I was messing around with some leverage index stuff after thinking about why Jason Kendall would ever, ever hit in a save situation, and this is what came out of my brain tonight.  Just wondering if this is something worth continuing with.  Notes were entered in note pad, so we'll see how neat they are here.

AT WHAT LEVERAGE INDEX DO YOU PULL A DEFENSIVELY MINDED PLAYER IN THE BOTTOM OF THE 9TH INNING EVERY TIME?

- WHEN DO YOU PLAY A BETTER FIELDER OVER THE BETTER HITTER
  -+50 R/150G FIELDING
     - +.31 R/G => +.0711 R/PA AT 4.34 PA/G (average)
     - AT 1 LI
       -.113 wOBA = -.0713 R/PA
       -.195 wOBA = .0000 R/PA
       -.277 wOBA = +.0713 R/PA
	  -SO A .082 DIFFERENCE IN WOBA ROUGHLY EQUALS A +50 R/150 DIFFERENCE IN FIELDING
  	  -SCALE IS LINEAR
              -+25 R/150 = .039 WOBA POINTS
              -+10 R/150 = .0156 WOBA POINTS
	      -ETC.

  -WE ASSUME THAT THE REGULAR FIELDER WILL HAVE A 1.00 pLI, BUT HOW MUCH BETTER DOES THE BENCH HITTER HAVE TO 
   BE TO DEMAND A PINCH HIT, BASED ON LI?
  	- CONSIDER THAT THE HITTER RECEIVES A ROUGHLY -.025 IMPACT TO HIS wOBA IN A PH ROLE
	- 4.9% OF GAMES GO TO EXTRA INNINGS - SO THEN THE PINCH HITTER OR SOME DEFENSIVE REPLACEMENT MUST TAKE
          THE FIELD
	- THEN RV(B-9) = LI*{[PH(wOBA)-.025) - STARTER(wOBA)]/1.15} 
          + xinLI*{.049*[(DREPLACE(DEF)/150/4.34)-STARTER(DEF)/150/4.34)]}
[Note by jhmoore, RV(B-9) is run value in the bottom of the 9th]
 
-SPECIFIC EXAMPLE
  -JASON KENDALL vs. MIKE RIVERA (OR ANGEL SALOME)
     -JASON KENDALL: wOBA (2008): .293
     -MIKE RIVERA: wOBA (2008): .309
	 -DIFFERENCE: .016
              -THE BREWERS ORGANIZATION SEEMS TO ASSUME ROUGHLY 15 RUN DIFFERENCE AT CATCHER (10 BY TZONE)
	      -15 R/150 = .246 wOBA POINTS
	      -IF THE ORGANIZATIONS ASSUMPTION IS TRUE (DEBATABLE), KENDALL SHOULD BE EVERYDAY CATCHER
              -BUT WHEN SHOULD KENDALL BE PINCH HIT FOR
	      -KAPLER wOBA FOR BREWERS 2008 - .362
	      -KAPLER(wOBA) - .025 = .337
              -RV(B-9) = LI*{[KAPLER(wOBA)-.025-KENDALL(wOBA)]/1.15} 
                + xinLI*{.049*(RIVERA(DEF)/150/4.34)-KENDALL(DEF)/150/4.34} 
                = LI*[.337-.293]/1.15 + 2.50*.049(-.0926) 
                = LI*[.0383] - .0113
              -RV(B-9) = LI*[.0383] - .0113, UNDER THE ASSUMPTION THAT EXTRA INNING LI WORKS OUT TO ~2.50
              -BREAKEVEN POINT
                  - 0 = LI*[.0383] - .0113
                  - LI*[.0383] = .0113
                  - LI = .0113/.0383
                  - LI = .295
              -SO ACCORDING TO THIS, THE ONLY SITUATION IN THE BOTTOM OF THE NINTH WHERE KENDALL SHOULD BAT 
               WITH A KAPLER-QUALITY PHAVAILABLE IS WHEN LI < .295, 
               OR 2 OUTS, BASES EMPTY 2 OUTS, NOBODY ON
	      -BUT IN THIS SITUATION, THE CHANCE OF EXTRA INNINGS IS MUCH MUCH LOWER THAN 4.9% (RE ~= .1), 
               SO THE CORRECT MOVE IS STILL TO PINCH HIT