Getting it Right: Solving GHSA challenges with math, Part IV

Welcome back. This is the fourth part in a series on answering the question "Which teams are deserving of a playoff invitation?" In it I'll outline a model I'll refer to as "Extended Standings" and over the course of the series I'll provide the exact details so anyone interested can independently verify the results.

Index: Part I | Part II | Part III

In Part III of this series, I (hopefully) established that including the margin of victory is a requirement for any playoff invitation model to be considered economical. If you haven't read that part yet, I encourage you to do so now. It's a great read :-)

At the close of that article I acknowledged that using margin of victory to rate teams is often controversial as it is believed to encourage poor sportsmanship between mismatched teams.

However, rather than oversimplify the model by reducing the margin of victory to a binary outcome, I suggested we instead use what I'll refer to as "Rothman Grading", which divides the win for a game between the two participants by using the margin of victory as a measure for the closeness of the game. We can then test Rothman Grading against our equitable principles.

As a refresher, our four equitable principles as outlined in Part I are:

• Winning should be valued more than margin of victory against the same set of opponents
• A larger margin of victory should be valued more than a smaller margin of victory against the same set of opponents
• Increasing margins of victory should diminish in value
• Stronger opponents should be valued more than weaker opponents

Since the opponent is not considered in Rothman Grading, the fourth equitable principle is not addressed, so we’ll investigate the first three.  Factoring in the opposition will be addressed when we return to discussing pairwise comparisons.

For reference in the following discussion, here's a set of common margins of victory and the outcomes using Rothman Grading:

The second equitable principle states "A larger margin of victory should be valued more than a smaller margin of victory against the same set of opponents". Using the chart (or the Excel spreadsheet as developed in Part III of the series) we see a ten point victory is worth more than a one point victory. In every case it is true that the larger of two margins of victory awards the winning team with a higher percentage of the win.

Perhaps a corollary is that teams who lose by smaller margins should be penalized less than teams that lose by larger margins. Intuitively it seems equitable to award a team that loses 21-20 with a larger portion of a win than a team losing 30-20 to the same opponent. Again, this is evidence from the chart above.

At any rate, Rothman Grading passes the test for the second equitable principle.

The third equitable principle states "Increasing margins of victory should diminish in value".

Using our chart (or the spreadsheet) again we can see that the first point of a win nets the winning team 76.50% of the win. To evaluate at this point of the chart alone, the third equitable principle is met since further points scored to extend the margin of victory, from one to infinity, can only net the team an additional 23.5% of a win at most.

However, let's look further down the continuum. If a team wins by seven points, they are awarded 85.0% of the win. An additional touchdown, now a 14-point victory, awards them with an additional 7.2%, and a touchdown beyond that, now a 21-point victory, awards them with an additional 4.1%, for a total of 96.4%. At this point, if the winning team were to score all the points for the remainder of the game, from 22 to infinity, in total they would be worth less than going from a two touchdown victory to a three touchdown victory.

Looking further down the continuum, there's even less incentive to run up the score. Margins of victory over 35 points are all within 0.7% of each other and approach the point of being practically indistinguishable as they increase.

In short, not only does Rothman Grading pass the test for the third equitable principle, but clearly points scored beyond a three touchdown victory are exceedingly marginal in value.

On a separate and perhaps philosophical note along these lines, I'd argue that under most circumstances increasing the margin from two touchdowns to three might be how a reasonably coached team would perform in the natural effort to secure a win as opposed to running up the score.  Stated another way, Rothman Grading seems intuitively equitable in that margins below three touchdowns are relatively distinguishable while margins above three touchdowns are relatively indistinguishable.

Now let's turn our attention to the first equitable principle, which states "Winning should be valued more than margin of victory against the same set of opponents".

To this point we've examined Rothman Grading in terms of individual games, but to test this principle we'll look at how Rothman Grading performs in the aggregate. In other words, how does Rothman Grading perform when looking at a team’s entire regular season?

First, let’s introduce the terms Adjusted Wins (adjW) and Adjusted Losses (adjL). For each game of a team’s season, we'll award adjusted wins and adjusted losses according to Rothman Grading for each game.

As an example, let's look at the scores from Clarke Central 1986 regular season:

In 1986 Clarke Central finished 10-0 with an average margin of victory of 21.7 points per game. For use in our Extended Standings we can also say they had 9.416 adjusted wins and 0.584 adjusted losses.

Notice Clarke Central's closest game was a 7-6 victory against their biggest rival, Cedar Shoals. We can perhaps perform one test right here by realizing that Clarke Central earned 0.765 adjusted wins for the 7-6 victory, but if they had instead defeated Cedar Shoals by infinity they could still have only gained an additional 0.235 wins.

But let’s push the test slightly further by looking closer at the details of the game.

In reality (or at least in my memory), the Gladiators lead the Jaguars 7-6 for the majority of the fourth quarter. However Cedar Shoals attempted a chip-shot field goal on the last play of the game. Fortunately (for the Gladiators anyway), All-State linebacker Doug Brewster broke through the line to block the kick and preserve the win.

But what if the Jaguars had made it?

Here we’ll substitute the 7-6 win over Cedar Shoals with a 7-9 loss. In that case, 1986 Clarke Central's regular season would have looked like this:

Now Clarke Central finishes 9-1 with an almost unperceivable change in their average margin of victory to 21.4 points per game. However, they now have 8.871 Adjusted Wins and 1.129 Adjusted Losses, a 0.545 swing.

As a reminder, even if Clarke Central had won by infinity against Cedar Shoals, they could only have gained 0.235 adjusted wins, so the 0.545 swing seems equitable given that it was a small change in the score with a fairly large impact on the outcome.

In fact, a successful Cedar Shoals field goal in this game would have cost Clarke Central more than twice as much as running the margin to infinity could have netted them.

Now let's look at a follow on scenario, where an infuriated Clarke Central team defeated each of their next two opponents by a ridiculously unrealistic 300-0 margin:

Now Clarke Central finishes 9-1 with an absurd average margin of victory of 76.4 points per game. However, their new totals are 8.962 Adjusted Wins and 1.038 Adjusted Losses, a slight 0.091 improvement but still 0.454 less than their actual 10-0 finish against the same opposition.