Pennants Added

In this year’s Annual, I wrote an essay attempting to rank the most valuable pitchers of all time. Though I devoted space to explaining his ranking, many people still complained that no rational person would rank Sandy Koufax as the 64th most valuable pitcher of all time, even if he did retire at 30.

Part of that has to do with the dichotomy between value and greatness, a concept which is often difficult to grasp. Greatness isn’t really tangible, and thus perhaps impossible to define, but value—well that can be evaluated much more easily. Value, after all, is simply a player’s contribution towards his team achieving the ultimate goal—winning the World Series.

Generally, sabermetricians measure value by evaluating how many runs or wins a player creates, and my method was no different in that regard, as I ranked pitchers based on the number of wins they contributed above a replacement-level player. A common critique of that method was that it gave too much credit to longevity, and not enough to a player’s peak. Frankly, I see no reason to give an arbitrary amount of extra weight to peak just so that the rankings correspond better to our expectations, but at the same time, I am willing to consider methodology changes, as long as they make sense.

Today, then, we will consider an idea known as Pennants Added.

Pennants Added is a concept that was first proposed by Bill James in The Politics of Glory. The basic idea is that wins above a given baseline are just a proxy for what we really care about: how many championships a player was worth. The Pennants Added approach has been lauded by many advocates of peak performance, the idea being that such a system would give players the proper credit for having huge seasons which catapult their teams into the playoffs.

There have been a couple studies in which the authors actually tried to construct a Pennants Added Measure—one by Michael Wolverton in Baseball Prospectus 2002, and another by Dan Levitt in By the Numbers (PDF). While the studies are commendable, I believe they are both somewhat flawed, so today, we will try to construct our own Pennants Added measure.

To do that, we need to know two things: One, how wins are distributed in baseball, and two, the probability of an n-win team making the playoffs. Essentially, we are asking: How much would a given player increase the odds of a randomly-chosen team making the playoffs?

I have restricted my dataset to the wild card-era, specifically, 1995 to 2005. For each team in those years, I calculated their wins per 162 games. If we plot the number of teams at each given win total, we get the following distribution:


The average team wins 81 game (duh), with a standard deviation of 12 wins (meaning two-thirds of all teams should win between 69 and 93 games). That’s actually a very important fact, because it will let us figure out the theoretical probability of an n-win team existing.

Essentially, a player will have a greater chance to being assigned to an average team than to a really good or bad one, since there are more teams with a close-to-.500 record than teams with triple-digit wins or losses, but he will also have some chance of being placed on an extreme squad as well. The probabilities will be calculated mathematically using the normal curve you see in the above graph.

So now we know what the chances of a team of any given quality existing, but we still need to know the chances of an n-win team making the playoffs. Again, we will construct a theoretical model, this time using a binary logistic regression. Essentially, what a binary regression does is ask whether or not each team in our dataset made the playoffs, and how many games it won. It then spits out a formula which determines a team’s odds of making the playoffs based on how games it wins. If we graph our results, this is what we get:


Essentially, what we see is that a team with less than 80 wins has almost no chance of making the playoffs (again, duh), while a team that wins 100-plus games will almost always get to the postseason (no shock there, either). At 89 wins, a team’s odds of making the playoffs are a little less than 50%; at 90, they’re more likely to make the playoffs than not.

Okay, so now we have these two graphs—where do we go from here?

First, we figure a random team’s chances of reaching the postseason. For example, a 90-win team will go to the playoffs 57% of the time. Now let’s say we add Albert Pujols to that team, and it wins 97 games. Now, its chances of reaching the playoffs are almost 98%. So Pujols has essentially added .41 postseason appearances all by himself.

However, the chances of a random team winning exactly 90 games are just 1-in-40, so multiply .41 by .025, giving us roughly .01 pennants added.

We repeat the exercise at every win level, from 0 to 162, and add up the results. It turns out that a seven Win Above Replacement (WAR) player would have .44 Pennants Added (technically, playoff appearances added, but we will use the terms interchangeably).

A Hardball Times Update
Goodbye for now.

In fact, let’s graph Pennants Added for all players between 0 and 10 WAR (which is pretty much the maximum a player can contribute in one season):


Looks like a pretty straight line, and in fact, it looks like the value of a win decreases at the very high WAR levels! We can better understand the value of each additional win by fitting a polynomial to this graph, and taking its derivative. In simple terms, we’re just finding how much each additional win is worth.


What do we see here? The value of an additional win peaks at four wins above replacement; four years of average performance (two WAR) is more valuable than one huge MVP year (eight WAR)! That is certainly unexpected. Pennants Added is supposed to help players who had great peaks, but instead the measure rewards players who are above-average for long periods of times.

So does that mean that Pennants Added is wrong? No, what it means is that our assumptions about peak are wrong. A high peak might be exciting to watch but it is not as valuable as a steady record of above-average performance.

If we focus on what really matters—winning the World Series—we should downgrade Sandy Koufax, and value someone like Bert Blyeleven that much more.

References & Resources
Two Notes:

(1) Some might be wondering if I’m not underrating the value of a great season by looking at the probability of a team making the playoffs instead of its odds of winning the World Series, which is what we really care about.

But the numbers from the past decade indicate that there isn’t really a difference between the two; in other words, the postseason is basically a crapshoot, and winning more games in the regular season does not increase a team’s chances of winning it all. Last year’s Cardinals, of course, are a great example of that fact, even though they were not included in my dataset.

(2) I didn’t want to use the word “derivative” in my article any more times than I already did, but I figure some people might be interested in why the value of each additional win above replacement declines after four WAR. It becomes pretty clear if we differentiate the formula for determining a team’s odds of making the playoffs:


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