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The Cardinals and Phillies are playing the last game of their Division Series tomorrow, with each team’s ace (Chris Carpenter and Roy Halladay) taking the mound. The game—and thus the series—may very well be settled by which team’s ace pitches better. But it might just come down to which team’s manager has the audacity to sit his ace down on the bench.

One game this series has already been strongly influenced by a decision to pinch-hit (or not) by each manager—Cardinals starter Jaime Garcia struck out with runners on first and second to end the sixth inning in Game Three, then stayed in to allow a three-run homer to Phillies pinch-hitter Ben Francisco in the seventh.

The trouble with pinch-hitting for your starter is that it forces you to go to your relievers early, and often that means bringing in the middle- or back-end of your bullpen rather than your top relievers. But this is an elimination game, meaning that both managers should be managing to win this one game without worrying about tomorrow. A team’s best relief options (including starters from earlier in the series) should be available if necessary, so a creative and determined manager should be able to find more quality bullpen innings than he would in an ordinary game.

So how can we identify situations in which a manager should be most willing to turn to a pinch-hitter? We can look at something called run expectancy, which examines the average number of runs scored in the rest of the inning, given how many runners are on base, which bases they occupy, and how many outs there are. I’ve provided two run expectancy tables side-by-side—one in which a pitcher bats in that plate appearance, and one in which a pinch-hitter bats in that plate appearance. Looking at historic data from 1993 through 2011:

RUNNERS

OUTS

RE_PIT

RE_PH

DIFF

___

0

0.50

0.49

0.00

___

1

0.21

0.25

0.04

___

2

0.05

0.09

0.04

1__

0

0.79

0.84

0.05

1__

1

0.38

0.50

0.12

1__

2

0.12

0.19

0.07

_2_

0

1.02

1.13

0.11

_2_

1

0.56

0.64

0.08

_2_

2

0.18

0.28

0.09

12_

0

1.46

1.40

-0.06

12_

1

0.69

0.87

0.18

12_

2

0.24

0.43

0.19

__3

0

1.34

1.39

0.06

__3

1

0.75

0.93

0.18

__3

2

0.22

0.35

0.14

1_3

0

1.57

1.74

0.17

1_3

1

0.86

1.10

0.24

1_3

2

0.26

0.44

0.18

_23

0

1.77

1.96

0.19

_23

1

1.10

1.34

0.24

_23

2

0.28

0.57

0.29

123

0

2.04

2.32

0.28

123

1

1.20

1.51

0.32

123

2

0.42

0.79

0.3

Speaking broadly, the situations where a pinch-hitter offers the most bang for the buck come with multiple runners on and at least one out. Especially with two outs, a key offensive weapon for the pitcher (the sacrifice bunt, or any other out in play that would advance the runners) has been removed, making the pitcher even more offensively challenged than he would be normally. In the situation that led to the Garcia strikeout earlier in the series, a typical pinch-hitter would be expected to score .19 more runs on average than the pitcher would if allowed to bat, basically doubling the expected number of runs.

Another way we can look at it is to assess a team’s chances of scoring at least one run. In the situation Garcia faced, a team scores at least one run 13 percent of the time when the pitcher bats, compared to 22 percent when the team utilizes a pinch-hitter.

How does that compare to what we would expect from the starter who stays in to pitch the following inning, relative to a reliever? When the starting pitcher goes out to pitch the seventh inning, his team will allow at least one run 29 percent of the time, while if the inning is started by a reliever the team will allow at least one run 28 percent of the time. It can be noted that Garcia is an above-average pitcher and had been pitching well so far, but that’s going to be true of starters who reach the seventh inning far more often than if we simply looked at all innings. Moreover, even when looking at starters who are pitching well enough to make it to the seventh inning, we see that a fresh reliever has a slightly better chance of having a scoreless inning than a starter who has already worked his way through the lineup two-three times.

It was unlikely that the Cardinals would push a run across in that situation no matter what Tony La Russa did, but it was nearly twice as likely had he put in a pinch-hitter instead of letting Garcia bat for himself. In addition to robbing the Cardinals of an extra chance to score runs, it’s possible that he may have increased his team’s odds of surrendering more runs the following inning as well. It would be unfair to assume that La Russa cost his team the game by leaving Garcia in—it doesn’t always work out the way it did there. But it is an advantage that a manager can’t afford to deny his team in a must-win game, even with his ace on the mound.