference fairly quickly.

Once I accomplished this, I needed to define a way to quantify the fairness of a given, random set of schedules. In Part One of the series referenced above, I explained my methodology for calculating strength of schedule. The basic principle is to calculate the expected number of wins that an average Power Five teams (as good as, for example, Indiana) would earn with any given schedule. It is trivial to make this calculation for all 14 Big Ten schedules by using Kenpom efficiency data to estimate win probabilities for any set of Big Ten games.

I also performed a corrected version of this calculation by artificially adjusting the strength of one average Big Ten team (Indiana) to instead be equal to that of the team whose schedule is being analyzed. This corrects for a potential bias in the numbers due to the fact that weak teams and strong teams dont play themselves.

In order to quantify the fairness of the entire Big Ten schedule, I calculated both the range and standard deviations of my calculated strengths of schedule (both regular and corrected) for all 14 individual schedules. In other words, I looked at the quantitative difference between the toughest and easiest schedules as well as the overall variance for all 14 schedules.

I then scanned the list of randomly generated schedules, searching for the one with the smallest standard deviation of corrected schedule strengths. I generated over 40,000 potential schedules and found the one with the smallest variance. This is perhaps the one of the most optimized schedule that can be created, based on the projected preseason strength of each team in 2020.

To refresh our memories, Table 1 below summarizes the actual 2020 Big Ten conference schedule. Relative to each row, the green cells represent the single-play home games while the orange cells are single-play road games. For example, MSU plays Wisconsin only once at home this year and only plays Maryland on the road.

Without doing any math at all, it is easy to see why Wisconsin has such a schedule advantage. The Badgers are the only Big Ten team with two scheduled games against the four projected weakest teams in the conference (Maryland, Penn State, Northwestern, and Nebraska). Furthermore, the Badgers draw Kenpoms No. 2 and No. 3 ranked teams (Ohio State and Michigan State) only once.

First, I was curious how the actual schedule compares to a set of randomly generated schedules, based on the fairness metrics described above. Figure 1 below makes this comparison.

As the histograms above shows, not all schedules are created equally. Just based on range alone (the difference between the hardest and easiest schedule) some full conference schedules can differ by over a game and a half in expected wins. However, it is possible to find schedules where the team with the easiest schedule has less than a half of a win advantage over the team with the most difficult schedule.

If the raw, uncorrected strength of schedule values are considered (the blue bars), the real 2020 schedule is quite a bit less fair than the an average, random schedule based on both range and variance. If the corrected values are used (the orange bars), the 2020 schedule is average.

However, it is clearly possible to do better. If I select the individual schedule that showed the lowest observed corrected strength of schedule variance (located all the way to the left in the histogram in the right panel of Figure 1) that schedule looks like this:

While it is always difficult to compare one wall of numbers to another, the schedule shown in Table 2 looks a lot more fair on its face compared to the real schedule. First of all, there is much better balance at the bottom of the table. The top eight Big Ten teams all play the collective bottom four conference teams a total of six times. No team plays all four of these teams twice (as Wisconsin does in the actual schedule) and only two teams (Rutgers and Minnesota) play this collection of teams a total of seven times.

A similar balance is also found at the top of the table. Every team which the exception of Illinois, Michigan, and Purdue, have exactly two single-play matchups among the top four projected Big Ten teams (Wisconsin, MSU, Ohio State, and Iowa). That all said, this schedule still seems challenging for Nebraska, which draws four single-play games amongst the bottom four Big Ten teams not named Nebraska.

A more quantitative comparison of the actual and optimized Big Ten schedules is shown below in Figure 2. In this case, the individual team schedules are ordered from easiest (left) to hardest (right) to make the comparison easier to see. Also, the left panel shows the raw strength and schedule calculation, while the right panel gives the corrected values which are what were used in the optimization.

The data in both panels shows a clear difference in the two full conference schedules. For the raw strength of schedule calculation (left panel) the difference between the easiest schedule and ninth easiest schedule is over half a win in the actual schedule, but less than a quarter of a win in the optimized schedule.

In both cases, there is a drop off in expected win for two most difficult schedules. In both the actual and optimized scenarios, these schedules belong to Northwestern and Nebraska, the two teams that project to be the weakest in the Big Ten this year. This is no coincidence, as once again those two teams suffer from not getting to play themselves.

Fortunately, the corrected strength of schedule appears to handle for this problem. As the right panel shows, the full range of expended wins for an average power five team (i.e. strength of schedule) only differs by slightly over 0.3 wins which seems to be about as fair of a schedule that can be created. In the actual schedule, this range is almost 0.8 wins.

One could make the argument that the it would be better to optimize the schedule based on the raw strength of schedule values as opposed to the corrected values. After all, the left panel still suggests that Northwestern and Nebraska draw the short straw. On some level that is true.

However, a schedule optimized based on the raw strength of schedule values would effectively be creating a schedule that is easier for the weaker teams and harder for the stronger teams. While this seems like a nice gesture, what happens if Nebraska is actually much better than expected? In this scenario, the Huskers would suddenly have an advantage, simply because they were under-valued in the preseason. For this reason, I believe that the corrected values are the best way to find the most fair Big Ten schedule.

That said, there are a few aspects of the schedule that are still perhaps not ideal. For example, Michigan State and Michigan only play each other once. The same is true of Purdue and Indiana. In reality, it would be better if these types of rivalry games would be protected. There may be other constraints on scheduling of which I am not aware.

In any event, it would be simple to modify my algorithm to exclude any random schedule that does not meet these criteria. I am confident that this method can and perhaps should be used to create a better Big Ten schedule. If anyone in the MSU athletic department or the Big Ten office is reading this and would like my assistance, send me a direct message. (I am sort of kidding... but not really).

The strength of schedule calculations discussed above provide a pathway to create a schedule that is mathematically more fair. The next logical question is if the impact of a more level playing field actually matters. I touched on this issue briefly in the original series, but I would like to revisit the topic now that we have a more balanced schedule to compare to the original one.

For this study, I once again ran a series of Monte Carlo simulations on the full Big Ten season using both the actual Big Ten schedule and the optimized schedule shown in Table 2 above. This simulation outputs the odds for each Big Ten team to win the regular season Big Ten title (outright or shared). For here out in order to keep things simple, I will focus only on the results of the simulation for Wisconsin (the Kenpom presumed favorite) and Michigan State.

In addition to the baselines simulation using preseason Kenpom efficiency margin values to assign each team a certain strength, I ran three additional simulations in order to try to separate the effect of the schedule from the effect of each teams strength. In one simulation, I artificially swapped the strength of MSU and Wisconsin. In effect, this simulates the effect of Wisconsin playing MSUs schedule while MSU plays Wisconsins schedule.

In the other two simulations, MSU and Wisconsin are assumed to be equal in strength, either both as good as Wisconsins preseason projection or both as good as MSUs preseason projection. This set of simulations was performed using both the actual schedule as well as the optimized schedule. The results are shown below in Figure 3.

This figure contains a ton of information. The best way to extract information is to make various comparisons between the different scenarios. If we start with the real schedule (left panel) we can see that in the baseline simulation (using the real Kenpom efficiency margin preseason data) Wisconsin has almost exactly a 20-percentage point advantage over Michigan State in the race for the Big Ten title. The is the same data that I presented in Part Two of my series.

If the two teams had schedules with equal difficulty, swapping the schedules should also swap the championship percentages. In the case of the optimized schedule, this is true within one percentage point. But in the actual schedule, there is a significant gap.

In the baseline case Wisconsins championship odds (38.5 percent) are notably higher than MSUs odds if their strengths are swapped (31.9 percent). In effect, this is equivalent to Wisconsin playing MSUs schedule, and it implies that Wisconsins schedule is worth about six and a half percentage points. A similar analysis of MSUs baseline odds (18.4 percent) to the odds for Wisconsin if they were equally as good as MSU (23.9 percent, the second red bar) gives a five and and half percentage point difference.

A similar story is told by the third and forth sets of bars. In these cases, MSU and Wisconsin have equal Kenpom efficiency margins, but in both cases, the Badgers odds are better (by seven percentage points and five and a half percentage points). It should also be noted that in the right panel of Figure 3 (the optimized schedule) if MSU and Wisconsin have the same efficiency margin, they also have almost identical odds.

Looking at the data from another point of view provides insight into how much of an advantage Wisconsin has simply because of the higher preseason efficiency margin. In other words, how much does an advantage in actual (or simulated) team strength impact the title odds?

This value can be estimated by comparing the first two red bars to each other or the first two green bars to each other in either panel. This is simply the difference in odds that each team would have with the same schedule and either their MSU or Wisconsins strength. The difference varies between 13.5 and 14.5 percentage points.

All of this data points to one basic fact: Wisconsins calculated 20-percentage point preseason advantage in the regular season odds is due to a one-third contribution of a schedule advantage (about 6.5 percentage points) and a two-thirds contribution (about 13.5 percentage points) from Wisconsins estimated efficiency margin advantage.

In addition, I feel that Figure 3 makes a fair case that if an ideal Big Ten schedule is constructed, this schedule advantage shrinks to zero. That said, I should also point out that in all four comparisons in the right panel of Figure 3, MSU actually has about a one-percentage point advantage over the Badgers, which appears to arise due to a combination of MSUs very slightly easier schedule in the ideal scenario, combined with the slightly larger home court advantage that Kempon assigns to the Breslin Center relative to the Kohl Center.

While the results of the analysis above are interesting (at least to people like me...and I assume anyone still reading this) there are still a few questions that remain. This entire analysis hinges on the idea that the preseason efficiency margin data is correct. The season is now several games in and already some teams have moved up or down. I showed above that Wisconsins preseason efficiency margin advantage (+1.79 compared to MSU) is worth about 13.5-percentage points. But, what happens when that value changes? How sensitive are the title odds to these numbers?

In order to clarify this, I ran one additional simple set of simulations. In this case, I fixed the efficiencies margins for all Big Ten teams and varied the efficiency margin of Wisconsin from a value of 18.00 (roughly the quality of a bubble team, ranked around No. 30 in Kenpom) all the way to to a value of 30.00 (roughly the quality of the No. 1 ranked team in Kenpom in any given year).

I then calculated both the expected win totals and regular season title odds for Wisconsin using the actual schedule. Those results are shown below in Figure 4. The same data is also shown for MSU (using their fixed, preseason efficiency margin as a reference).

Note that the actual preseason values for each team are shown with the large, solid data point. The data in Figure 4 looks about like one would expect it to look. A bubble team (with an efficiency margin of 18) would be expected to win just over 10 games in conference play and have less than a 10 percent chance to win the Big Ten regular season, while a team ranked No. 1 in Kenpom (with an efficiency margin of 30) would be expected to win close to 16 games and would have over an 80 percent chance to win the Big Ten.

The correlations are not truly linear (especially for the title odds) but I included the linear fit equations in each plot for reference. In both cases, the slopes provide a good rule of thumb for the sensitivity of expected wins and title odds. Specifically, for every 1.00 improvement in efficiency margin, a teams expected win total will increase by about 0.44 wins and the title odds will improve by about 6.7 percent for a team in a conference like the Big Ten in 2020.

Note that the second number is consistent with Wisconsins roughly 13-percentage point schedule-independent advantage mentioned above, considering their almost 2.00 lead in efficiency margin over the second best Big Ten team. Also note that Wisconsins schedule advantage is worth almost exactly 1.00 in efficiency margin.

The artificial change in Wisconsins strength makes a relatively small impact on MSUs expected wins and even odds. MSU only faces Wisconsin once, so it makes sense that MSUs expected win total is almost unaffected. As for the title odds, MSUs odds decrease gradually as the strength of the Badgers increases. Even in the cases where Wisconsin is really good, MSUs odds only drop by eight to ten percentage points. Basically, MSU still mows their own grass, more or less.

Finally, this analysis begs the question of the general correlation between expected wins and regular season title odds. That correlation is shown below and was derived for the original baseline simulation of all 14 Big Ten teams combined with the Wisconsin sensitivity analysis discussed above.

Again, this plot makes a lot of sense. If a teams expected win total is around 10 wins or less, the odds to win the Big Ten are very low (five percent or less). Those odds increase fairly linearly to close to 90 percent as the expected win total approaches 16. Recent history suggests that the regular season Big Ten champs usually win roughly 16 games in a 20-game schedule. Furthermore, the slope of the line suggests that every whole win improvement is worth about 17 percentage points in championship odds.

At this point, I think that it is safe to say that I have beaten the preseason Kenpom data to a bloody pulp. Fortunately, the Big Ten season is right around the corner, and the analytical tools at my disposal will allow for a real time tracking of expected wins, regular season championship odds, and Big Ten tournament seeding and odds.

The first Big Ten game tips off on Sunday, Dec. 13, and I plan to give a brief update on the numbers which reflect the changes that have occurred since the preseason data was released. I will then provide updates following most MSU games. That is all for now. Until next time, enjoy, and Go Green

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Big Ten Basketball Mathematical Analysis, Addendum: The Ideal Schedule - The Only Colors