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October 14, 2004

Lies, Damned Lies

Using the Golden Run Ratio

by Nate Silver

One of the classic managerial dilemmas--perhaps the classic managerial dilemma--is whether to remove a starter who is still throwing pretty well for a middle reliever in the sixth, seventh or eighth inning. This decision is relatively straightforward if the sole concern is winning the game in question. You trot your butt out to the mound, the pitcher grabs his crotch a couple times and spits out his chaw and tells you that his arm still feels real good, and then you sit back down in the dugout ask the pitching coach what's really going on. If the pitching coach tells you that the guy throwing in the bullpen is likely to be more effective than the starter, you make the switch. Otherwise, you leave the dude out there. Certainly, it's possible to make an error of judgment now and then--figuring out just how much fatigue will reduce a pitcher's effectiveness is a guessing game of sorts--but fundamentally, the problem is simple.

Things become much more complicated if you are concerned not only about the outcome of the current game, but also the outcomes of future games. If your ace has given you a 8-0 lead after seven innings with a pitch count of 110 or so, common sense dictates that you pull the guy before the eighth. You sacrifice some very small amount of expectation in the near-term if you replace him with an inferior alternative--maybe you'll win 99.68% of the time instead of 99.72%--but the future benefit of keeping him healthy and well-rested for subsequent outings well outweighs this.

It isn't always that easy, though, especially in the post-season when pitchers are frequently asked to throw high-pressure innings against good offenses on short rest. Nor is the impact on future games easy to determine. If Roy Oswalt is going to be marginally less effective in Game 5 because he went an extra inning in Game 2, how much will that impact Houston's probability of winning the ballgame? How much should the impact on Game 5 be discounted since there might not be one at all?

Those questions are vexing to answer, but we can try and quantify the impact that removing a better pitcher for a worse one will have on a game whose outcome is in immediate doubt. Let's take Ron Gardenhire's decision to remove Johan Santana in Game 1 of the ALDS as an example. Santana, in spite of a thin, two-run lead and the appearance of having plenty left in his tank, was pulled in favor of Juan Rincon after seven innings and 93 pitches. Rincon, it is true, had a pretty good season, but was not quite as good as his ERA suggests, and nothing along the lines of a post-All Star Break Santana; the Twins were giving a little something up by making the switch. For purposes of this problem, let us assume the following:

• Santana's "true" ERA against an average lineup is 2.00.
• Rincon's "true" ERA is 3.50.
• Joe Nathan, who will pitch the ninth regardless of what happens in the eighth, has a "true" ERA of 2.50.