Fun With Sabermetrics/Fun With Not Studying For Finals

Just some musings I had tonight - we'll see if they're worth delving into more later.

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