by Kevin Harlow
June 28, 2004
Bill James created the original Game Score as a quick and easy calculation to determine the quality of a pitcher’s start.
GS = 50 + Outs + 2*(IP-4) + K – 2*H – 4*ER – 2*UER – BB
This measure is not independent of fielding, however, given the H, ER, and UER terms. Fielding independence is the focus of defense independent pitching statistics (DIPS). What would a DIPS-based game score look like? Also, why not make the revised game score equivalent to a support neutral win probability?
The first step is to have win probabilities with average offensive support per inning and runs given up. For this I took advantage of the heavy lifting Rob Wood did in his Win Values articles, specifically the win probabilities with average pitching in Part 5.
http://www.baseballthinkfactory.org/files/main/article/robw_ood_2002-08-19_0/
To convert this data into what I needed, I simply subtracted each value from 1. I put these data points into Excel and ran a linear regression of win probability versus IP and RA.
W% = .5 + 0.049516*IP – 0.08712*RA
Actually, the linear regression was run on just the data where the win probability was between 0.1 and 0.9. The standard error was 0.040. If you multiply by 100, and round off the coefficients, the result is starting to look a lot like a game score.
GS = 50 + 5*IP – 9*RA
To make this a DIPS-based GS, a DIPS-based estimate of RA is needed. Since a primary requirement of a game score metric is ease of calculation, I decided a linear weights estimate of RA was probably the way to go. A somewhat reasonable linear weights run estimator is
RA = 0.33*BB + 0.5*S + 0.8*D + 1.1*T + 1.4*HR – 0.1*Outs
Based on typical batting event rates a reasonable coefficient for ball-in-play hits (BIPH – that includes S, D, T) is 0.6. From the GS above, we see that we’re going to multiply by 9 so the RA coefficients should be adjusted.
RA = 3*BB/9 + 5.4/9 * BIPH + 13/9*HR – 0.9*Outs/9
The only change has been to the HR coefficient, changing it from 1.4 to 1.44, which is still a typical value. Substituting into the GS equation gives
GS = 50 + 5*IP - (3*BB + 5.4*BIPH + 13*HR – 0.9*Outs)
We really don’t want BIPH, for a DIPS-based metric we want the average expected BIPH given the number of BIP.
BIPHavg = $Havg*(Outs + BIPHavg – K)
BIPHavg ~= 0.4*(Outs – K)
Replacing BIPH in GS with BIPHavg gives
GSDIPS = 50 + 5*IP - (3*BB + 5.4*0.4*(Outs-K) + 13*HR – 0.9*Outs)
GSDIPS = 50 + 5/3*Outs - (3*BB + 2.16*Outs – 2.16K + 13*HR – 0.9*Outs)
GSDIPS = 50 + 5/3*Outs - (3*BB + 13*HR – 2.16*K + 1.26*Outs)
Now, forcing the square peg into the round hole as creatively as possible, we’ll “trade” 0.16 strikouts for 0.7 outs. Not quite an even trade, but it’s pretty close. Mmmm, fudge!
GSDIPS = 50 + 5/3*Outs - (3*BB + 13*HR – 2*K + 4/3*Outs)
GSDIPS = 50 + 1/3*Outs + 2*K – 3*BB – 13*HR
GSDIPS = 50 + IP + 2*K – 3*BB – 13*HR
And there you have it – a DIPS-based pitcher game score equivalent to support neutral win probability that is easier to calculate and with fewer terms than James’ original Game Score!
Notes: Did James originally intend the Game Score to approximate win probability? The (2*K – 3*BB – 13*HR) term is the same thing as FIPS that was independently and originally noted by TangoTiger as a DIPS-based pitcher measure
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