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Mitchel Lichtman, or MGL, has been doing sabermetric research and writing for over 20 years. He is one of the authors of The Book: Playing the Percentages in Baseball, and cohosts The Book blog, www.insidethebook.com. He consulted for the St. Louis Cardinals from 2004 to 2006, as well as other majorleague teams. He holds a B.A. from Cornell University and a J.D. from the University of Nevada Boyd School of Law. Most of the time these days you can find him on the golf course.
In Game 6 of the World Series, Texas scored a goahead run in the top of the fifth inning. With two outs and the bases loaded, Colby Lewis, Texas’ starter, was allowed to bat for himself. He struck out to end the frame and pitched 1 1/3 subsequent innings. St. Louis eventually won the game in 11 innings.
In Game 1, with runners on first and third, two outs, and a 22 tie in the bottom of the sixth, Chris Carpenter was due up for the Cardinals. This time, La Russa pinchhit with Allen Craig, who singled in the goahead run, and St. Louis went on to win the game, 32.
In Game 1 of the NLDS, with Arizona losing 10, Kirk Gibson, the DBacks’ manager, sent his starter, Ian Kennedy, to the plate in the top of the sixth inning. Kennedy pitched another 1 2/3 innings and would allow one more run and take the loss for the Snakes.
Finally, in Game 3 of the STL/PHI series, with two outs in the bottom of the sixth inning of a scoreless game and runners on first and second, Jaime Garcia batted for himself, struck out, and the Phillies went on to win the game 32. Garcia pitched only one more inning, giving up a single, double, and a tworun homer (after an illfated IBB to Carlos Ruiz).
What all these scenarios have in common should be obvious. In a close game in the middle or late innings, with the starting pitcher due to bat, the manager must decide whether to pinchhit for him. In all but one of the above cases, the pitcher was allowed to hit. In no case did the pitcher remain in the game for more than two innings subsequent to his PA.
How often does a similar situation arise? In NL games from 20052010, the starting pitcher was due to bat in the fifth or sixth inning or later, after completing at least five innings on the mound (thus becoming eligible for a win), with a leverage index (LI) greater than 1.5 (based only on the score, inning, bases, and outs, using a 9.0 rpg environment), a total of 2,687 times. That is 448 per year or 28 per team per season (once every six games or so). In 43.6 percent of those occasions, the pitcher batted for himself. The average number of subsequent innings pitched was 1.42. Amazingly, 5.5 percent of the time the starter who was just allowed to hit was taken out before retiring another batter! An additional 10.0 percent of the time, he pitched less than an inning.
Not surprisingly, one of the biggest determining factors as to whether the starting pitcher is allowed to hit is his runs allowed thus far. Here is a chart detailing the relationship between runs allowed and how often a pitcher bats for himself (and how many subsequent innings he pitches) in highleverage situations:
Pitcher has completed at least five innings and the LI > 1.5 (NL games ’05’10)
Runs allowed 
N 
Starter allowed to hit 
Average subsequent IP 
0 
314 
72.3% 
1.79 
1 
678 
56.0% 
1.47 
2 
702 
42.7% 
1.34 
3 
558 
31.4% 
1.21 
4 
279 
23.3% 
1.06 
5 or more 
156 
16.0% 
.64 
Home/Road 
N 
Starter allowed to hit 
Road 
1253 
35.9% 
Home 
1434 
50.3% 
Score differential 
N 
Starter allowed to hit 
Winning 
334 
64.4% 
Losing 
1413 
35.4% 
Tied 
940 
48.6% 
Inning 
N 
Starter allowed to hit 
Average subsequent IP 
5 
616 
82.6% 
1.70 
6 
747 
53.1% 
1.34 
7 
922 
24.4% 
1.06 
8 
339 
10.3% 
.83 
9 
57 
7.0% 
0 
10+ 
6 
33.3% 
.83 
Finally, let’s take a quick look at which teams over the last six years allowed their pitchers to bat (with the LI > 1.5) the least and the most after throwing at least five frames (2010 is also listed):
Team 
Starter allowed to hit (2010) 
FLO 
49.2% (35.0) 
CIN 
48.6% (51.4) 

47.2% (50.0) 
SLN 
47.1% (38.7) 
COL 
46.5% (40.0) 
CHN 
45.9% (48.4) 
NYN 
44.8% (48.4) 
ATL 
43.7% (34.5) 
LAN 
43.3% (48.6) 
PHI 
43.0% (42.1) 
MIL 
42.2% (54.5) 
HOU 
41.5% (40.9) 
PIT 
40.9% (39.4) 
ARI 
39.5% (57.1) 
SDN 
38.0% (42.9) 
WAS 
36.5% (33.3) 
Let’s try and list what criteria and information managers presumably use in order to facilitate these decisions.
 The starter’s pitch count, number of batters faced, and any other information related to whether or not he has anything left in the tank or it might be a good time to remove him anyway (for example, he typically has low pitch counts, is coming off an injury or is injured, or is pitching on short rest).
 How important the PA is—essentially, the game’s leverage index (LI). In a close game, especially with the batting team behind in the late innings, the leverage is usually high. If there are runners on base and/or there are two outs, the leverage may be higher.
 Who is available to pinchhit and whether the manager wants to “burn” a pinch hitter at this stage in the game.
 Who is available to relieve the starter and what effect that might have on the bullpen in general.
 Is it a sacrifice bunt or possible squeeze situation?
 How good is the starting pitcher (e.g., is he an ace?) and how has he been pitching thus far?
 Psychological and developmental considerations, such as the effect on a pitcher’s confidence, ego, or learning process if he is taken out of a game early, especially if he is pitching well.
 Whether or not it is a good time for a reliever to come in the next half inning, based on the opponent hitters coming to bat in that inning (e.g., is the lineup turning over for the third or fourth time, do several batters have the platoon advantage over the starter, is the heart of the lineup coming up?).
 Is the pitcher a good hitter?
What if we ignored the bullet points above (and anything else a manager might use to facilitate his decision) and simply went by one, simple rule, the starting pitcher’s ego (and possibly bank account) be darned? What if we instructed every majorleague manager that he is never to let his starter come to bat when the LI is greater than 1.5 and he has already pitched at least five innings? “Heresy,” you say—especially if that starter is Roy Halladay or Tim Lincecum (or any other workhorse ace).
First, let’s look at what kind of pitchers we’re talking about. Remember that 44 percent of the time certain pitchers are allowed to bat, and 56 percent of the time a pinch hitter is called upon. Of course, it’s not just the identity and talent of the pitcher that dictates the decision.
By definition, the average pitcher in the NL during this time frame (’05’10), or any league and year, has a .260 TAv against. We’ll call these “.260 pitchers” to denote their talent/performance. However, when a starter is on the mound, his TAv is .265; for a reliever, it is .250. (All pitcher batting is removed, as are IBB and SH.) Remember that. It is a very important piece of data. Pound for pound, the average reliever is a better pitcher than the average starter. So why don’t relievers start (and starters relieve)? Because if relievers had to start a game and throw five or more innings, they would do considerably worse (to the tune of around one run per nine innings, or 26 points in TAv). Why is that? Because relievers get the benefit of (typically) facing each batter only one time per game, they generally pitch when it is colder (in night games, which are most games), they can throw harder with fewer pitches in their repertoire, and they are less likely to get fatigued.
This is one feather in the “replace your starter as soon as possible, especially when he comes to bat in a highleverage situation” cap. The other feather, of course, is the fact that you can replace him with a much, much better hitter. We’ll get to that feather in a minute.
Now, just because the average reliever is “better” than the average starter, that doesn’t mean that the average reliever is better than the average starter who bats for himself in the sixth or seventh inning of a close ballgame. Let’s take a look at the talent of those starters who are allowed to bat, and those who were sent to the showers.
Allowed to bat 
TAv for that year = .258 
Pinchhit for 
TAv for that year = .262 
For pitchers who bat for themselves, prior to their PA, their TAv against is .221. For pitchers who are removed for a pinch hitter, it is .243. (Why are those numbers so low since the average pitcher is .260? Both groups are selfselected—they don’t include pitchers who were taken out of the game on defense. In other words, if you survive either to hit for yourself or be taken out for a pinch hitter after pitching at least five innings, you were, on the average, pitching well.) So pitchers who are allowed to hit and remain in the game are indeed pitching extremely well, while those who are removed are not pitching nearly as well (although still pitching well, as noted above). The 22point difference in TAv represents around 0.8 runs per nine innings.
Also, starters who hit for themselves averaged facing 21.7 batters (excluding IBB, SH, and INT)—the equivalent of throwing around 83 pitches (at 3.8 pitches per PA). Those who were pinchhit for faced 26.1 batters, or around 100 pitches. Even though the former group has faced fewer batters and thrown fewer pitches, and thus, is more likely to have something left in the tank, I think it is reasonable to presume that managers also think they are likely to continue pitching well based on the fact that they have already pitched spectacularly well, at least according to the numbers. A .221 TAv is equivalent to an ERA of roughly 2.20 in a league that averages 4.00. The $64,000 is, “Did they continue to pitch well?”
According to The Book (of which yours truly is one of the authors), when a starting pitcher starts off a game like gangbusters (retiring the first nine batters in order), it has little predictive value. The exact quote from The Book is, “You can’t tell if a pitcher is on based solely on the results of the first nine batters he faces.” On the other hand, we also found that when a starter is getting lots of outs late in a game, he tends to have very good subsequent performance.
Also, in this thread on The Book blog, I presented some research that indicated that pitchers who were “on” and were allowed to pitch the seven, eight, and ninth innings did not pitch particularly well in the seventh and eighth innings. (The ninth inning, as you’ll see if you read the entire post, is a special circumstance, and one cannot just look at starter performance as measured by TAv or wOBA against.) Here is the money quote from that blog entry:
When I looked at the 0710 data for all starters and not just elite ones, so that I had a much large sample size, they did not pitch very well in the 8th, but pitched exceptionally well in the 9th. You would expect that if starting pitchers who were “on” continued to be “on,” we would see this effect in the 8th (and 7th) as well as the 9th inning, although perhaps not to the same degree.
In any case, it is easy enough to see how these starters pitched after they were allowed to bat. Remember that, on average, the starter pitched another 1.42 innings after his stint at the plate. What was the average TAv against over these 1.42 innings, again, after removing all pitcher batting? Remember, these were .258 pitchers for the season who had pitched to the tune of .221 so far in the game. They were also facing the lineup for the third time on the average; if there was no carryover effect, we would actually expect them to have pitched worse than their normal .258, all other things— like the park, weather, and opposing batters—being equal.
So how did they pitch? (We’re going to exclude the ninth inning for the reasons stated above.) They pitched to the tune of a .251 TAv against (after adjusting for opponent batter pool)—better than we expected, but quite a bit worse than their .221 prior to being allowed to hit. Coincidentally, .251 is almost exactly the same as the average reliever, who is a .250 pitcher. So leaving your “hot” starter in the game yields no advantage over replacing him with an average reliever, unless he is a considerably aboveaverage pitcher.
What about when the starter was taken out of the game for a pinch hitter? How did the relievers pitch in the very next inning? They allowed a TAv of .243, which is around what you would expect from a lateinning reliever.
So let’s recap the last few paragraphs. When a manager allows his starter, who is an overall .258 pitcher but has thus far pitched at a .221 level, to bat in a highleverage situation (LI > 1.5) after he has pitched at least five full innings, he pitches at a .251 level (TAv) for the next 1.42 innings, on the average. When the starter is removed in the same situation, relievers pitch at a .243 clip for at least the next inning. Clearly, there are some starters who are good enough to post a sub.243 TAv later in the game, but remember we are asking the question, “What if we were to remove all starters when they have completed at least five innings on the mound and they are due to bat in a situation where the LI is greater than 1.5?” Whatever our answer is, we can perhaps leverage that onesizefitsall strategy by letting some pitchers (aces) bat for themselves and continue to pitch. Of course, you are still giving up the value of using a pinch hitter, which is almost the whole point of the strategy. In other words, a pitcher would have to pitch a heck of a lot better than .243 or .250 in order to justify allowing him to bat. This brings us to our next, vitally important question:
How much do we gain when we replace our pitcher with a pinch hitter when the leverage is high? A simple way of computing an answer is to figure the average run value difference between a pitcher and a pinch hitter batting, and multiplying that result by the average LI. The average LI in our “starting pitcher coming to bat” situation is 2.29. When the pitcher ends up batting, it is 2.13, and when a pinch hitter is used, it is 2.41. So while managers are certainly using leverage to make their decision, they are letting their starters bat in some pretty highimpact situations (the average is 2.13, so there are many situations which are considerably higher than that).
How can we approximate the average gain in run expectancy (RE) from pinchhitting for our pitcher? First, let’s establish the hitting level of the pitchers who are allowed to bat. The average pitcher who is sent to the plate has a career OPS of 397. The average pitcher who is lifted has a career OPS of 372. Starting pitchers in general are at 383, so one of the criteria for the “hit/don’t hit” decision appears to be the hitting prowess of the pitcher. The approximate “line” for a 397 (OPS) hitting pitcher, per 500 PA (no SH or IBB), is:
Single 
Double 
Triple 
NIBB+HP 
SO 

63.8 
13.3 
1.0 
2.6 
21.3 
162.6 
3.5 
Single 
Double 
Triple 
HR 
NIBB+HP 
SO 
ROE 
69.7 
20.8 
2.6 
10.5 
50.8 
110.4 
4.7 
Base Runners 
0 out 
1 out 
2 outs 
XXX 
.091 
.000 
.001 
1XX 
.202 
.152 
.009 
X2X 
.047 
.049 
.045 
XX3 
.001 
.015 
.026 
12X 
.046 
.058 
.078 
1X3 
.012 
.023 
.030 
X23 
.003 
.012 
.012 
123 
.003 
.029 
.055 
The only thing left to do, in order to estimate the average gain by pinchhitting for the starter, is to compute the difference in RE between our average pitcher and an average pinch hitter for each of the 24 bases/outs states, using the “batting lines” above, and multiply that difference by their respective frequency of occurrence.
Here is an example which will make that last sentence a lot clearer:
Bases loaded and two outs, typically a very highleverage situation, occurred 5.5 percent of the time that a pitcher batted for himself. Using the stat line for our aboveaverage hitting pitchers, and a “miniMarkov” calculation, we get a resultant RE of .596. In the same situation with a pinch hitter at the plate (using his stat line above), we get an RE of .837. So our pinch hitter gain is .241 runs. Multiplying this by .055, the frequency with which this bases/outs state occurred in our sample, we get .0133.
We do this for all of the bases/outs states and add all of the resultant numbers (like the .0133 above) together. This gives us our average gain by pinchhitting for the starting pitcher in these highleverage situations in the late innings.
Here is the same chart as above, with the pinchhitting gain in runs (RE of pinch hitter minus RE of pitcher batting) added for each bases/outs state:
Base Runners 
0 out 
1 out 
2 outs 
XXX 
.091/.091 
.000/.067 
.001/.039 
1XX 
.202/.154 
.152/.126 
.009/.082 
X2X 
.047/.115 
.049/.112 
.045/.112 
XX3 
.001/.234 
.015/.210 
.026/.103 
12X 
.046/.219 
.058/.185 
.078/.162 
1X3 
.012/.188 
.023/.188 
.030/.091 
X23 
.003/.135 
.012/.144 
.012/.162 
123 
.003/.282 
.029/.301 
.055/.241 
Also remember that each team averages around 28 of these decisions per season, and the starter ends up batting around 12 of the 28 times. Multiplying 12 times .032 wins yields a gain of .384 wins per season per team by virtue of this simple strategy.
If one were to argue that such a strategy might tax a bullpen or hurt the confidence or ego of a team’s starting pitchers, remember that the average subsequent IP whenever a starter is not removed for a pinch hitter is only 1.42. That means that we would be transferring a total of 12 * 1.42, or 17 IP, from the starters to the bullpen, an average of around three fewer innings per starter and perhaps one or two more innings per reliever. This hardly seems like a crisis.
In addition, a manager can leverage this strategy and thus invoke it less often (and increase the overall gain in wins) by balancing the gain from pinchhitting with the true talent of the starter. For example, ace starters are around .1 runs per inning better than an average starter. If a manager expects him to pitch for another two innings on the average (say it is only the fifth or sixth inning, he has been pitching well, and has thrown only 75 or 80 pitches), we might expect that starter to gain around .2 runs times, perhaps, an average LI of 1.5 while pitching (these decisions typically come in close games), for a total of .3 runs. If the gain from pinchhitting (including leverage, of course) is less than .3 runs, then the manager can stick with his ace starter. Similarly, if an average of worse starter, regardless of how he has been pitching thus far, is due to bat in a situation where the gain from a pinch hitter is large, it is a clear case of sending the pitcher to the showers, after congratulating him on a job—albeit an abbreviated one—well done. There is also no law that precludes a manager from invoking this strategy in the fourth or fifth inning (or even earlier!), before the starter has pitched at least five frames, especially if he is pitching poorly such that the manager is not likely to receive much flack for his early hook. By leveraging or expanding our general strategy, a team can add a half win per season, maybe more.
Let’s go through the 2010 season and see which teams/managers could have gained the most and least wins from invoking our general strategy of removing every starter for a pinch hitter after he has pitched at least five innings and is due to bat in a highleverage (LI > 1.5) situation. For each occurrence, we’ll use the actual potential gain in runs, based on the bases/outs state times the actual leverage, in order to figure the total loss (in missed opportunities) for the season. In other words, I’m simply adding up all the potential losses from not pinchhitting.
I am also going to add in the potential loss or gain from extending the starting pitcher. If he is a belowaverage starter, there is additional loss to the team, since I am assuming that he could have been replaced by a better pitcher—a leagueaverage reliever (.250 TAv against). If the starter is better than a leagueaverage reliever, then the loss from allowing him to bat will be mitigated by the difference between his seasonal TAv against (actually six points better—I’ll assume he is “on” that day) and that of the average reliever, multiplied by the number of expected innings after the PA (two more if the PA occurs in the fifth or sixth inning and one more if the PA occurs in the seventh or later). In other words, this exercise will reward those managers who leverage or mitigate their decisions by the talent of the starting pitcher, and further penalize those who allow belowaverage starters to come to the plate.
I’ll add two more entries in the last two columns: one, “leveraged wins gained,” which allows excellent starters to remain in the game (if the pinchhitting gain is less than the difference between the starter’s expected runs allowed and an average reliever’s expected runs allowed over the estimated number of subsequent IP), and two, the same “leveraged wins gained,” but expanding the criteria to allow a starter to be removed from the game after pitching only four innings (rather than five).
Before looking at this chart, perhaps you can guess which teams/managers might be the best or worst at allowing their starting pitchers to bat for themselves in the middle and late innings when the game is on the line.
2010 Team 
N general/N leveraged/N leveraged(4) 
Total wins lost 
CHN/Pinella and Quade 
19/17/31 
.64/.67/1.51 
ATL/Cox 
13/11/27 
.40/.42/1.25 
COL/Tracy 
10/9/20 
.38/.38/1.07 
MIL/Macha 
13/11/23 
.47/.51/1.04 
CIN/Baker 
16/14/24 
.55/.55/1.01 
SLN/LaRussa 
14/9/19 
.40/.42/.96 
LAN/Torre 
15/12/22 
.44/.47/.95 
WAS/Riggleman 
14/13/25 
.49/.50/.91 
PIT/Russell 
17/16/24 
.46/.48/.87 
SFN/Bochy 
12/10/21 
.33/.35/.85 
FLO/Gonzalez and Rodriguez 
14/13/24 
.32/.33/.80 
ARI/Hinch and Gibson 
11/10/23 
.33/.34/.73 
PHI/C. Manuel 
13/12/19 
.32/.33/.72 
HOU/Mills 
10/8/19 
.25/.31/.72 
SDN/Black 
13/11/19 
.28/.30/.61 
NYN/J. Manuel 
7/4/12 
.13/.15/.46 
If you look at the last entries in the third column, you’ll see that allowing starters to be removed after pitching only four innings greatly increases the effectiveness of our strategy. Five teams could have added more than a win in expectancy, and one loveable loser, more than a win and a half! (The downside to this expanded strategy includes a heavier workload for the bullpen, depriving your starters of an occasional win, and “burning” more pinch hitters early in the game.)
The last chart is our expanded, leveraged strategy, for each team, 20052010:
20052010 Team 
N leveraged(4) 
Total wins lost 
MIL 
127 
5.7 
SLN 
136 
5.6 
SDN 
144 
5.5 
WAS 
144 
5.5 
SFN 
137 
5.3 
PHI 
122 
5.2 
CHN 
125 
5.2 
ATL 
124 
5.1 
NYN 
120 
5.0 
PIT 
125 
5.0 
HOU 
129 
4.9 
ARI 
127 
4.9 
COL 
123 
4.6 
FLO 
128 
4.5 
LAN 
115 
4.4 
CIN 
117 
4.3 
Thank you for reading
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Interesting analysis, though, and worth some exploration. And who would have thought that St. Louis would have suffered second most severely in the NL, over the last several years, as a result of Tony La Russa NOT messing with his bullpen?
McClellan, to take one, had a very substantial firsthalf/secondhalf split in 2011. He threw too many innings as it was. Piling on more innings in the second half of the season, when he became the long guy after starting for the first half, would not have ended well. And it wasn't going to be the LOOGYs absorbing those extra innings, whether the calculation talks about "an average reliever" or not.
Let's first start with what could be the "limit" for a team's bullpen. Last year, the Braves led the NL in ERA for bullpen AND were #2 in most IP (at 522). Fine, the rookie stars on the Braves are an exception you say. The Nationals had almost as many IP (520), with an ERA of 3.20, 4th in the league. The Pirates led with most IP (526), and their reliever ERA 3.76) was a bit worse than the league (3.59).
If you take the top 4 in the NL in IP by relievers (average of 521 IP), the average ERA for those teams was 3.36.
It seems to me that we can say that the "limit" to a team's bullpen is 520530 innings, and that's really being conservative.
Since MGL is saying that we want to add 17 innings to the bullpen, then any team that had less than 500 innings on its bullpen would be able to figure out how to get those 17 innings. Nine of the 16 teams had under 484 innings, and therefore, would EASILY be able to add 17 innings to its bullpen.
Reds, Cubs, Rox were all on the 500 inning bubble, so they'd have to stretch a bit to get to 17 innings, but even doing so, they'd still end up with fewer innings than the leagueleader in Pirates.
Only 4 of the 16 teams were close to that 520530 limit that they couldn't add another 17 innings.
I should also note that the league leader in IP in 2010 was the Nationals (at 546 innings) with a low ERA of 3.35. The Padres in 2009 led with 571 innings, and also a better than league average ERA (3.75).
Therefore, you can make the case that every NL bullpen in 2011 had plenty of room to add 17 innings.
So, what Bill is saying sounds nice, and seems reasonable, but when you look into it, you see that it's not a concern most of the time.
Certainly the manager can add to the requisites for using this or a similar strategy (a quicker hook for the starter when he comes to bat in high leverage situation) that he has a reliever available who is better than the expected performance of the starter.
BTW, there are many examples where the starter who was allowed to hit was an awful starter (since the average starter in that bucket was .258 (TAv against), there are plenty of much worse ones), such that even a replacement reliever would be better. Surely in those cases, Bill's objection doesn't hold much water, again assuming that the bullpen in general is not overly taxed from recent overuse.
Again, managers can leverage this strategy by using it less when their bullpen has been taxed lately and using it more when they need work or are not overly tired. Managers probably use this anyway as one of their criteria for letting the pitcher hit or not.
I suspect that the 3 principle reasons that managers would be resistant to this strategy, from most to least important (to them), are:
1) They think that their .221 thus far pitcher will continue to pitch at near that level.
2) They underestimate the value of the pinch hitter, especially in certain situations, like a 1 out bunt situations (BTW, an aboveaverage hitting pitcher bunting with 1 out is a terrible strategy  even so with many 0 out situations).
3) They are fearful of overtaxing their bullpen.
Although the Braves' pen led in ERA and innings last year, it seems the workload for Kimbrel et al caught up with them the last few weeks of the year.
I have to agree though..it is annoying when the pitcher is allowed to bat and then removed in the next halfinning if the first batter reaches base. I seem to remember Torre doing that a lot when he was with LA.
That is a good question. So let's orchestrate an answer. You seemed to have made your mind up already, based on what? Your gut instinct? Your gut has nothing to do with it.
You already have the batting line for an above average pitcher. Kuroda is a below average one, so that would have to be adjusted. You would then have to create a batting line for one of those fine batters you list as the Dodger PH'ers. Then you have to do some work to see the difference depending on the bases/outs state that you are contemplating. Then you would take that difference and multiply it by the LI, which is based on the score, inning, bases, and outs. Now you would need to compare that to the number of runs you are gaining or losing by replacing Kuroda with Guerrier or MacDougal for the average number of innings that Kuroda will pitch after he bats (I suspect around 1.25).
Have you done all that? If not, how would you know the answer to your own question?
One of the (many) ways that you have come to an incorrect conclusion regarding your own answer is by assuming that Kuroda is better than those two relief pitchers in true talent. When a manager has to make a decision like that, he obviously doesn't know how these pitchers are going to pitch for the remainder of the season, so he has to actually do a projection in his mind (or on paper I suppose).
Let's quickly look at a projection for Kuroda and Guerrier:
Bill James
Kuroda 3.57 ERA
Guerrier 3.34
MGL
Kuroda 3.79 nERC
Guerrier 3.57 mERC
Hmmm. It seems like Guerrier is the better pitcher right off the bat. So, we would surely like to replace Kuroda with Guerrier when Kuroda is facing the lineup for the 3rd time, whether he comes to bat or not. So we don't really need to go through all of those calculations above...
http://www.insidethebook.com/ee/index.php/site/article/starters_and_relievers_in_the_9th_inning_and_score_differential/