THE BOOK--Playing The Percentages In Baseball

The numbers to back up MGL’s theory (and why LI is so super-cool):

In the bottom of the 9th inning, down by two runs, runner on 1B and 2 outs, the chance of winning is 5% (0.0501).  The Leverage Index is 1.68, meaning that we’ve still got a game here, despite the low chance of winning.  (This is why baseball is great.)

If you get a walk, the chance of winning goes up to 0.0977, or +.0476.  The standard walk is +.03 wins, so that gives us a walk LI of .0476 / .03 = 1.6.  We see that a walk, compared to the average outcome (LI = 1.68 in this situation) is about typical in terms of giving it extra or lesser worry than you normally would.

If you get a single, and let’s say that it’s 50/50 if you get to 1B and 3B or 1B and 2B, the chance of winning goes up to 0.1015, or a gain of +.0514.  The standard single is around +.045 wins, so the 1B LI is .0514/.045 = 1.1.  While giving up a single is still a bad thing, it’s not as bad relatively speaking.  A single is always worse than a walk.

If you get a double the chance of winning goes up to 0.155, or a gain of +0.1049.  Compared to the standard double of +.075 wins, that’s a 2B LI of 1.4.  Again, compared to the standard LI of 1.68, it’s not a real killer.

On a triple, the chance of winning goes up to 0.1722, or a gain of +.1221 wins.  Compared to the standard triple of +.100 wins, that’s a 3B LI of 1.2.

So, where do we stand so far?  The overall LI is 1.68.  But by event we have:
1.6 BB
1.4 2B
1.2 3B
1.1 1B

All we have left are HR and outs.  First the out: you get the out, the batting team loses the game, hence loses .0501 wins.  The standard out is worth .027 wins, so the LI for the out is 1.9.

And the HR?  That ties the game with 2 outs and bases empty, for a win expectancy of .539, or a gain of +.4889 wins.  The standard HR is worth +.13 wins.  That gives us a whopping LI for the HR of .4889/.13 = 3.8

3.8 HR
1.9 outs
1.6 BB
1.4 2B
1.2 3B
1.1 1B

So, when you pitch in this situation, you pretty much want to do whatever it takes to avoid giving up a HR.  And you want to do this far more than in other game situations.  And the idea of giving up a non-HR hits shouldn’t be that bad (again, relative to your other options in this game situation and other game situations).

Tango, very nice way of quantifying the concept.  Do any of you (the readers) think that the manager or pitching coach talked to the pitcher and the catcher about this concept?  Or did they just say, “Shake it off...”

If the batter knows that any fastball he sees will be on the outter portion of the plate, unless it is a mistake, doesn’t the value of throwing on the outer portion of the plate go way down?

I’m not sure that I buy the analysis here. It’s certainly true that a proper strategty favors throwing pitches that can’t be hit hard. But it would seem that the reason pitches on the outside of the plate typically aren’t hit hard is because the hitter has to cover the inside of the plate. So the proper strategy also involves forcing the hitter to cover the inside of the plate too.  That means a non-negligible chance of a fastball on the inside.

In game theoretic terms, pitch selection always involves a mixed strategy. But you can never tell what mixed strategy a person is using from one obseervation (assuming there’s more than one rational mixed strategy.)

You can’t tell the pitcher’s strategy from one pitch.

#3, yes, which is why I said this:

“Now, according to game theory, you rarely want to eliminate any pitch/location...”

That being said, it is possible for the optimal percentage of a certain strategy (type of pitch in this case) to be zero.  I am not saying that it IS zero (for an inside fastball), but it could be.

“You can’t tell the pitcher’s strategy from one pitch.”

Right, absolutely.  I’ve also said the same thing many times.  In fact, if you asked the pitcher or catcher about that pitch, he may say, “Well, I am only going to throw that pitch in that situation 5% of the time, and you happened to observe that 5%.”

More importantly is the discussion about how how the pitching strategy changes with the game situation.  All successful pitchers (most of them at least) need to know and utilize that type of thinking process.

And it still may be that the optimal percentage for the inside fastball was 0%. I don’t know.  It woulds certainly depend on the batter. Given that it was a rookie, it would be more likely that you would not have to “trick him” as much (by mixing up your pitches)....

Using Pitch FX, I looked at the past results of a 2-2 pitch.  MGL is almost right.  The pitcher also wants to keep the ball down.

http://www.fangraphs.com/blogs/index.php/game-theory-end-of-game-pitch-location/

Very nice analysis, Jeff.

#5

Looks like very low and very outside is a great place to pitch in most of these circumstances.  That would explain some of Glavine’s success, for example.

Anyway, very interesting stuff.

Really enjoyed reading this article on Friday, but I realized something amiss. I know the actual situation was only the vehicle to a deeper examination of pitch selection in game situation in general, but in reality, Melancon was just following the scouting report. The knock on Paul Goldschmidt IS that he has trouble with the inside fastball.

Obviously, he didn’t have any trouble with this one, but the inside heat is ostensibly the biggest hold in Goldschmidt’s swing.

Very nice work Jeff.

It was not that Goldschmidt hit the inside fastball. It is that Malancon tried to pitch it inside and he missed by just a little and the pitch ended up almost right down the middle or middle-in, which is the worst location for a fastball if you are trying to avoid the HR.

That is the problem with trying to throw the inside fastball in general.  As I said in the original post, it tends to have a bi-modal outcome (I am proffering an educated guess about that) - trimodal if you include the HBP. If you nail your intended location, it is a great pitch, especially if you have a good fastball.  If you miss toward the plate a little, it tends to have a very bad result.  And obviously in that situation (tying or winning run at bat), you would rather have a higher chance of allowing a hit or a walk if the hit is not a HR.

Again, whether it is OK to sometimes throw the inside fastball to that batter from that pitcher in that exact situation, and if yes, how often is optimal, is not something I can answer.  The point was how your pitching location and pitch type strategy changes with the game situation.

Certainly that pitch should be thrown less often in that situation than if you were just trying to get an out and didn’t care if you gave up a walk or a HR.  Again, it might be zero.  I don’t know and I don’t think anyone knows. I have seen many pitchers who absolutely avoid the inside pitch in that situation.

Thanks Mike and MGL. 

I think there can be more done on this front. I will probably look at 3-2 counts some time in the future.

Thanks for the response, Mitchel!

BTW, and I’ve talked about this from time to time over the years, old timers decry that pitchers have gotten away from pitching inside over the years.  They have, because, at least until recently, it is correct for pitchers not to pitch inside as much.

Pitching inside is not very effective anymore, at least during the steroid era - maybe it will come back a little.  The reasons are the following:

1) In the old days, the strike zone was much higher so when a pitcher threw high and tight it would get called a strike if the batter did not swing. No more. If a pitch is thrown high and tight, the batter might as well take it for a ball.

2) Batters are much stronger, again, at least during the steroid era, so that the inside pitch is much more likely to be hit for a home run than 30 or 40 years ago.

3) If you throw inside and hit a batter, you might get thrown out of a game or endanger your own teammates.

4) With more runners on base, the home run is not so great an outcome for the pitching team. Again, with run scoring (and base runners) down a lot, this reason is negated.

MGL - In addition to the reasons that you give above I would add that it seems more batters are hitting from an open stance which allows easier adjustment to the inside pitch.