Sunday, December 21, 2008

How should we value closers?

J.C. Bradbury, a man who writes a blog that combines two of my favorite subjects, baseball and economics, has put up a couple of posts so far this offseason decrying what he believes are the inordinately large salaries being given to closers.

His chief problem is that relievers, even the best ones, save so many fewer runs than starting pitchers that it seems ridiculous to offer relievers salaries that even approach the salary of a comparably elite starter. The usual response to this is that relievers pitch in "high leverage" situations, which is to say that the best relievers often pitch in situations where each run is of magnified importance. I think we can all agree that it is less damaging to give up a run when you are leading by 15 in the top of the ninth than when the game is tied in the bottom of the ninth.

J.C. does not agree that leverage, which amounts to a simple usage pattern for relievers, should factor in to their salaries:
I have considered the impact of leverage, but I don’t think leverage can explain the vast differences in my estimates and what is happening in the market. Leverage is a product of outside factors when a pitcher faces the same rules during all times of the game. The quality of his pitching is the same in the 5th inning as it is in the 9th.
He concludes (emphasis mine):
Another factor is that better pitchers in earlier innings affect the leverage in later innings. So, a good starter preventing runs as an impact on reducing leverage later in the game by creating bigger leads. I’m not sure exactly how to value that. So, I believe that the proper method is to treat all pitcher innings the same, while acknowledging that some elite relievers have some extra value in that they could be used in more valuable spots. But this value doesn’t necessarily come from when they pitched in the past.
J.C. is certainly right that it doesn't matter when a reliever has pitched in the past. All that matters is how a reliever can be used in the future. I will also admit that I myself have struggled with how to deal with the question of leverage when assessing player value, both in a past sense (how much reliever X contributed to a team's success) and in a future sense (how much reliever X will be worth to me next year).

I want to leave aside the question of past performance for now. I think that J.C. is in error to not factor leverage into future player value. Let me offer an analogy.

Let's say that I have been offered a prize of $1,000,000.00 if I can win a race from New York to Los Angeles. After a couple days of driving, I'm only a couple hours from Los Angeles and comfortably in the lead, by perhaps an hour or two. Suddenly, I find that I have a flat tire. Worse, I have no spare, since I dropped it to lighten my load and increase my fuel economy. I'm stuck in the middle of nowhere with no way to get going again.

But wait! Just when it seems that all hope is lost, a man pulls up beside me with tires for sale. He knows about my predicament and the prize waiting for me at the end of the race. If I act now, my lead will still be comfortable and I can cruise to the finish line in peace.

So I ask my savior, "How much for a spare tire?" He replies, "Only $750,000.00."

Obviously that tire did not cost that much to make, nor would it be worth that much at just about any other point in the race. Nonetheless, that tire is almost certainly worth that much at this point in time because I have no other options. I can either accept a prize of $250,000.00 or receive nothing. In fact, one might say that my enterprising benefactor is giving me a good deal, since I should probably be willing to accept almost up to the entire $1,000,000.00.

The situation is slightly more complicated when there are multiple tires (or relievers) on the market and multiple people bidding for them, but the principle is the same. How you intend to use a tool and the situation in which you might use it most certainly impacts how much that tool is worth, even if that tool is a pitcher.


D.Cous. said...

An interesting post. Are you by any chnace familiar with the Diamond-Water Paradox, which you have restated nearly verbatim?

Just curious.

Unfortunately, I don't know enough about pitching and "leverage" to make any intelligent comment on the subject. Thanks for the post, though.

John Lynch said...

I had not heard of this paradox, but thanks to your link to the subject via The Font of All Knowledge I know have heard of it.

It's not really necessary to know much about leverage. All you really need to know is what I said in the post: some runs impact outcomes more than others (perhaps I should say that the marginal value of a run allowed varies greatly according to game situation). Thus, it makes sense to use your best pitchers when the impact of the next run is the highest.

Jeremiah said...

I partially agree with you. I think it is worth spending more money on the tire at the end of the race than at the beginning, but not because the tire is somehow worth more now. Near the finish, there is a greater probability of winning than at the beginning, so the probability of winning times the reward is how valuable the tire is. The same concept applies to relievers.

John said...


I'm not sure how this means that you only "partially" agree with me. That seems to me to be the essence of my argument.

I will note that you say two different things in your post that seem to contradict each other:

"not because the tire is somehow worth more now"

"the probability of winning times the reward is how valuable the tire is"

If I interpret that "now" in the first quote to mean "near the end of the race" then what you seems to be saying that the tire is not magically worth more (quote one), but yet it is (quote two).

I think people get hung up on the idea that goods, whether tires or baseball labor, have some intrinsic worth. It just ain't so. That tire is certainly worth more to my hapless would-be-race-winner than it is to almost anyone else. That this worth greatly exceeds the cost to produce the tire is immaterial.

Jeremiah said...


That's a good point about objects not having intrinsic worth. Their value is measured by what someone is willing to give up for them.

What I meant to indicate is that the tire does not produce different results late in the race. The distance traveled at the end of the race is just as important to winning as the distance traveled at the end. However, the value of the tire to the driver changes because of the situation: he has a very good chance of winning a whole bunch of money if he buys the tire, but a very small chance of winning without the tire. However, if he was 500 miles behind the leader, the tire would not be as valuable to him. Similarly, if it were early in the race, and he had a modest lead, the tire would not be as valuable to him because there would still be a significant chance that he would not win even with the new tire.

In baseball, a run prevented early in the game has the same effect on the final outcome as a run prevented late in the game. As the game nears its conclusion, there are fewer opportunities for scoring, which means that run prevention will have a bigger impact on the probability of winning. I think this is different than saying that runs scored or prevented in late innings have a bigger impact on the outcome.

Hopefully this makes sense. I'm thinking it through as I write.

John said...


Properly accounting for leverage is one of the harder tasks in valuing baseball players, particularly pitchers, in my opinion.

On the one hand, you're right: it doesn't matter when you score a run. They all count the same in the end. Winning a game 4-3 with a walk off grand slam nets the same result as winning a game by scratching out four runs over the first six innings.

On the other hand, it's also totally true that you'd much rather give up a run in the top of the first inning game than in a tie game in the bottom of the ninth.

This is why I think it's important to separate the ability of a pitcher to prevent runs from his usage pattern. If I am looking at acquiring a pitcher, all I should be concerned with is his ability to prevent runs. It will be up to me (and my manager) to use him to prevent runs where they will have the most impact.

Nonetheless, because an "ace reliever" can be utilized in higher leverage situations, the runs that he prevents will net me more wins than a comparable starting pitcher. Therefore, I will probably have to pay for those wins even though they have to do with the usage pattern I impose on the pitcher and not in the pitcher's innate ability. That's the point of my parable, I guess. It's no credit to the tire that it happens to be incredibly important at that point in the race. Nonetheless, I would be a fool not to pay through the nose for it. The same is true of top relievers.

To put it another way, you should absolutely not pay a reliever for wins gained through past usage patterns. You should pay him based on the number of wins he can provide you based on the usage pattern you intend to employ. Because reliever runs prevented will be leveraged by any rational manager and starting pitcher runs prevented will not, you should pay more per run prevented for a reliever. This is where I quibble with J.C. If you try to pay relievers as if their runs prevented were not leveraged (like starting pitchers), you will always underbid for the best relievers. Your opponents, who bid on the number of wins they stand to gain, will bid correctly and end up with a far superior AND correctly valued bullpen.

All that being said, there's still no way you should pay the best relievers as much as the best starters. Of course, you also don't see that happening. Great starters make more than great relievers. Good starters make more than good relievers. However, because relievers can and will be leveraged, the difference between starter and reliever pay is not as great as would be indicated by the difference in the number of runs each group would prevent. I think in general the baseball market has correctly assessed the relative value of starters and relievers.

Yeah, I'm rambling too. Thanks for posting!