There are 3 viable options.
- Let the pitcher hit swinging away
- Have the pitch bunt runners over
- Pinch hit for the pitcher
Which of the three you choose largely depends on who you have available on the bench, how close the same is and who the pitcher is. If we assume it's late game, the score is tied, and the pitcher is only going to pitch one more inning (with a slightly better than average bullpen), then we can go from there.
Now, with the pitcher up 3rd in the order, there are three possibile situations:
- None on, 2 out
- One on 1 out (runner on 1/2/3)
- Two on, 2 out.
I'm going to assume three things:
- The pinch hitter has about a 50% chance of improving the run frequency of the inning (walk, hit, sac fly)
- The pitcher will succeed on a sac bunt 90% of the time (it's not this high)
- The pitcher swinging away will advance the run frequency of the inning about 20% of the time.
Let's examine the possible run frequencies and run expectancies of our starting state:
none on 2 out: 0.117 (number of runs you're likely to score)
- In this case it doesn't matter who hits. The pitcher should probably do so if he will be better than a reliever. The run expectancy is already so low, that the unlikely case of a homerun from either party is the only thing that will truly improve it (obviously that's an automatic run). The most likely outcome from a pinch hitter, a single or a walk, only really raises the run expectancy to .251 runs. Still quite unlikely
1 on, 1 out: .573/.725/.983 (runs you're expected to score). Let's look at our three scenarios here.
Let the pitcher hit: pitchers generally get on base at less than .200, and their most likely result is a K, so let's look at where that leaves our run expectancies: .251/.344/.387. If the pitcher manages to advance the runner we have .344/.378/1.117(since a run scores). It the pitcher manages to get a single, that will provides us a lot more possibilities, more than I can list here (view the matrix to see for your self). However, for the sake of argument, let's assume that a pitcher getting a hit always leads to exactly 1 run. So let's see on average, the pitcher striking (or otherwise failing to advance the runner) out drops .4, advancing the runners drops .1 and getting a hit raises it by .3. If we assume that the pitcher has a 50% chance of not advancing the runner, a 20% chance of a hit, and a 30% chance of advancing the runner. Than the average change is .2 * .3 - .5 * .4 - .3 * .1 = -.17
A sac bunt is a bit easier to figure. Going from 1 out to 2 is always bad for run expectancy and should be avoided. Sac bunting is not an option.
For the pinch hitter, we can use the same formula we used for the pitcher, and tweak the odds. He is much more likely to get a hit or draw a walk, being that he's a bench player, we'll call it .35. Add in another 30% chance to advance the runner .35 * .3 - .3 * .4 - .35 * .1 = -.05.
two on, no out: 1.904/2.052/2.417. This is a huge situation, and an automatic out does not automatically kill your rally here. However, it does lessen the chance of scoring multiple runs pretty significantly. A sac bunt is still a terrible play (the only time it's even close to even is, with 0 outs, getting a runner from second to third). This is a definite spot for a pinch hitter, as the higher contact rate and much higher OBP will lead to better results.
So suffice to say, if there is a runner on, it's almost always better to pinch hit in this circumstance. No matter who pitches the 7/8/9 the odds are good you are going to need another run to keep pace and two to win. So it's a good idea to pinch hit for your pitcher if you even have a chance to score.