What Leverage Index Do Relievers Deserve?

Mariano Rivera pitches 65 innings a year and makes \$15M.  Sure, they're really really good innings, but the other reason he's worth so much in only 65 inning is because his manager can pick the spots when he pitches.  In geek-speak, he pitches in a lot of high-leverage situations, making his innings worth as much as 115 "regular" innings.

Nobody's going to argue that Mo deserves to be a closer and rack up a 1.8 Leverage Index (LI) year after year.  But not all managers are rational and not all relievers are used in the proper situations.  Take, for example, Joe Borowski and Heath Bell.  Over the past few years, Borowski's been a closer with a FIP over 4.00, while Bell's been a setup man with a FIP close to 3.00.  Giving each pitcher credit for the average LI in which they were used makes them look similarly valuable, while almost anyone in their right mind would rather have Bell.

One way to solve that dilemna is to still give relievers credit for the leverage of important late innings (so they can be compared fairly to other positions), but to use an LI based on the role they deserve instead of based on when their managers chose to use them.

I first discussed the idea of "deserved leverage index" (which Harry wants to call deLI, mmm) with Justin Inaz and Tom Tango over a year ago, but it got dropped.  I'm bringing it back.  Tom's best idea was to use a model that looks like deLI=(lgFIP/FIP)^1.5, where the 1.5 could be messed around with.  Using a 4.10 average FIP for relievers, you'd get:

FIP    deLI
2.50    2.0
2.75    1.7
3.00    1.5
3.25    1.4
3.50    1.2
3.75    1.1
4.00    1.0
4.25    1.0
4.50    .9
4.75    .8
5.00    .8

(deLI should probably be capped at the high end at 1.8ish, by the way.)

Now, Joe Borowski's 4.25 FIP spits out a 1.0 deserved LI, and Heath Bell's 3.25 FIP deserves a 1.4 LI, a step above setup man, but not an elite closer.  To me, that seems a lot more fair when we attempt to quantify their performance, at least in a forward-looking analysis.

That's where I'm at.  How does it look?  Any other approaches you'd use?

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