Akousmatikoi Win Estimators, pt. 3: Differential-Based Simplifications

Simplifying the Pythagorean estimate by focusing on run differential is not as intuitive as using run ratio, since of course Pythagorean constructs are based on the latter rather than the former. The upfront calculus is messier, the relationships harder to explain – I’ve covered all this before, and so I went back to my previous work rather than go through the hassle of re-deriving it. However, while the calculus is messier, the end result is simpler, and give you relationships that you might actually choose to use in place of the full Pythagorean treatment if you want something quick and simple to punch into a calculator.

The easiest way I’ve found to demonstrate this approach (which is not to say that a simpler derivation doesn’t exist) is to use the following definitions. To make this easier to follow, I’m going to define R as R/G and RA as RA/G:

RR = R/RA

RD = R - RA

RPG = R + RA

Given these relationships, we can relate run ratio and run differential using RPG:

RR = (RD + RPG)/(RPG – RD)

If you need a proof of that, replace RD and RPG with the equations above and you will see that:

RR = (R – RA + R + RA)/(R + RA – (R – RA)) = (2*R)/(2*RA) = R/RA

In the last installment, we differentiated Pythagorean win ratio with respect to run ratio; here, I want to differentiate Pythagorean winning % with respect to run ratio, which will look slightly messier. Starting from the Pythagorean relationship:

W% = RR^x/(RR^x + 1)

we differentiate to get:

dW%/dRR = ((RR^x + 1)*(x*RR^(x – 1)) – RR^x*(x*RR^(x – 1)))/(RR^x + 1)^2

= (x*RR^(x – 1))*((RR^x + 1) – RR^x)/(RR^x + 1)^2

dW%/dRR = x*RR^(x – 1)/(RR^x + 1)^2

That’s well and good, but it doesn’t tell us anything about the relationship between Pythagorean W% and run differential. To bridge that gap, we can differentiate run ratio with respect to run differential and multiplying this result with dW%/dRR which we just derived:

(dW%/dRR)*(dRR/dRD) = dW%/dRD

Since we know that RR = (RD + RPG)/(RPG – RD), we get:

dRR/dRD = ((RPG – RD)*1 – (RD + RPG)*(-1))/(RPG – RD)^2

= 2*RPG/(RPG – RD)^2

If you slogged through any of my previous treatments of this topic, I must apologize – I missed some simplifications of both of these formulas before. The final math worked out the same, but it was needlessly difficult to follow. In any event, we now have:

dW%/dRD = (x*RR^(x – 1)/(RR^x + 1)^2) * (2*RPG/(RPG – RD)^2)

= 2*RPG*x*RR^(x -1)/((RR^x + 1)^2*(RPG – RD)^2)

This ends up being expressed in terms of marginal wins per margin run. The classic sabermetric presentation is marginal runs per margin win (Runs Per Win, ala the rule of thumb that 10 runs = 1 win). So we can take the reciprocal to get this formula for Runs Per Win from Pythagorean:

Pythagorean RPW = (RR^x + 1)^2*(RPG – RD)^2/(2*RPG*x*RR^(x - 1))

Before moving forward, one thing I should note is that this function does not allow us to match the Pythagenpat W% at a given point for a set of inputs. For example, if you plug in 5 runs scored and 4 runs allowed, you will get a dW%/dRD of .1071. You might then reasonably assume that if you take the team’s run differential of 1 times .1071 plus a y-intercept (which by definition would be .5 since Pythagorean will estimate a .500 W% when R = RA), you will get a restatement of the team’s Pythagorean W%. But in fact you will get .6071, while Pythagorean would estimate 5^2/(5^2 + 4^2) = .6098. The differences will be more extreme if you put in more extreme teams.

Alas, I do not have a simple mathematical explanation for why this is the case. However, I will note that we don’t need calculus to calculate the actual Runs Per Win value from Pythagorean for any given set of R, RA, and x that we input. We can simply calculate this by noting that:

W% = RD/RPW + .5

Plugging in Pythagorean relationships and solving for RPW:

R^x/(R^x + RA^x) = RD/RPW + .5

R^x/(R^x + RA^x) - .5 = RD/RPW

RPW = (R – RA)/(R^x/(R^x + RA^x) - .5)

For our 5 R/4 RA team, this results in (5 – 4)/(5^2/(5^2 + 4^2) - .5) = 9.1111 RPW or .1098 wins/run, which of course is the right answer. In terms of simplifying the Pythagorean relationship, though, this is useless – all we’ve done is rearrange terms to calculate runs per win for a given set of inputs. How we could use this to produce a flatter win estimator is to eliminate the use of a team’s R and RA figures and instead replace with a function that only considers the scoring level (i.e. RPG).

This is what the rule of thumb that 10 runs = 1 win does, substituting a general rule for specifics about the team’s actual location on a run/win curve with respect to the marginal value of an additional run scored or allowed. As such, since it’s establishing a rule that will be applied to all teams, it makes sense to center it at the point which will be closest to an average team – at the point where R = RA.

In other words, we will be developing a RPW equation that can be applied generally, but will be defined based on the relationship at the point where R = RA for a given RPG. Using our formula above for RPW based on rearrangement of terms in the Pythagenpat relationship, we can substitute R = RA wherever we see one of those terms and...reduce the equation to 0/0, as the denominator R – RA equals 0 when R = RA, and the numerator R^x/(R^x + RA^x) - .5 = 0 when R = RA.

However, this is where the equation for RPW derived using calculus can step in, and tell us what the theoretical RPW value is at that point. Recall from above that:

RPW = (RR^x + 1)^2*(RPG – RD)^2/(2*RPG*x*RR^(x – 1))

If we assume that R = RA, then RR = 1 and RD = 0, and this simplifies nicely to:

RPW = (1^x + 1)^2*(RPG – 0)^2/(2*RPG*x*1^(x – 1))

= 2^2*RPG^2/(2*RPG*x) = 4*RPG^2/(2*RPG*x) = 2*RPG/x

The first immediate implication is that for our special Pythagorean case where x = 2, RPW = RPG. Since the general case is:

W% = (R/G – RA/G)/RPW + .5

RPW = RPG is equivalent to saying that (after all of the game denominators cancel out):

W% = (R – RA)/(R + RA) + .5

What if x is a constant other than 2, like the value of x = 1.847 that minimizes RMSE for expansion-era major league teams? Then RPW = 2*RPG/1.847 = 1.083*RPG, and we could say that:

W% = (R/G – RA/G)/(1.083*(R/G + RA/G)) + .5

= (1/1.083)*(R/G – RA/G)/(R/G + RA/G) + .5

= .923*(R – RA)/(R + RA) + .5

More generally:

W% = (x/2)*(R – RA)/(R + RA) + .5

This form is one that was proposed by Ben Vollmayr-Lee as .91*(R – RA)/(R + RA) + .5 (I’ve rewritten his formula to match the format I’m using), which would imply a Pythagorean x = 1.82. I would suggest that the Kross equations and the Vollmayr-Lee equation are the ultimate in terms of simplified win estimators from the Akousmatikoi family (again, Kross and Vollmayr-Lee did not start from Pythagorean as we have; by including these estimators in the Akousmatikoi family, I only mean to suggest that they are mathematically related to Pythagorean, not that their creators didn’t independently discover them).

Remember that for the expansion era, the average RPG is 8.83, which would imply that the long-term RPW value is approximately 1.083*8.83 = 9.56; close enough to ten that you can see why we might have a rule of thumb, although ten runs would imply a 4.5% higher scoring context (10/1.083 = 9.23) than observed in the expansion era.

We could also use a hybrid approach, in which we allow each team’s RPW according to the formula that applies when R = RA to vary based on their RPG, but not on how that RPG breaks down into runs scored and allowed. In order to do this, we’d return to RPW = 2*RPG/x, but instead of setting x equal to a constant, use a custom value for x. Of course, my suggested value would be the Pythagenpat estimate of x, namely:

x = RPG^z, where z = .282 for now (value that minimizes RMSE for the expansion era)

Substituting this equation for x, we find a general case for a variable z that:

RPW = 2*RPG/(RPG^z) = 2*RPG^(1 – z)

Or for the specific case that z = .282:

RPW = 2*RPG^.718

We could further flatten this equation by approximating it with a linear function. Recall from the last section that we can write a tangent line in the form:

y – y1 = m(x – x1) where x1 and y1 and the x and y values for the point in question, and m is the slope of the curve at x1.

To apply this approach to develop a linear approximation of the above equation, we first need the slope of the RPW function 2*RPG^(1 – z). Differentiating with respect to RPG yields 2*(1 – z)*RPG^(-z).

Let’s center this at the point corresponding to our expansion-era averages, so x = 1.847 (For the eagle-eyed readers or those checking my math (always welcomed!) I’m choosing to use the value that minimizes RMSE to be consistent with earlier applications rather than the value of 1.848 that corresponds to 8.83 RPG using the equation directly). In this case x1 will be 8.83 RPG, and y1 = 2*8.83^.718 = 9.555 At 8.83 RPG, m will be 2*(1 - .282)*8.83^(-.282) = .777, so we have:

RPW – 9.555 = .777*(RPG – 8.83)

which simplifies to:

RPW = .777*RPG + 2.694

We’ve now developed two RPW estimates, using only RPG as a dependent variable, one with a y-intercept and one without, by trying to flatten the Pythagorean relationships wherever possible. Which is more accurate? One would assume that it’s the version with y-intercept, but even if it is, how much more accurate for normal teams, and how does this tangent line based approach compare with the best fit for an equation of the form RPW = m*RPG + b? Those are questions we’ll explore in the final installment.

References

Ben Vollmayr-Lee’s article on win estimation formulas:

http://www.eg.bucknell.edu/~bvollmay/baseball/pythagoras.html

Ralph Caola published multiple articles on using differentiation with the Pythagorean formula, as well as an (to the best of my knowledge) unpublished article he shared with me on double the edge.

His articles can be found in the 11/2003, 2/2004, and 5/2004 issues of By the Numbers.

https://sabr.org/research/statistical-analysis-research-committee-newsletters/

Kevin D. Dayaratna and Steven J. Miller explored the relationship that RPW = 2*RPG/x in the 5/2012 issue of BTN. I had known and used that one for a long time, thanks originally to a post by David Glass on rec.sport.baseball. Unfortunately a quick search did not yield a live link to Glass’ post.

Akousmatikoi Win Estimators, pt. 2: Ratio-Based Simplifications

We will begin our endeavor to simplify/”flatten” the Pythagenpat exponent by looking at approaches that maintain the use of run ratio as the chief independent variable in the W% estimate. Before jumping into that, I should note that we could think of the first flattening as being moving from a variable exponent like Pythagenport/pat to a fixed exponent. However, since the latter came first historically, and is easier to explain conceptually, I didn’t approach it in that manner.

We could also make flattening the Pythagenpat exponent itself the first step. My definition of “flatten” for the sake of this discussion is to replace exponents with multiplication where possible. We could start by trying to convert z = RPG^.282 into a linear formula. I’ve skipped this step because we would still be left with exponents when we go to calculate the winning percentage. While simplifying the equations will generally cost us some theoretical and a tiny bit of empirical accuracy, it will gain us ease of calculation. Replacing RPG^.282 with a linear equation wouldn’t really make the calculation any easier, but more importantly I don’t think it would result in an interesting alternative methodology to estimate W%. It would just result in a very slightly easier to calculate, less accurate Pythagenpat equation.

I previously wrote the general Pythagorean relationship as:

W% = R^x/(R^x + RA^x)

but note that we could equivalently define win ratio (W/L = WR) as:

WR = RR^x where RR = run ratio = R/RA

I will alternate between these two ways of writing the equation depending on whichever is most convenient for what we’re trying to do. In this case, I want to see what happens if we get rid of the exponent. The approach I will take is to replace the current function with a simplified function that produces the same result for a particular point. Of course we cannot replace the function with another that will produce the same results at all points, or even expect to find one that would produce the same results at multiple points. But we will be able to find a function that produces the same result at a given point.

Mathematically, this will the tangent line to the curve at that point. At that point, the tangent line intersects the curve and has the same slope as the curve. We will determine the slope by differentiating the function, and we will then determine the tangent line using the point-slope equation for the line as a starting point (to me, this is the most intuitive way to write the equation of a line, and if necessary we can simplify later). The point-slope equation of a line is:

y – y1 = m(x – x1)

where x1 and y1 and the x and y values for the point in question, and m is the slope of the curve at x1.

I’m going to switch to referring to the Pythagorean exponent as “a”, so that it doesn’t get confused with x, our independent variable (which is run ratio). So if we want the tangent line for the equation WR = RR^a, we first differentiate with respect to run ratio to get:

dWR/dRR = a*RR^(a – 1)

Now we just need to determine x1 and y1. Since we are going to be applying simplified win estimation formulas across the entire spectrum of possible team performance, it makes the most sense to look at a team with R = RA, that we expect to have a .500 W%. Picking the average will likely result in the most accurate simplified equation over the entire spectrum of teams.

Of course, by simplifying the equation, we will lose accuracy (at least when the result of our simplified equation is compared to the “parent” equation – we hope in this case that the Pythagorean form is more accurate or else the entire premise of Akousmatikoi win estimators is moot). However, the simplified equation will match the parent equation precisely at chosen point, and will produce very similar results near the chosen point, so picking a point in the center of the distribution should maximize accuracy.

So, if R = RR, then RR equals one, and so our slope is simply equal to a, which is the Pythagorean exponent. Our x value is RR, which is 1, and our y is the WR corresponding to a RR of 1, which is 1 for any value of a as WR = RR^a. So in point slope form:

y – 1 = a*(x – 1)

which can simplify to

y – 1 = a*x – a

y = a*x – a + 1

Remembering what y and x represent in this case:

WR = a*RR – a + 1

For a fixed Pythagorean exponent a = 2:

WR = 2*RR – 2 + 1 = 2*RR - 1

This relationship suggests that if a team scores 10% more runs than it allows, it should win 20% more games than it loses. In the 1984 Baseball Abstract, Bill James wrote:

Another method that I have never tested but which I suspect would work as well as the others would be just to “double the edge”; that is, if a team scores 10% more runs than their opponents, they should win 20% more games than their opponents. If they score 1% more runs, they should win 2% more games. That method would probably work as well or better than the Pythagorean approach.

To my knowledge that’s the extent of James’ writings on this subject, so I can’t say whether he either explicitly or implicitly inferred “double the edge” from the Pythagorean formula, or whether he came across it some other way. Either way, it can be directly related back to his own Pythagorean method.

If WR = a*RR – a + 1, and we already know that by definition W% = WR/(WR + 1), then we can convert this into a W% estimate as:

W% = (a*RR – a + 1)/(a*RR – a + 1 + 1) = (a*RR – a + 1)/(a*RR – a + 2)

For the special case of a = 2, this becomes:

W% = (2*RR – 2 + 1)/(2*RR – 2 + 2) = (2*RR – 1)/(2*RR) = 1 – 1/(2*RR) = 1 – 1/(2*R/RA)

= 1 - RA/(2*R)

This special case was noted by Bill Kross, and got a brief callout in The Hidden Game of Baseball. Kross also noticed that this method would not produce the same result for teams that had inverse runs and runs allowed. A team that scores 5 and allows 4 runs would have an estimated W% of 1 - 4/(2*5) = .600, but a team that scores 4 and allows 5 would have an estimated W% of 1 – 5/(2*4) = .375.

So Kross proposed that that for the case in which runs scored < run allowed, the W% would be estimated as R/(2*RA), which would produce 4/(2*5) = .400 for the the team scoring 4/allowing 5. Not only is it satisfying to get a consistent result for the two sides of the same coin, this modification significantly improves the accuracy when comparing empirically comparing estimated to actual W%s.

Expressing this inversion in terms of the general case above, in a case where R < RA, the estimated WR would be:

WR = 1/(a*1/RR – a + 1) = 1/(a/RR – a + 1)

and the W% would be:

W% = 1/(a/RR – a + 1)/(1/(a/RR – a + 1) + 1)

There are some ways to make that look nicer, but I don’t think any of them are sufficiently nice to bother with here. For the specific case when a = 2, Ralph Caola has suggested this formula as a clean way to boil the Kross equations down to one line:

W% = (R - RA)/(R + RA + ABS(R - RA)) + .5

You might be reading this and objecting “I thought you were going to simplify the Pythagorean relationship, but nothing about the equation with all of those reciprocals above looks simpler”. That is true – other than the special case when a = 2 and the Kross equations apply, this is not an easier way to calculate an estimated winning percentage provided you have a modern calculator or computer. However, it is “simpler” mathematically in the sense that we have eliminated exponents. Of course, in so doing we have lost some accuracy, particular for extreme cases. Next time, instead of starting with run ratio, we’ll start with run differential and see what shakes out of Pythagorean and how it compares to methods that have been developed independently of Pythagorean.

Akousmatikoi Win Estimators, pt. 1: Pythagorean

This series will be a brief review of the Pythagorean methodology for estimating team winning percentage from runs scored and runs allowed, and will examine a number of alternative winning percentage estimators that can be derived from the standard Pythagorean approach. I call it a “review” because I will not be presenting any new methods – in fact, not only was everything I plan to cover discovered and published by other sabermetricians, but it is all material that I have already written about in one form or another. When recently posting old articles from my Tripod site, I saw how poorly organized the section on win estimators was, and decided that I should try to write a cleaner version that focuses on the relationship between the Pythagorean approach and other mathematical forms for win estimators. This series will start from the assumption that Pythagorean is a useful model; I don’t think this is a controversial claim but a full treatment would need to establish that before jumping into mathematical offshoots.

By christening his win estimator the “Pythagorean Theorem” due to the three squared terms in the formula reminding him of the three squared terms Pythagoras discovered defined the dimensions of right triangles, Bill James made it irresistible for future writers to double down with even more ridiculous names. I am sure any students of Greek philosophy are cursing me, but I am calling this the “Akousmatikoi” family of win estimators because Wikipedia informs me that Akousmatikoi was a philosophical school that was a branch of the larger school of Pythagoreanism based on the teachings of Pythagoras. A rival branch, the Mathematikoi school, was more focused on the intellectual and mathematical aspects of Pythagorean thought, which would make it a better name for my purposes, but even I think that sounds too ridiculous. I’ve also jumbled the analogy as James’ Pythagorean theorem is the starting point for the Akousmatikoi family of estimators but Pythagoras begat this school of philosophy, but not the other way around. Of course, James’ Pythagorean theorem really has nothing to do with Pythagoras to begin with, so don’t think too hard about this.

Before I get started, I want to make certain that I am very clear that I’m introducing nothing new and that while I will derive a number of methods from Pythagorean, the people who originally discovered and published these methods used their own thought processes and ingenuity to do so. They did not simply derive them from Pythagorean. I will try to namecheck them throughout the series, but will also do it here in case I slip up – among the sabermetricians who developed the methods that I will treat as Pythagorean offshoots independently are Bill Kross, Ralph Caola, and Ben Vollmayr-Lee.

I also want to briefly address the win estimators that are in common use that are not part of what I am calling the Akousmatikoi family. The chief one that I use is Cigol, which is my implementation of a methodology that starts with an assumed run distribution per game and calculates W% from there (I say “calculates” rather than “estimates” because given the assumptions about per game and per inning run distribution functions, it is a logical mathematical derivation, not an estimate. Of course, the assumptions are just that). Cigol is very consistent with the results of Pythagenpat for teams across a wide range of scoring environments, but is its own animal. There are also approaches based on regression that offer non-Akousmatikoi paths to win estimates. If you regress on run differential or run ratio, your results will look similar to Akousmatikoi, but if you take the path of Arnold Soolman’s pioneering work and regress runs and runs allowed separately, or you use logistic regression or another non-linear methodology, your results won’t be as easily relatable to the Akousmatikoi methods.

It all starts with Pythagorean, which Bill James originally formulated as:

W% = R^2/(R^2 + RA^2)

The presence of three squared terms reminded James of the real Pythagorean theorem for the lengths of the side of right triangle (A^2 = B^2 + C^2) and gave us the charmingly wacky name for this method of win estimation. James would later complicate matters by noting that a lower exponent resulted in a slight increase in accuracy:

W% = R^1.83/(R^1.83 + RA^1.83)

Later research by Clay Davenport and Keith Woolner would demonstrate that a custom exponent, varying by run environment, would result in better accuracy in extreme situations. Pete Palmer had long before demonstrated that his linear methods increased in accuracy when considering run environment; “Pythagenport” brought this insight to Pythagorean, which we’ll now more generally express as:

W% = R^x/(R^x + RA^x)

Where Pythagenport estimates x = 1.5*log(RPG) + .45, where RPG = (R + RA)/G

Davenport and Woolner stated that the accuracy of Pythagenport was untested for RPG less than 4. A couple years later, David Smyth had the insight that 1 RPG was a situation that could only occur if the score of each game was 1-0, and that such a team’s W% would by definition be equal to R/(R + RA). This implies that the Pythagorean exponent must be 1 when RPG = 1. Based on this insight, Smyth and I independently developed a modified exponent which was constructed as:

x = RPG^z

where z is a constant generally in the range of .27 - .29 (I originally published as .29 and have tended to use this value out of habit, although if you forced me to pick one value and stick to it I’d probably choose .28)

This approach produced very similar results to Pythagenport for the RPG ranges tested by Davenport and Woolner, and returned the correct result for the known case at RPG = 1. It has come to be called “Pythagenpat”.

Using Cigol, I tried to develop a refined formula for Pythagorean exponent using data for truly extreme temas. I loosened the restriction on requiring x = 1 when RPG = 1 to be able to consider a wider range of models, but I wasn’t able to come up with a version that produced superior accuracy with a large dataset of actual major league team-seasons to the standard Pythagenpat construction. My favorite of the versions I came up are below, which I won’t dwell on any longer but will revisit briefly at the end of the series. The first is a Pythagenpat exponent that produces a Pythagorean exponent of 1 at 1 RPG; the second is a Pythagorean exponent that does not adhere to that restriction.

z = .27348 + .00025*RPG + .00020*(R - RA)^2

x = 1.03841*RPG^.265 + .00114*RD^2

There are several properties of a Pythagorean construct that make it better suited as a starting point (standing in for the “true” W% function, if there could ever be such a thing) than some of the other methods we’ll look at. I have previously proposed a list of three ideal properties of a W% estimator:

1. The estimate should fall in the range [0,1]
2. The formula should recognize that the marginal value of runs is variable.
3. The formula should recognize that as more runs are scored, the number of marginal runs needed to earn a win increases.

As we move throughout this series, we will make changes to simplify the Pythagenpat function in some ways; in my notes I called it “flattening”, but that’s not a technical term. Basically, where we see exponents, we will try to convert into multiplication, or we will try to use run differential in place of run ratio. As we “flatten” the functions out, we will progressively lose some of these ideal properties, with the (usual) benefit of having simpler functions.

Throughout this series I will make sporadic use of the team seasonal data for the expansion era (1961 – 2019), so at this point I want to use this dataset to define the Pythagorean constants that we’ll use going forward. Rather than using any formulaic approach, I am going to fix x and z for this period by minimizing the RMSE of the W% estimates for the teams in the dataset. I will also use the fixed Pythagorean exponent of 2 throughout the series as it is easy to calculate, reasonably accurate, widely used, and mathematically will produce some pleasing results for the other Akousmatikoi estimators.

Using this data, the average RPG is 8.83, the value for x that minimizes RMSE is 1.847, and the z value that minimizes RMSE is .282. Note that if we used the average RPG to estimate the average Pythagorean exponent, we’d get 1.848 (8.83^.282), which doesn’t prove anything but at least it’s not way off.

Palmerian Park Factors

The first sabermetrician to publish extensive work on park effects was Pete Palmer. His park factors appeared in The Hidden Game of Baseball and Total Baseball and as such became the most-widely used factors in the field. They continue in this role thanks to their usage at Baseball-Reference.com.

Broadly speaking, all ratio-based run park factors are calculated in the same manner. The starting point is the ratio of runs at home to runs on the road. There are a number of different possible variations; some methods use runs scored by both teams, while others (including Palmer’s original methodology in The Hidden Game use only the runs scored by one of the teams (home or road)). The opportunity factor can also vary slightly; many people just use games, which is less precise than innings or outs. The variations are mostly technical rather than philosophical in nature, so they rarely get a lot of attention. Park factor calculations are accepted at face value to an extent that other classes of sabermetric tools (like run estimators) are not.

Among park factors in use, the actual computations in Palmer’s are the most unique, so I thought it would be worthwhile to walk through his methodology (as stated in Total Baseball V) and discuss its properties.

In the course of this discussion, I will primarily focus on the actual calculations and not the inputs. Where Palmer has made choices about what inputs to use, how to weight multiple seasons, and the like, I will not dwell, as I’m more interested in the aspects of the approach that can be applied to one’s own choice of inputs. Palmer uses separate park factors for batting and pitching (more on this later); I’ll focus on the batting ones here.

Palmer generally uses three years of data, unweighted, as the basis for the factors. There are some rules about which years to use when teams change parks, but those are not relevant to this discussion. The real meat of the method starts by finding the total runs scored and allowed per game at home and on the road, which I’ll call RPG(H) and RPG(R).

i (initial factor) = RPG(H)/RPG(R)

I will be using the 2010 Colorado Rockies as the example team here, considering just one year of data to keep things simple. Colorado played 81 games both home and away, scoring 479 and allowing 379 runs at home and scoring 291 and allowing 338 on the road. Thus, their RPG(H) = (479 + 379)/81 = 10.593 and RPG(R) = (291 + 338)/81 = 7.765. That makes i = 10.593/7.765 = 1.364 (I am rounding to three places throughout the post, which will cause some rounding discrepancies with the spreadsheet from which I am reporting the results).

The next step is to adjust the initial factor for the number of innings actually played rather than just using games as the denominator. This step can be ignored if you begin with innings or outs as the denominator rather than using games. The Innings Pitched corrector is:

IPC = (18.5 - Home W%)/(18.5 - (1 - Road W%))

Palmer explains that 18.5 is the average number of innings batted per game if the home team always bats in the ninth inning. Teams that win a higher percentage of games at home bat in less innings due to skipping the bottom of the ninth. The IPC seems to assume that in all games won by the home team, they do not bat in the bottom of the ninth.

Colorado was 52-29 (.642) at home and 31-50 on the road (.383), so their IPC is:

IPC = (18.5 - .642)/(18.5 - (1 - .383)) = .999

The initial factor is divided by the IPC to produce what the explanation refers to as Run Factor:

RF = i/IPC

For the Rockies, RF = 1.364/.999 = 1.366

The next step is the Other Parks Corrector (OPC). The OPC “[corrects] for the fact that the other road parks’ total difference from the league average is offset by the park rating of the club that is being rated.” The glossary explanation may be confusing, but the thought process behind it is pretty straightforward--a team’s own park makes up part of the league road average, but none of the team’s own road games are played there. Without accounting for this, all parks would be estimated to be more extreme than they are in reality.

The OPC is figured in the same manner as I do it in my park factors; I borrowed it from Craig Wright’s work without realizing that Palmer had done the same mathematical operation, but its derivation is fairly obvious and the equivalent appears in multiple park factor approaches. Let T equal the number of teams in the league:

OPC = T/(T - 1 + RF)

Basically, OPC assumes a balanced schedule, so each team’s schedule is made up half of games in its own park (hence the use of RF) and half in the other parks, of which there are T - 1. For a sixteen team league (and thus Colorado 2010):

OPC = 16/(16 - 1 + 1.366) = .978

The next step is to multiply RF by OPC, producing scoring factor:

SF = RF*OPC

For Colorado, SF = 1.366*.978 = 1.335

If all you were interested in was an adjustment factor of the park effect’s on scoring, rather than specific adjustments for the team’s batters and pitchers, this would be your stopping point. The scoring factor is the final park factor in that case, and with the exception of the Innings Pitched Corrector, it is equivalent to the approach used by Craig Wright, myself, and many others. (My park factors are then averaged with one to account for the fact that only half of the games for a given team are at home, but that only obscures the fact that the underlying approach is identical, and Palmer accounts for that consideration later in his process).

It is when the other factors are adjusted for that the math gets a little more involved. The first step is to calculate SF1, which is an adjustment to scoring factor:

SF1 = 1 - (SF - 1)/(T - 1)

For the 2010 Rockies:

SF1 = 1 - (1.335 - 1)/(16 - 1) = .978

While I am writing this explanation, I must stress that it is a walkthrough of another person’s method. I cannot fully explain the thought process behind it and justify every step. I decided to include that disclaimer at this point because I don’t understand what the purpose of SF1 is, as it is mathematically equivalent to OPC. Why it needed to be defined again and in a more obtuse way is beyond me.

In any event, the purpose of SF1 is to serve as a road park factor. If a team plays a balanced schedule and we take as a given that the overall league park factor should be equal to one, but we determine that its own park has a PF greater than one (favors hitters), then it must be the case that the road parks they play in have a composite PF less than one. That’s the function of the OPC, and of SF1. For the rest of this post, I will refer to OPC rather than SF1 when it is used in formulas.

Palmer uses separate factors for batters and pitchers to account for the fact that a player does not have to face his teammates. By doing so, Palmer’s park factors make their name something of a misnomer as they adjust for things other than the park. (One could certainly make the case that the park factor name is constantly misapplied in sabermetrics, as we can never truly isolate the effect of the park, and the sample data we have is affected by personnel and other decisions. Palmer takes it a step further, though, by accounting for things that have nothing to do with the park.) The park effect is generally stronger than the effect of not facing one’s own teammates, since a team plays in its park half the time but only misses out on facing its own pitchers 1/T percent of the time assuming a balanced schedule.

One can argue about the advisability of adjusting for the teammate factor at all, and if so it certainly is debatable whether it should be included in the park factor or spun off as a separate adjustment. I would find the later to be a much better choice that would result in a much more meaningful set of factors (both for the park and teammates), but Palmer chose the former.

The separate factors are calculated through an iterative process. One must know the R/G scored and allowed at home and away for the team (I’ll call these RG(H), RG(R), RAG(H), and RAG(R) for R/G at home, R/G on the road, RA/G at home, and RA/G on the road respectively). Additionally, one must know the average RPG for the entire league (which I’ll call 2*N, since I often use N to denote league average runs/game for one team). For the Rockies, we can determine from the data above that RG(H) = 5.913, RG(R) = 3.593, RAG(H) = 4.679, RAG(R) = 4.173, and N = 4.33.

The iterative process is used to calculate a team batter rating (TBR) and a team pitcher rating (TPR, not to be confused with Total Player Rating, another Palmer acronym). These steps are necessary since the strength of each unit cannot be determined wholly independently of the other. Just as the total impact of park goes beyond the effect on home games and into road games (necessitating the OPC, but not being of strong enough magnitude to swamp over the home factor), so the influence of the two units on another are codependent.

The first step of the process assumes that the pitching staff is average (TPR = 1). Then the TBR can be calculated as:

TBR = [RG(R)/OPC + RG(H)/SF]*[1 + (TPR - 1)/(T - 1)]/(2*N)

For Colorado:

TBR = [3.593/.978 + 5.913/1.335]*[1 + (1 - 1)/(16 - 1)]/(2*4.33) = .936

The first bracketed portion of the equation adds the teams’ adjusted (for park) R/G at home and on the road, using the home (SF) or road (OPC) park factor as appropriate. The second set of brackets multiplies the first by the adjustment for not facing the team’s pitching staff. The difference between the team pitching and average (TPR - 1) is divided by (T - 1) since playing a true balanced schedule, the team would only face those pitchers in 1/(T - 1) percent of the games anyway. If the team’s pitchers are above average (TPR < 1), then the correction will increase the estimate of team batting strength.

Then the entire quantity is divided by double the league average of runs scored per game by a single team. This may seem confusing at first glance, but it is only because the first bracketed portion did not weight the home and road R/G by 50% each. The formula could be written as the equivalent:

TBR = [.5*RG(R)/OPC + .5*RG(H)/SF]*[1 + (TPR - 1)/(T - 1)]/N

In this case, it is much easier to see that the first bracket is the average runs scored per game by the team, park-adjusted. Dividing this by the league average results in a very straightforward rating of runs scored relative to the league average. Both TBR and TPR are runs per game relative to the league average, but in both cases they are constructed as team figure/league figure. This means that the higher the TBR, the better, with the opposite being true for TPR.

The pitcher rating can then be estimated using the actual TBR just calculated, rather than assuming that the batters are average:

TPR = [RAG(R)/OPC + RAG(H)/SF]*[1 + (TBR - 1)/(T - 1)]/(2*N)

For the Rockies, this yields:

TPR = [4.173/.978 + 4.679/1.335]*[1 + (.936 - 1)/(16 - 1)]/(2*4.33) = .894

The first estimate is that the Rockies batters, playing in a neutral park and against a truly balanced schedule, would score 93.6% of the league average R/G. Rockie pitchers would be expected to allow 89.4% of the league average R/G. At first when I was performing the sample calculations, I thought I might have made a mistake since the Colorado offense was evaluated as so far below average, but I was forgetting that they would appear quite poor using a one-year factor for Coors Field. My instinct as to what Colorado’s TBR should look like was informed by my knowledge of the five-year PF that I use.

Now the process is repeated for three more iterations, each time using the most recently calculated value for TBR or TPR as appropriate. The second iteration calculations are:

TBR = [3.593/.978 + 5.913/1.335]*[1 + (.894 - 1)/(16 - 1)]/(2*4.33) = .929

TPR = [4.173/.978 + 4.679/1.335]*[1 + (.929 - 1)/(16 - 1)]/(2*4.33) = .893

Repeating the loop two more times should ensure pretty stable values. Theoretically, there’s nothing to stop you from setting up an infinite loop. I won’t insult your intelligence by spelling the second pair of iterations suggested by Palmer, but the final results for Colorado are a TBR of .929 and a TPR of .993.

So far, all we’ve done is figured teammate-corrected runs ratings for team offense and defense. Our estimate of the park factor remains stuck back at scoring factor. All that’s left now is correcting scoring factor for 1) the teammate factor and 2) the fact that a team plays half its games at home and half on the road. This will require two separate formulas--a batter’s PF (BPF) and a pitcher’s PF (PPF), which are twins in the same way that TBR and TPR are:

BPF = (SF + OPC)/[2*(1 + (TPR - 1)/(T - 1)]

PPF = (SF + OPC)/[2* (1 + (TBR - 1)/(T - 1)]

For Colorado:

BPF = (1.335 + .978)/[2*(1 + (.893 - 1)/(16 - 1)] = 1.165

PPF = (1.335 + .978)/[2*(1 + (.929 - 1)/(16 - 1)] = 1.162

The logic behind the two formulas should be fairly obvious at this point. Again, the multiplication by two in the denominator arises because the two factors in the numerator were not weighted at 50% each. You can write the formulas as:

BPF = (.5*SF + .5*OPC)/[1 + (TPR - 1)/(T - 1)]

PPF = (.5*SF + .5*OPC)/[1 + (TBR - 1)/(T - 1)]

Here, you can see more clearly that the numerator averages the home park factor (SF) and the road park factor (OPC). The numerator could be your final park factor if you wanted to account for road park but didn’t care about teammate effects. The denominator is where the teammate effect is taken into account, and in the same manner as in TBR and TPR. If TPR > 1, BPF goes down because the batters did not benefit from facing the team’s poor pitching. If TBR > 1, PPF goes down pitchers benefitted from not facing their batting teammates.

If one does not care to incorporate the teammate effect, then Palmer’s park factors are pretty much the same as any other intelligently designed park factors. This should not come as a surprise, because as with many implementations of classical sabermetrics, Palmer’s influence is towering. The iterative process used to generate the batting and pitching ratings is pretty clever, and incorporates a real albeit small effect that many of us (myself included) often gloss over.

Musings on Positional Adjustments

This is an old post that I never published. It is sort of an attempt to justify why I use offensive positional adjustments, which is an even more dated position today than it was when I wrote it. In re-reading it, though, I thought my comments about zero-level defense were at least somewhat pertinent (if not particularly insightful) given Bill James' current effort at developing "Runs Saved Against Zero".

This post is not intended to be a comprehensive discussion of the issue of position adjustments; it will just quickly sketch out a system to classify adjustments and then I’ll offer a few of my opinions on them. There is a lot more that could be said and most of it could and has been said more eloquently by others.

The most important technical distinction between position adjustments (which I’ll shorten to PADJ sometimes) is which type of metric is used to set them--offensive or defensive. This distinction is well-known and gets a lot of attention. One that is talked about less is the difference between explicit and implicit position adjustments, and while people who get their hands dirty with various rating systems are well aware of implicit position adjustments, the average reader presented with a metric might gloss over them.

Explicit position adjustments are obvious and are acknowledged as being position adjustments. The first well-known example of their usage was in Pete Palmer’s linear weights system. They have also been used in VORP, just about every implementation of WAR, and many other metrics.

Implicit position adjustments usually crop up in the work of Bill James, although there are other metrics out there that utilize them. An implicit position adjustment is not really implicit in the truest sense of the word--they are obviously position adjustments if you look at them and consider what their function is. James likes to hide them in his fielding systems.

James’ metrics have always attempted to measure absolute wins and losses. I’ve always maintained that this is a fool’s errand, and that absolute wins and losses only make sense on the team level, not the player level. Most sabermetricians are in general agreement on this, and construct systems that are built to yield results against some baseline.

This is especially true for defensive metrics, whether for pitching for fielding. Absolute metrics (such as runs created) are tempting to apply to individual batters because there is a theoretical minimum on the number of runs a player can create (zero, of course), and such a performance represents the worst possible performance. There is no such cap on the poor performance of a defense; a team could allow an infinite number of runs. The only real cap on the poor performance of an individual fielder is the number of balls that are hit to the locations on the field that fall under his responsibility.

As such, it is impossible to develop a true zero baseline metric to evaluate pitchers or fielders (one can certainly argue that it’s impossible for batters as well, but the existence of the theoretical floor makes it undeniably more tempting). You have to start by comparing to a non-zero baseline (average being the most straightforward), but the problem is compounded for fielders by the fact that it’s also impossible to directly compare fielders at different positions. The fielding standards, be it in fielding average, range factor, or more complex methods vary wildly from one position to another. While all fielders have the same objective (record outs and prevent the opponent from scoring), the primary ways in which fielders at different positions contribute to the common goal are very different.

That pretty much leaves comparing a player to the average fielder at his position as the only viable starting point for the developer of a fielding metric. As is, the results are not satisfactory for inclusion in a total value metric, because they implicitly assume that an average fielder at any position is equal in value to an average fielder at any other position.

There is no one with any degree of baseball knowledge that believes this to be true. Everyone agrees that an average shortstop is harder to find than an average first baseman--that is, the pool of available players that can adequately field shortstop is much smaller than the pool of adequate available first basemen. This basic truth is sometimes obfuscated by silly hypotheticals (i.e. “if you didn’t have a catcher, every pitch with a runner on base would be a passed ball” and “without a first baseman, it would be nearly impossible to convert a groundball into an out”), but serious people agree on this.

So what is one to do about this problem? You have to do something--you cannot have a functioning estimate of total value that pretends first basemen and shortstops are equal in fielding value. The easiest answer is a position adjustment.

While James attempts to express all of his value metrics relative to an absolute baseline, he of course can’t pull off a clean implementation. His solution, in both his early 1980s Defensive Winning Percentage and his more recent Win Shares, is to develop a fielding winning percentage for each position and convert this to wins and losses (the terminology and procedure is a little different in Win Shares but that’s a long story).

To make the conversion to from a rate to a total of wins and losses, James multiplies by a number of games for which each position is assumed to be responsibility. Positions on the left side of the defensive spectrum are assigned less responsibility than those on the right side…and thus this is an implicit position adjustment.

In pointing this out, I don’t mean to suggest that James is in any way dishonest in describing his systems--the assigned games are clearly defined in the system and aren’t hidden. The characterization I’ve offered of these adjustments as “implicit” is therefore not really accurate. The real difference between James-style position adjustments and the ones I’ve defined as “explicit” is that explicit adjustments either add or subtract a set number of runs dependent upon a player’s position or apply a different baseline to their rate in converting to a value stat.

The other major characteristic that defines a position adjustment’s type is whether it is an offensive PADJ or a defensive PADJ. The categories are not black and white--many positional adjustments incorporate subjective weighting of various factors, which could include offensive performance by players at a position, the range of offensive performance, the performance of fielders at multiple positions, comparisons of average salary as a stand-in for how teams value players, subjective corrections that the developer feels better matches the results of the system to common sense--but usually the primary basis can be identified as either offensive or defensive.

Offensive position adjustments have fallen out of favor recently, although there are still some people using them (including me). The offensive PADJ originated with Pete Palmer, who used it as part of his linear weights system. The other most prominent use came in Keith Woolner’s VORP.

Defensive positional adjustments are a more recent phenomenon, but are key to both the Fangraphs and Chone WAR methodology. Tango Tiger was the driving force behind their development, and Chone has also done his own research to establish the adjustments for his WAR.

Before deciding how to construct a position adjustment, it’s a good idea to take a step back and figure out why you need a position adjustment at all. Taking the reason behind your metric for granted is a path to just slapping numbers around indiscriminately and failing to model baseball reality. From my perspective, the only real reason that a PADJ is necessary is that it is essentially impossible to measure a player’s fielding value independent of his position. Therefore, one has to have a way of comparing the value of fielding performances across positions--a position adjustment.

A common misperception regarding all position adjustments among people not that well-versed in sabermetrics is that they provide a bonus “just for playing the position”. While I suppose that might be technically true in the sense of calculation, the underlying need for such an adjustment is discussed above. If one does not believe in applying a positional adjustment, and accepts the use of defensive metrics baselined to an average fielder at the position, then they must conclude that, as a group, the most valuable players are those at left-side of the spectrum positions. Or, in other words, that the overall average value of players at a given position is strictly a function of their aggregate offensive production.

It is possible to complicate the question of position adjustments by talking about baselines (particularly replacement level) and other considerations, but at the heart of the issue is the need to compare the value of a shortstop -5 fielding runs relative to an average shortstop to a third baseman +10 runs relative to an average third baseman to an average first baseman.

Such a viewpoint suggests that a defensive PADJ is the way to go, since the sole reason for needing the adjustment is consideration of defense. So while the overwhelming positive of a defensive PADJ is that it is defined in the terms that necessitate the entire endeavor, it also carries a few negatives.

One is the difficulty of accurately evaluating fielding value, even within the constraints of one’s own position. While it is quite possible that any biases or methodological errors will balance out when aggregated over a large population of players, it would nonetheless be more comforting to begin from metrics in which one had a great deal of confidence.

Another key issue is that the pool of players who log times at multiple positions, while relatively large when comparing similar position groups (particularly outfielders, but also middle infielders, corner infielders, etc.), there is a much smaller available sample of players who play very different positions, at least in the same or adjacent seasons. And catcher? Forget about it--Bill James even left catcher off of the defensive spectrum due to the difficulty of comparing it directly to the positions whose occupants stand in fair territory.

Players that move positions introduce all kinds of selective sampling issues as well. Consider the problem comparing positions where left-handers are de facto ineligible, and the fact that players moved off a position are more likely to have been stopgaps. For a more complete discussion of these issues (and an all-around good discussion of PADJ issues), see Colin Wyers’ article at Baseball Prospectus.

Thus, to avoid strange conclusions, defensive position adjustments are always going to require a little subjective massaging. That’s not necessarily a bad thing--the construction of any metric requires subjective decisions made on the part of the developer--but it makes them inherently high maintenance.

Of course, offensive position adjustments are best employed with a measure of caution as well. The pros of offensive adjustments are that they are very easy to work with. Offensive statistics are more reliable than fielding statistics, require much more basic data to calculate, and are available throughout the entire history of the game. Rather than having to compare performance of players across positions, one can at least start by simply looking at the average performance of all players at a particular position.

An offensive PADJ implicitly assumes that teams allocate their talent in such a manner that the average player at any position is equal to the average player at any other position--alternatively stated, that the offensive value gap between positional averages is equal to the defensive value gap. This is certainly never 100% truly the case for any sample period, particularly for single years. Offensive PADJs based on one year of data or other short stretches should be viewed with a great deal of skepticism.

Another problem lurking is what Wyers, in the linked article, refers to as the “Mays problem”--the existence of supremely talented players that excel at both hitting and fielding. Such players might be superstars at any position (ignoring handedness and other impediments), even first base, thanks to their hitting alone but are able to handle the defensive rigors of right-side defensive spectrum positions. While more ordinary players offer a package of offensive and defensive skills that limits their possible fielding positions commensurate to their offensive production, these players are playable anywhere. There are also potential issues with the very worst players at a position.

The Mays problem skews offensive positional averages, so Wyers proposes using an alternative offensive PADJ that adjusts the overall positional average for the gap between the upper and lower median of observed performance at the position. This approach (and other similar algorithms that could be offered) is novel but involves subjective choices similar to those necessitated by defensive PADJs.

The offensive PADJ will surely fail at lower levels of baseball thanks to the Mays problem--the best high school players, for instance, are often the cleanup hitter, ace pitcher, and center fielder or shortstop when not pitching. Such all-around stars are also more common in college ball or in earlier, less developed major leagues than they are in the modern major leagues with their high overall quality of play and relatively strong competitive balance. An offensive PADJ approach will surely break down at those low levels without serious alterations.

There are other relevant issues to discuss with respect to position adjustments, such as their relationship to replacement level and the manner in which they are applied (That is, if they should be used to change the baseline to which performance is compared or if they should be assigned as a lump sum based on playing time. The possible answers to this question is also closely tied to one’s choice of offensive or defensive adjustment), but those will have to wait for some other time.

End of Season Statistics, 2020

At first I wasn’t sure if I would go through the trouble of putting together year-end statistics for a 60-game season. This reticence was due entirely to the small sample size, and not to any belief that the season itself was “illegitimate” or an “exhibition” or any of the other pejoratives that have been lobbed against it. On the contrary, I was thrilled that we got a season at all given the immense social and political pressure aligned against any attempt by human beings to engage in voluntary economic activity. 2020 has provided a glimpse into the near future, and it is indeed a boot stamping a human face forever.

But I digress. 60 games is a very small sample size when one is accustomed to 162, but the reason I decided to do this exercise anyway was simple: I wanted to better understand what happened in those 60 games. I did decide to cull some of the categories I usually look at, mostly to reduce the amount of effort necessary on my part to produce the statistics. Additionally, I had to make some changes to park factors and comparisons to league average which I will describe below.

It is common sabermetric practice to compare player performance to the league average. This is always a simplification of reality, and there are alternative options. Bill James argued in the original Historical Baseball Abstract that the true context for a player’s performance was his team’s games. This is true as it goes, and it conveniently allowed James to sidestep the issue of developing a complete set of historical park factors. On the other hand, we understand that the run environment for a given team is shaped by the quality of their hitters and pitchers. A batter on a team with excellent pitching will benefit in any kind of comparison to average from his teammate’s suppression of runs, but so would a player from another team were they to switch places. Since the usual goal of these exercises is to facilitate context-neutral (this is a loaded term, as there are many shades to a claim of context neutrality which I will not address here) comparisons between players, we typically decide that it is preferable to place the player’s performance in a wider league context, even if that context now includes the events of thousands of games in which the player in question did not participate.

We could split the difference, and one could argue that perhaps we should. We could build a custom “league context” for each team based on their schedule. Were the schedule perfectly balanced, this would not be necessary; alas, the general trend in MLB has been for schedules to become more unbalanced over time. We could, in a typical season, construct a context in which to evaluate Indians players in which 19/162 of the non-Indians portion consists of the Twins, and another 19/162 for each of the Royals, White Sox, and Tigers, and 6/162 for the Reds, and 7/162 for the Yankees, and 6/162 for the Orioles, etc. based on the number of games each team actually plays with each opponent. This is opposed to the league average, in which we simply compare to the entire AL, which implicitly assumes balance and ignores the existence of interleague games altogether.

I am not advocating for all of this complexity, and the approach I just sketched out is insufficiently refined to work in practice. The point I’m trying to make is that the league context is not exactly right, but it is a useful approximation, and with a semi-balanced schedule it makes sense.

When does it not make sense? In 2020, when there is not any semblance of a balanced schedule. The games played by teams in the AL Central bear no relation to those played by teams in the AL East or the AL West, because there are no games or opponents in common. To compare to the AL or NL average in 2020 is not a useful simplification – it is an outright abrogation of any attempt to accurately model the world as it is.

Thus I will be replacing the traditional AL/NL breakdown of the statistics with an East/Central/West breakdown. All comparisons to league average will compare to one of the three divisions, rather than the two leagues. Of course, 2/3 of any given team’s games were against their league brethren and only 1/3 against teams from the other circuit, so this is still a simplification of reality – but no more so than the use of league average in a normal season with an unbalanced schedule. In fact, in a way of looking at things which in my opinion is damning to the wildly unbalanced schedule used in a typical season, teams played close to the same share of their games against their intra-”league” opponents than they normally do (for this example, treating the NL Central as intraleague opponents):

Of course, there are problems associated with using the three divisions as leagues for the purpose of statistical comparisons. The big one is that we all know that league quality is not necessarily equal, or even close to equal, between the AL and NL; even less so as you divide the teams further into E/C/W, partly due to making the units even smaller. I ignore this when dealing with AL/NL; for instance, in ranking players by runs above average, I don’t try to account for league strength, and I’ve also ignored it here. This is a distortion – if we for the sake of argument assume that the Central is weaker than the East, a player who is +10 runs relative to the Central would be less valuable in a truly context-neutral comparison than one who is +10 runs relative to the East. This goes for comparisons to replacement level as well, as one of the assumptions of replacement level comparisons is that a replacement player can be obtained at zero marginal cost and thus is equally available to any team in MLB.

Usually I gloss over these simplifying assumptions without discussion; I wanted to call them out here because the non-traditional approach makes the simplifications more glaring. In short, I will pretend there are three leagues for all calculations that require the use of league average, but I will still group the reports into AL/NL.

The other thorny issue for 2020 is park factors. One complication is hinted at by the discussion above regarding balanced schedules; what happened in games at Yankee Stadium in 2020 is completely irrelevant to the park factor for Safeco Field. The set of parks that comprise road games for any team in 2020 is very different than that which fed their historical park factor calculations.

But you can also go crazy trying to be too precise on a question like this, so I have gone for what I consider to be a less than optimal but defensible approach. It is highly subjective, but it makes sense to me, and the whole purpose of these reports is for me to calculate the statistics that I want to see – if I wanted someone else’s ideal statistics, I could save a lot of effort and find that in other places.

The first step in estimating park factors was to choose a home/road runs ratio to use as a starting point for each park. In so doing, I completely threw out 2020 data except when it was absolutely necessary (the new park in Texas and the Buffalo park used by the Blue Jays). This was simply to avoid having to deal with not just a different number of “home” and “road” games by team, but also the issue of seven-inning games, artificially inflated runs totals from extra inning games, and the like. Hopefully these are just temporary issues, although it seems there is momentum for the scourge of the latter infecting the game on a permanent basis. Should they become permanent issues, I will think of solutions – but it’s not worth it for this 60 game look.

So what I’ve done is for each extant major league park, I’ve used up to four years of data from 2016-2019 (up to because some parks didn’t exist for the entire span) and calculated the ratio of home RPG to road RPG. Then I have regressed this ratio. The regression coefficients were based on an updated study I did of the slope of the linear regression equation using home/road RPG ratios from 1-5 years of data as the independent variable and the home/road RPG ratio for the subsequent season as the dependent variable. This is similar to how the regression coefficients I have used in the past were derived, but the previous ones were only rules of thumb rather than the actual coefficients, and were based on a study by MGL and not my own work. The results are broadly similar, although they give slightly less weight to 4-5 years of historical data. I will do a proper write-up of the study and provide the dataset at a later date, but for now it must suffice to say that I used those results to come up with an equation based on games which is .1364*ln(G/162) + .5866 (the rounded results I’m actually going to apply based on count of seasons is .59 for one year, .68 for two years, .74 for three years, and .78 for four years).

Let me walk through the park factor calculations for the Blue Jays. They played 26 games at home, which had a 11.35 RPG, and 34 on the road, which had a 9.38. So their initial home/road run ratio is 1.209, but over 60 games we only weight that at .1364*ln(60/162) + .5866 = .451 (of course, their actual ratio should be given even less credibility because the equation inherently assumes a 50/50 split of home and road games). So the run ratio we use as a starting point for their park factor is 1.209*.451 + 1.00*(1 - .451) = 1.094.

To finish it off for application to full season statistics, we need to take halve the difference between the calculated ratio and 1.0. Well, usually we halve it, since half of teams’ games are at home or close enough. But in 2020 there was enough weirdness in this regard (especially for the Blue Jays) that I used the team’s actual percentage of home games. In the Blue Jays case this is 26/60, so their final park factor is 1.094*26/60 + 1*34/60 = 1.04

There are a couple things I could have done to refine this a little, but when rounding to two decimal places they don’t have a huge impact. One would be to use something other than 1.00 as the “road” park factor, and another would be to ensure that the average for each of the E/C/W is 1.00, or doing the latter only as a shortcut. Since they round to 1.00 for each of the E/C/W when rounded to two places, that’s close enough for me. We also could have also used inning as a denominator rather than games, but it’s more work than I’m willing to put in for analyzing a sixty-game season.

All data was gathered from various pages on Baseball-Reference. The League spreadsheet is pretty straightforward--it includes league totals and averages for a number of categories, most or all of which are explained at appropriate junctures throughout this piece. The advent of interleague play has created two different sets of league totals--one for the offense of league teams and one for the defense of league teams. Before interleague play, these two were identical. I do not present both sets of totals (you can figure the defensive ones yourself from the team spreadsheet, if you desire), just those for the offenses. The exception is for the defense-specific statistics, like innings pitched and quality starts. The figures for those categories in the league report are for the defenses of the league's teams. However, I do include each league's breakdown of basic pitching stats between starters and relievers (denoted by "s" or "r" prefixes), and so summing those will yield the totals from the pitching side. I have broken this down by E/C/W and A/N/MLB.

I added a column this year for “ActO”, which is actual (rather than estimated) outs made by the team offensively. This can be determined from the official statistics as PA – R – LOB. I have then replaced the column I usually show for league R/G (“N”) with R/9, which is actually R*27/ActO, which is equivalent to R*9/IP. This restates the league run average in the more familiar per nine innings. I’ve done the same for “OG”, which is Outs/Game but only for those outs I count in the individual hitter’s stats (AB – H + CS) ,“PA/G”, which is normally just (AB + W)/G, and “KG” and “WG” (normally just K/G and W/G) – these are now “O/9”, “PA/9”, still “KG”/”WG” and are per 27 actual outs.

The Team spreadsheet focuses on overall team performance--wins, losses, runs scored, runs allowed. The columns included are: Park Factor (PF), Winning Percentage (W%), Expected W% (EW%), Predicted W% (PW%), wins, losses, runs, runs allowed, Runs Created (RC), Runs Created Allowed (RCA), Home Winning Percentage (HW%), Road Winning Percentage (RW%) [exactly what they sound like--W% at home and on the road], R/9, RA/9, Runs Created/9 (RC/9), Runs Created Allowed/9 (RCA/9), and Runs Per Game (the average number of runs scored an allowed per game). For the offensive categories, runs/9 are based on runs per 27 actual outs; for pitching categories, they are runs/9 innings.

I based EW% and PW% on R/9 and RA/9 (and RC/9 and RCA/9) rather than the actual runs totals. This means that what they are not estimating what a team’s winning percentage should have been in the actual game constructions that they played, but what they should have been if playing nine inning games but scoring/allowing runs at the same rate per inning. EW%, which is based on actual R and RA, is also polluted by inflated runs in extra inning games; PW%, which is based on RC and RCA, doesn’t suffer from this distortion.

The runs and Runs Created figures are unadjusted, but the per-game averages are park-adjusted, except for RPG which is also raw. Runs Created and Runs Created Allowed are both based on a simple Base Runs formula. The formula is:

A = H + W - HR - CS

B = (2TB - H - 4HR + .05W + 1.5SB)*.76

C = AB - H

D = HR

Naturally, A*B/(B + C) + D.

In order to actually estimate runs scored accurately for 2020, one would need to use .83 as the B multiplier. When I first saw the discrepancy between actual and estimated runs, I was alarmed; the formula isn’t perfect, of course, but usually it doesn’t vary much from year to year. Then I realized that the biggest reason for this was likely the distortion caused by extra inning games. As such, I’ve kept with my standard formula, but we won’t be able to compare a player’s estimated runs directly to the league average. Keep in mind that any statistic based on actual runs is also contaminated. Should something like the current extra inning rule become a permanent fixture in MLB, it will be necessary to make adjustments to every metric that uses runs as an input. The extent to which the easy extra inning runs distort the statistics is something I did not fully grasp until actually sitting down and going through the year end stats exercise.

The easy runs are everywhere, and they cannot easily be removed - should the rule become permanent, I think the easiest solution will be to make “regulation” runs the starting point, and then tack on extra inning runs less the run expectancy for a man on second, nobody out times the number of extra innings. Even that is a poor solution, as it only exacerbates the distortions caused by the early termination of innings with walkoffs that Tom Tango noted some time ago. Since extra innings will be fewer under such a rule, a higher percentage of them will be walkoff-truncated than otherwise would be the case.

There are also Team Offense and Defense spreadsheets. These include the following categories:

Team offense: Plate Appearances, Batting Average (BA), On Base Average (OBA), Slugging Average (SLG), Secondary Average (SEC), Walks and Hit Batters per At Bat (WAB), Isolated Power (SLG - BA), R/G at home (hR/G), and R/G on the road (rR/G) BA, OBA, SLG, WAB, and ISO are park-adjusted by dividing by the square root of park factor (or the equivalent; WAB = (OBA - BA)/(1 - OBA), ISO = SLG - BA, and SEC = WAB + ISO).

Team defense: Innings Pitched, BA, OBA, SLG, Innings per Start (IP/S), Starter's eRA (seRA), Reliever's eRA (reRA), Quality Start Percentage (QS%), RA/G at home (hRA/G), RA/G on the road (rRA/G), Battery Mishap Rate (BMR), Modified Fielding Average (mFA), and Defensive Efficiency Record (DER). BA, OBA, and SLG are park-adjusted by dividing by the square root of PF; seRA and reRA are divided by PF.

The three fielding metrics I've included are limited it only to metrics that a) I can calculate myself and b) are based on the basic available data, not specialized PBP data. The three metrics are explained in this post, but here are quick descriptions of each:

1) BMR--wild pitches and passed balls per 100 baserunners = (WP + PB)/(H + W - HR)*100

2) mFA--fielding average removing strikeouts and assists = (PO - K)/(PO - K + E)

3) DER--the Bill James classic, using only the PA-based estimate of plays made. Based on a suggestion by Terpsfan101, I've tweaked the error coefficient. Plays Made = PA - K - H - W - HR - HB - .64E and DER = PM/(PM + H - HR + .64E)

For the individual player reports, I only bothered with starting pitchers and batters, since the sample size for relief pitchers was minuscule. For starting pitchers, I included all pitchers with at least six starts. The categories presented are stripped down from prior years, and I included all in one spreadsheet rather than splitting up by league.

For starting pitchers, the columns are: Innings Pitched, Estimated Plate Appearances (PA), RA, ERA, eRA, dRA, KG, WG, G-F, %H, RAA, and RAR. RA and ERA you know--R*9/IP or ER*9/IP, park-adjusted by dividing by PF. The formulas for eRA and dRA are based on the same Base Runs equation and they estimate RA, not ERA.

* eRA is based on the actual results allowed by the pitcher (hits, doubles, home runs, walks, strikeouts, etc.). It is park-adjusted by dividing by PF.

* dRA is the classic DIPS-style RA, assuming that the pitcher allows a league average %H, and that his hits in play have a league-average S/D/T split. It is park-adjusted by dividing by PF.

The formula for eRA is:

A = H + W - HR

B = (2*TB - H - 4*HR + .05*W)*.78

C = AB - H = K + (3*IP - K)*x (where x is figured as described below for PA estimation and is typically around .93) = PA (from below) - H - W

eRA = (A*B/(B + C) + HR)*9/IP

To figure dRA, you first need the estimate of PA described below. Then you calculate W, K, and HR per PA (call these %W, %K, and %HR). Percentage of balls in play (BIP%) = 1 - %W - %K - %HR. This is used to calculate the DIPS-friendly estimate of %H (H per PA) as e%H = Lg%H*BIP%.

Now everything has a common denominator of PA, so we can plug into Base Runs:

A = e%H + %W

B = (2*(z*e%H + 4*%HR) - e%H - 5*%HR + .05*%W)*.78

C = 1 - e%H - %W - %HR

cRA = (A*B/(B + C) + %HR)/C*a

z is the league average of total bases per non-HR hit (TB - 4*HR)/(H - HR), and a is the league average of (AB - H) per game. I’ve used the MLB average for both this year, and have defined a as the league average of (AB – H) per 9 innings rather than per game.

Also shown are strikeout and walk rate, both expressed as per game. By game I mean not nine innings but rather the league average of PA/G (although for 2020 I’m actually using the major league average AB+W per 9 innings which was 37.9). I have always been a proponent of using PA and not IP as the denominator for non-run pitching rates, and now the use of per PA rates is widespread. Usually these are expressed as K/PA and W/PA, or equivalently, percentage of PA with a strikeout or walk. I don’t believe that any site publishes these as K and W per equivalent game as I am here. This is not better than K%--it’s simply applying a scalar multiplier. I like it because it generally follows the same scale as the familiar K/9.

To facilitate this, I’ve finally corrected a flaw in the formula I use to estimate plate appearances for pitchers. Previously, I’ve done it the lazy way by not splitting strikeouts out from other outs. I am now using this formula to estimate PA (where PA = AB + W):

PA = K + (3*IP - K)*x + H + W Where x = league average of (AB - H - K)/(3*IP – K), using one value for the entire majors

Then KG = K*Lg(PA/G) and WG = W*Lg(PA/G).

G-F is a junk stat, included here out of habit because I've been including it for years. It was intended to give a quick read of a pitcher's expected performance in the next season, based on eRA and strikeout rate. Although the numbers vaguely resemble RAs, it's actually unitless. As a rule of thumb, anything under four is pretty good for a starter. G-F = 4.46 + .095(eRA) - .113(K*9/IP). It is a junk stat. JUNK STAT JUNK STAT JUNK STAT. Got it?

%H is BABIP, more or less--%H = (H - HR)/(PA - HR - K - W), where PA was estimated above.

The two baselined stats are Runs Above Average (RAA) and Runs Above Replacement (RAR). Different baselines are used for starters and relievers, although these should be updated given the changes in usage patterns that have taken place since I implemented the adjustment in 2015. It was based on patterns from the previous (circa 2015) several seasons of league average starter and reliever eRA. Thus it does not adjust for any advantages relief pitchers enjoy that are not reflected in their component statistics. This could include runs allowed scoring rules that benefit relievers (although the use of RRA should help even the scales in this regard, at least compared to raw RA) and the talent advantage of starting pitchers. The RAR baselines do attempt to take the latter into account, and so the difference in starter and reliever RAR will be more stark than the difference in RAA.

For 2020, given the small sample sizes and the difficulties of looking at the league average of actual runs, I decided to use eRA to calculate the baselined metrics. So they are no longer based on actual runs allowed by the pitcher, but rather on the component statistics. Since I’m using eRA rather than dRA, I am considering the actual results on balls in play allowed by the pitcher rather than a DIPS approach. Also, the baseline used is based on the division average eRA, not the league average.

RAA (starters) = (1.025*Div(eRA) - eRA)*IP/9

RAR (starters) = (1.28*Div(eRA) – eRA)*IP/9

RAA and RAR are then rescaled so that the three divisions are all on a (roughly) equal win basis. This is a step I don’t normally take – I usually publish the figures expressed in units of runs, without any adjustment for win value. Normally, I intend for the statistics to be viewed in the context of the particular league-season in question, and thus no adjustment for the win value of runs is necessary. However, with three different run contexts being used in each league, it is necessary in 2020. Why did I not convert them to WAR?

1. I have always preferred, when possible, to leave things denominated in runs, if for no reason deeper than its easier to work without the decimal place; I’d rather write “37 RAR” than “3.7 WAR”. 2. In order to actively describe the figures as WAR, I would have to make what is not an altogether obvious decision about what the runs per game context used to determine runs per win actually is. It can’t be based on figures that include easy runs without any adjustment. For the purpose of these statistics, I did not want to try to figure out a corrected runs figure. That is something that will be necessary if the rule becomes permanent.

So what I did instead was simply take the same figure I used as the baseline for RAA or RAR, the eRA of the division as a whole, and divided the initial estimate of RAR by it, then multiplied by the major league average eRA. This puts all the divisions on an equal basis if one assumes 1) that the division/MLB average eRA is a reasonable substitute for whatever the true run scoring rate for the league should be (and I think it is a reasonable substitute) and 2) given the assumption in 1, it means the practical assumption we are making is that RPW = RPG (multiplying by 2 is lost in the wash), which is not the most accurate way to estimate RPW but is a very simple one that also has the neat property of corresponding to assuming a Pythagorean exponent of 2. It is not what I would do as a best practice, but I think it is an acceptable approximation under the circumstances. So the final formulas for RAA and RAR are:

RAA = (1.025*Div(eRA) - eRA)*IP/9/Div(eRA)*MLB(eRA)

RAR = (1.28*Div(eRA) – eRA)*IP/9/Div(eRA)*MLB(eRA)

All players with 100 or more plate appearances (official, total plate appearances) are included in the Hitters spreadsheets (along with some players close to the cutoff point who I was interested in). Each is assigned one position, the one at which they appeared in the most games. The statistics presented are: Games played (G), Plate Appearances (PA), Outs (O), Batting Average (BA), On Base Average (OBA), Slugging Average (SLG), Secondary Average (SEC), Runs Created (RC), Runs Created per Game (RG), Hitting Runs Above Average (HRAA), Runs Above Average (RAA), and Runs Above Replacement (RAR).

PA is defined simply as AB + W+ HB. Outs are AB - H + CS. BA and SLG you know, but remember that without SF, OBA is just (H + W + HB)/(AB + W + HB). Secondary Average = (TB - H + W + HB)/AB = SLG - BA + (OBA - BA)/(1 - OBA). I have not included net steals as many people (and Bill James himself) do, but I have included HB which some do not.

BA, OBA, and SLG are park-adjusted by dividing by the square root of PF. This is an approximation, of course, but I'm satisfied that it works well. The goal here is to adjust for the win value of offensive events, not to quantify the exact park effect on the given rate. I use the BA/OBA/SLG-based formula to figure SEC, so it is park-adjusted as well.

Runs Created is actually Paul Johnson's ERP, more or less. Ideally, I would use a custom linear weights formula for the given league, but ERP is just so darn simple and close to the mark that it’s hard to pass up. I still use the term “RC” partially as a homage to Bill James (seriously, I really like and respect him even if I’ve said negative things about RC and Win Shares), and also because it is just a good term. I like the thought put in your head when you hear “creating” a run better than “producing”, “manufacturing”, “generating”, etc. to say nothing of names like “equivalent” or “extrapolated” runs. None of that is said to put down the creators of those methods--there just aren’t a lot of good, unique names available.

RC = (TB + .8H + W + HB - .5IW + .7SB - CS - .3AB)*.310

RC is park adjusted by dividing by PF, making all of the value stats that follow park adjusted as well. RG, the Runs Created per Game rate, is RC/O*25.5. I do not believe that outs are the proper denominator for an individual rate stat, but I also do not believe that the distortions caused are that bad. (I still intend to finish my rate stat series and discuss all of the options in excruciating detail, but alas you’ll have to take my word for it now).

The baselined stats are calculated in the same manner the pitcher stats are, except here using the division’s RG as the reference level, then with an adjustment to bring the three divisions onto the same win-value basis by using the ratio of division RG to MLB RG:

HRAA = (RG – divRG)*O/25.5/divRG*MLB(RG)

RAA = (RG – divRG*PADJ)*O/25.5/divRG*MLB(RG)

RAR = (RG – divRG*PADJ*.73)*O/25.5/divRG*MLB(RG)

PADJ is the position adjustment, based on 2010-2019 offensive data. For catchers it is .92; for 1B/DH, 1.14; for 2B, .99; for 3B, 1.07; for SS, .95; for LF/RF, 1.09; and for CF, 1.05. As positional flexibility takes hold, fielding value is better quantified, and the long-term evolution of the game continues, it's right to question whether offensive positional adjustments are even less reflective of what we are trying to account for than they were in the past. But while I do not claim that the relationship is or should be perfect, at the level of talent filtering that exists to select major leaguers, there should be an inverse relationship between offensive performance by position and the defensive responsibilities of the position. Not a perfect one, but a relationship nonetheless. An offensive positional adjustment than allows for a more objective approach to setting a position adjustment. Again, I have to clarify that I don’t think subjectivity in metric design is a bad thing - any metric, unless it’s simply expressing some fundamental baseball quantity or rate (e.g. “home runs” or “on base average”) is going to involve some subjectivity in design (e.g linear or multiplicative run estimator, any myriad of different ways to design park factors, whether to include a category like sacrifice flies that is more teammate-dependent).

The replacement levels I have used here are very much in line with the values used by other sabermetricians. This is based both on my own "research", my interpretation of other's people research, and a desire to not stray from consensus and make the values unhelpful to the majority of people who may encounter them.

Replacement level is certainly not settled science. There is always going to be room to disagree on what the baseline should be. Even if you agree it should be "replacement level", any estimate of where it should be set is just that--an estimate. Average is clean and fairly straightforward, even if its utility is questionable; replacement level is inherently messy. So I offer the average baseline as well.

For position players, replacement level is set at 73% of the positional average RG (since there's a history of discussing replacement level in terms of winning percentages, this is roughly equivalent to .350). For starting pitchers, it is set at 128% of the league average RA (.380), and for relievers it is set at 111% (.450).

The spreadsheets are published as Google Spreadsheets, which you can download in Excel format by changing the extension in the address from "=html" to "=xlsx", or in open format as "=ods". That way you can download them and manipulate things however you see fit.

2020 League

2020 Park Factors

2020 Teams

2020 Team Defense

2020 Team Offense

2020 Starters

2020 Hitters

In Defense of the Astros

I am glad to be writing this post as an irrelevant digression rather than a timely attempt to wrap one’s mind around the possibility of a team with a 29-31 regular season record winning the World Series. My standard outlook when watching the playoffs is to pull for the better team. This is not an ironclad rule, as I do allow my various prejudices (fandom, a favorite player, a least favorite player, style of play, etc.) to override the general principle. I also do this despite being unable to be truly surprised by any outcome of a seven-game series between any two legitimate major league baseball teams. That it does not at all surprise me when inferior teams win series against superior teams, and while it can occasionally be fun to see this happen just to see the egg on the face of people who wildly overestimate the probability of the better team winning (for me, the 2010-11 Phillies were the archetype of this phenomenon), I generally find it disappointing.

After Houston tied the series at three, though, and as I read some online reactions to the possibility of them appearing in the World Series, I decided it would be worthwhile to explain at length why I do not feel that they would be an illegitimate champion should it come to pass. The fact that it did not come to pass makes this discussion academic, but look at where you are. One thing that should be said upfront is that given the emotion generated by the Astros, it is impossible to disentangle how much of the backlash to the possibility of them winning was rooted in genuine concern about the legitimacy of the outcome of the MLB season (concerns that, I have argued, would have been much better channeled towards the possibility of Miami advancing deep into the playoffs), and how much is simply rationalization for the often disproportionate reaction to the sign stealing scandal. There are also people who are looking for any reason to discount the 2020 season, whether out of allowing the perfect to be the enemy of the good (I just cannot relate to people who would rather see no season than 60 games; neither can the bank accounts of the players), or out of darker authoritarian and totalitarian impulses. Arguments borne from there are outside the scope of this piece – this is about the implications of a 29-31 team appearing in or winning the World Series.

The first question that should be asked is “How good are the Astros, really?” My answer is – almost certainly better than 29-31. Why do I say that?

1. The Astros underlying performance was better than their win-loss record

The Astros scored 4.80 and allowed 4.72 runs per nine innings, which in a world of nine-inning games would be expected to result in a winning percentage of .508. The Astros tallied 4.58 runs created and 4.50 runs created allowed per nine innings, which in that same world would also be expected to be allowed in a winning percentage of .508.

Of course, this is not significantly or in any way meaningfully different than their actual W% of .483, but if you have a psychological hangup on a sub-.500 team winning a pennant, their underlying performance was ever so slightly better than that of a .500 team, and to the extent that these tiny differences mean anything, they should mean more the smaller the sample size of games actually is.

2. The Astros win-loss record would likely have been over .500, even at their same level of performance, had a normal schedule been played

Last year, the AL West was the best division in MLB, with an average schedule-adjusted W% of .527 (of course, the Astros as the best team in baseball contributed much to this). The NL West was third at .510. While such calculations cannot be made for 2020 because of the lack of inter-geographic games (we can add the playoffs in, but they don’t provide a sufficient sample to be relied upon, although it is hard not to notice the complete playoff decimation of the Central), my guess is that playing a western-only schedule was tougher than the average schedule faced by a MLB team. This also suggests a slightly better than .500 true performance.

3. Most importantly, the Astros’ true talent was significantly better than that of a .483 team

Some time before the season, I grabbed the win projections from three sources that I think do quality work on these sorts of questions: Baseball Prospectus, Clay Davenport, and Fangraphs. Their win projections for the Astros were 36, 36, and 35. Respectively, that put them second, first, and tied for first in the AL (I should note, I didn’t record the date I jotted down these numbers, so they may not have represented the final estimate from each source).

Of course, things changed, most notably the injuries to Justin Verlander and Yordan Alvarez, but these are insufficient to take a 35 win team to under .500. Some Houston hitters (particularly Jose Altuve) had poor seasons that might have recalibrated estimates of their true talent, but how much stock can you put in sixty games of additional data, and there are players going the other direction too (Framber Valdez is a big one). It seems highly likely that the Astros underlying true talent was that of a winning team.

I note this not to self-aggrandize (heavens knows I’m wrong about these things more than I’m right), but rather to cover myself in case anyone looks back at my 2020 Predictions post, I actually did not pick the Astros to make the playoffs under the traditional format, but did once the field was expanded. While I did not expound on my logic, I did think the Astros were more vulnerable than the purely objective approaches did, mainly because I thought that a starting staff anchored by two pitchers older than me was not a foolproof plan (even if they are Hall of Famers to be). I actually picked the Rays to win the AL pennant, which didn’t take any particular insight but viewed in the wrong light could discredit the opinions I’m expressing here.

If you now will accept the premise that the Astros 1) performed like a winning team 2) would have been a winning team against a normal schedule and 3) could expect to perform better over any stretch of future games due to their underlying talent, are the Astros in fact a beneficiary of a sixty-game season – or a victim? I would argue the latter – the 2020 Astros were a good team that happened to have a mediocre record, largely caused by a season with a schedule that was too short and too unbalanced.

I don’t know what the purpose of a playoff system is (actually I do – it’s to make money). Is it an attempt to reward the teams that performed the best in the regular season? (I’m pretty sure it’s not). Is it an attempt to identify the best team, treating the regular season only as a means of culling the herd? (I think many fans feel this way, which I think is crazy). In reality it is likely some combination of all three. But if the point of a playoff system is to use the regular season to cull the herd, then attempt to identify the best team, I am far from convinced that the Astros should not have been a beneficiary of such a system. And while I would generally consider such a system foolhardy, it is certainly more defensible when the regular season used to do the culling is only sixty games – we should have much less confidence that the results of the sixty games should count for more than the results of the playoffs than we normally would be.

To put this into quantifiable terms, let’s suppose that we decide that in order for a champion to be legitimate, they have to meet some minimal standard of competence. We could then test the worthiness of a champion by calculating the probability that a team that met that minimum standard of competence would have compiled a record equal or worse than the team in question.

For example, let’s say that we decree that in order to be a worthy champion, a team should be at least a true .556 W% team. This is of course an arbitrary value; I’ve chosen it because it corresponds to a 90 win team, which is a value I have always subjectively thought of as marking a legitimate contender. So under this argument, we will accept the playoff result as legitimate, to the extent that the champion it produces was in fact a 90 win quality team.

Using the binomial distribution, it is simple to calculate the probability that a team with a given W% would have a record of 29-31 or worse. The probability of a .556 team going 29-31 or worse over sixty games is 15.8%. Contrast this with the Rays, who were 40-20, which would happen with probability 97.0%.

What would an equivalent performance to Houston’s 29-31 be over the course of 162 games? A .556 team would win 83 or fewer with a probability of 14.9% and 84 or fewer with a probability of 18.9%. The 2006 Cardinals won the World Series after going 83-78 (darn that missing game; the probability for 83-78 is 17.0%). So if you demand that your world champion be at least a .556 team, you could plausibly argue that the Astros would have been the least worthy champion in history. (I would in fact argue that the Astros were more worthy under this criteria based on the considerations discussed above, particularly strength of schedule since it remains grounded in actual wins and losses rather than component performance or true talent).

But .556 may well be setting the bar too low. Is it too much to demand that a world champion be a .600 team (equivalent to about 97 wins)? One could certainly argue that it is not – the champion should be a team that has demonstrated excellence, not simply a contender.

The probability that a .600 team would win 29 or fewer out of 60 is 4.4%. Conveniently, the probability that a .600 team would win 86 or fewer out of 162 is 4.4%. So under this formulation, the Astros could be seen as an equally worthy champion to a team that went 86-76 over a full season. The 2006 Cardinals are joined by the 1987 Twins (85-77) in not clearing this bar, and the 2014 Giants (88-74) were close as well. Of course, in all cases I would argue that we should at least adjust actual wins for strength of schedule, but I think this suffices to make the point. One can make a very logical case that a 29-31 team would not in fact have been the least worthy World Series winner in history. (I am only going to hint at the low hanging fruit offered by the sixty-game record of the reigning world champions who denied a much more worthy team called the Astros in 2019).