An improvement to the baseball statistic “Pythagorean Wins”

2.1An example using pythagorean wins

Suppose we have three teams, which for simplicity we will just call 1, 2, and 3. They play a round-robin of one game apiece against each other team, so each team plays 2 games total. Suppose that the results are as follows: Team 1 beats Team 2 by a score of 1-0; Team 1 beats Team 3 by a score of 2-1; and Team 3 beats Team 2 by a score of 3-1. That means we have:

We can use these numbers to compute the Pythag-orean winning percentage of each team:

Since each game results in one win and one loss for each team playing, the average winning percentage is supposed to be one-half, or . 500. That means that we should expect the three fractions above to add up to 3/2. However, the actual sum is1359850.

If we want to consider Pythagorean wins instead of percentages, all we would do is multiply each of those fractions by 2 before adding them. Since we are multiplying by the same constant, this is the same as if we multiplied the final answer by 2, which clearly shows that the total number of Pythagorean wins for the three teams is equal to1359425. Since this fraction is greater than 3, the Pythagorean wins statistic has claimed that the three teams “should have won” more games than were actually played between all of them.

Remark 1. This was a simplified example to illustrate the math, but does it carry over into real-life computations for large samples? The data in the Appendix confirm very clearly that it does. In three out of five cases, the Pythagorean formula claims that the teams “should have won” more games than were played, and in the other two, it claims that they “should have won” fewer. It never actually recovers the exact total number of games played.

2.2The mathematics behind the example

The example illustrates the following mathematical dilemma: suppose we are given any amount of ordered pairs

and we are told that they are related in the following way: This is clearly the case if these ordered pairs represent the runs scored and runs allowed totals of baseball teams playing in the same league, because whenever one team scores a run it must be allowed by someone else, though it may be uncertain which other team allowed the run.

The problem, clearly shown in the example, is that when we fix an exponent a ≠ 0 and take a sum of the form

we have no idea, a priori, what that sum will be—but for the Pythagorean wins statistic to make sense, it should be N/2.

We of course have no assurance that this will ever be the case, and this is why the Pythagorean wins statistic runs afoul of the basic expectation property. The equal relationship between total runs scored and total runs allowed cannot be used to prove anything about what the sum must be equal to, because the individual denominators can vary so much.

3.1A special case

There is one time, however, when there is no problem at all, and we actually can prove that the sum equals N/2, as we hope it does. That is when there are exactly two teams, that only play each other, and four pieces of data: R₁, r₁, R₂, and r₂. In this case, it is obviously true that R₁ + R₂ = r₁ + r₂, and the reason it is so obvious is because in this case, we can say something even stronger: namely, that R₁ = r₂ and R₂ = r₁. This is clear because when there are only two teams, one team must allow all the runs that the other team has scored, and vice versa.

With this stronger assumption, we can now show that for any exponent a,

Since the sum is 1, this clearly implies that in this exceptional case, the sum of the two teams’ Pythagorean win totals will equal the number of games played between the two teams.

3.2The pairwise formula

The special case above suggests a simple improvement to the formula: all we have to do is split our runs-scored data so that we are considering Pythagorean wins on a team-by-team basis, and not a cumulative basis over all teams. To be more specific: instead of using a list of ordered pairs of the form (R_j, r_j), where R_j is the total runs scored by team j and r_j is the total runs allowed, we would instead use an entire matrix

where each entry R_ij is the number of runs scored by team i against team j. This matrix includes the cumulative data: for any team k, In other words, the sum of any row is the total number of runs scored by a team, and the sum of any column is the total number of runs allowed by a team. (This is not an original idea; matrices of this form have already been computed for most, if not all, baseball seasons in history. For example, the seasons in (Major League Baseball n.d.) contain such matrices, which is how the data in Section 4 and the Appendix were computed.)

The traditional Pythagorean wins formula is, for any team k, to take the percentage

and multiply that by the total number of games played against all teams. Presenting the formula in this way shows that this formula features taking sums first, then raising those sums to a power.

The proposed improvement would do essentially the opposite: for each team k, it would take each percentage

multiply that by the number of games team k played against team j, and then add up all those numbers for all other teams j to get the projected win total for team k. As a formula, if we set G_kj to be the number of games team k played against team j, the pairwise Pythagorean projection of team k’s win total is In other words, instead of the sum over all teams being done first, this method takes individual teams’ Pythagorean win totals against other individual teams, and does the sum over all teams last. That explains the “pairwise” part of the name of the formula.

As shown earlier, since R_jk is exactly the number of runs allowed by team k against team j, and since G_kj = G_jk, it is necessarily the case that two teams’ pairwise Pythagorean win projections against each other will add up to the total number of games played between the two teams:

It then follows that the total number of pairwise Pythagorean wins for all teams will sum to the total number of games played between all teams. This is the property that the original formula was lacking.

Remark 2. As pointed out earlier, a key property of an expectation is that the sum of individual components is supposed to equal the total expectation. Earlier we showed that the Pythagorean formula fails this criterion for entire leagues as a whole when broken up by team; but this section implies that it also fails the same criterion for individual teams when broken up by opponent. (For examples, see every computation in the Appendix.) Viewed in this light, the pairwise formula simply replaces the “total” with the sums of the individual components.

Remark 3. We have just proved the fact that the sum of all teams’ pairwise Pythagorean win totals in a single league must equal the total number of games played. This is equivalent to stating that if we look at the differences between pairwise Pythagorean wins and actual wins over all teams in a league, those differences must sum to 0. The examples in the Appendix show the sums of differences for five leagues. When we showed earlier that the traditional Pythagorean formula does not necessarily keep the sum of Pythagorean wins equal to the total number of games played, we were also showing—as is confirmed in the Appendix—that the sum of differences is not necessarily 0, while here we have shown that it is for the pairwise method.

4.1Pairwise computation: An example

Before delving into the entire data set, it may be illuminating to look at an example of the pairwise computations compared to the traditional formula in a specific case. In this section we will consider one specific team from the Appendix: the 1978 California Angels.

The Angels’ real record was 87-75. A look at the Appendix shows that the traditional formula with exponent 2 projected them to win approximately 84 games, meaning they finished about 3 games above their projection. When this happens, it implies the a = 2 formula will automatically be more accurate than the traditional a = 1.83 formula in this case (proving this assertion is a calculus exercise that we will omit), so for this example we will only examine the pairwise formula with a = 2. Another look at the Appendix shows that this formula projected the Angels to win approximately 85.4 games, a higher total than the traditional formula and, because the Angels outperformed both projections, a more accurate prediction. But why exactly is the pairwise formula more accurate for this team?

The answer can be found in the following breakdown of the Angels’ row and column of the 1978 AL run matrix (which can be found in (Major League Baseball n.d.)). For each opponent, the table shows their actual win-loss record, runs scored and runs allowed, their Pythagorean winning percentage (to 4 decimal places), and their Pythagorean win total (to 4 decimal places), all against that opponent.

The last two entries are left out because computing using the total run data and adding up the pairwise win totals are not equal (so we do not choose only one of them to fill the table in). Of interest to us in this case is why the pairwise projection is more accurate, and to help see this, let us focus on two particular lines of the table: the Chicago White Sox and the Oakland Athletics.

The Angels played 15 games each against both of these teams—30 games total. We can see from the table that the Angels finished 17-13 combined against these two teams. It is also easy to read off from the table that the Angels projected to win (approximately) 6.5661 games against the White Sox and 11.5760 games against the A’s, for a pairwise total of 18.1421 pairwise Pythagorean wins. This suggests that against these two teams only, the Angels were 1.1421 games below their projection.

What happens when, instead of projecting and then adding first, we combine run totals first? The Angels scored a total of 75 + 57 = 132 runs against these two teams, and allowed 85 + 31 = 116. Therefore they would be projected to win

games against these two teams.

Although it appears the traditional Pythagorean formula is far more accurate, this is a red herring. (Remember, overall, the pairwise formula is going to be more accurate over all teams. This is why it is a red herring.) The important fact is that the traditional formula projects the Angels to be more than a game below where the pairwise formula projects them. Why is this?

The reason can be explained as follows: the Angels showed more dominance in games that were lower-scoring. Against the A’s, their ratio of runs scored to runs allowed is very high—almost 2. Against the White Sox, that ratio is slightly less than 1. However, the number of runs scored overall is different. In the 15 games in which the Angels were not as dominant—against the White Sox—160 total runs were scored. But against the A’s, only 88 total runs were scored! It is now clear why the traditional Pythagorean winning percentage is relatively low—we have “softened” the effect of the games against the A’s, in a way claiming that they should matter less because fewer total runs were scored in those games.

To see this clearly, let us imagine that instead of 57-31, the runs scored and allowed totals against the A’s were 114-62. The ratio is the same; the pairwise Pythagorean projections would not be affected by this. But the traditional projection clearly will be; now the modified win projection will be

Notice that now, when the totals of runs scored are 160 and 176 instead of 160 and 88, the traditional formula is closer to, though still not equal to, the pairwise formula.

This leads to the following question: which of these should we consider as being more “correct”? Is it more correct to say that we should in fact tilt our projection towards the results in which more total runs were scored? Perhaps we might argue that the distribution of 88 runs can be more subject to “luck” than the distribution of 160 runs, and therefore it should matter less.

However, standing against this argument is the fact that both sets of run totals were amassed in sets of 15 games, the same amount of time. Therefore we might argue instead that there is something about the games between the two different pairs of teams that makes the distribution of runs—and therefore wins—different, and we should only be combining the data at the latest stage possible. If we believe this argument, this is a feature of the pairwise Pythagorean formula that is advantageous over the traditional formula. The evidence that the pairwise formula seems to have a tendency to be more accurate overall (see below) lends support to this line of argument as well.

4.2Pairwise computation: Results

The previous sections attempted to make a theoretical argument that the pairwise formula should have less error than the traditional formula. However, we also have the ability to test the formula with real data. So how does the pairwise method compare to the conventional method at generating predicted win totals? This section will provide evidence towards answering this question.

The table below shows summary data for every Major League season from 1960–1990, excluding 1981 (which had fewer games than other seasons due to a player strike). Next to each league is listed the number of teams in that league, followed by four different numbers. Each of these is the root mean square of the deviations between the teams’ actual wins and their projected wins using one of four formulae. The leftmost column is the root mean square for the traditional Pythagorean formula with exponent 2; next is the root mean square for the traditional Pythagorean formula with exponent 1.83; next is the root mean square for the pairwise Pythagorean formula with exponent 2; and finally, the rightmost column is the root mean square for the pairwise Pythagorean formula with exponent 1.83. The last line contains the total number of teams and all four root mean squares when the teams are considered as a single data set. All figures are rounded to three decimal places.

A few potentially useful observations about the table:

In the Appendix, five of these 60 seasons are broken down team-by-team to give the reader a sense of how the formulae compare for individual teams, and also to numerically demonstrate some of the assertions made in the previous sections.