Do batters learn during a game?

David Smith, Retrosheet’s head guru guy, has a bunch of studies on the Retrosheet research page. This is his earliest one, now ten years old, called “Do Batters Learn During a Game?”

David analyzed 12 years’ worth of “Baseball Workshop” data (Retrosheet didn’t yet have those seasons at the time, I guess), and split batter performance by whether it was the batter’s first, second, third, or fourth plate appearance against the starter. Here’s the results for batting average; the OBA and SA charts would be similar.

___________PA_______BA_
1st ... 419870 ... .259
2nd ... 401917 ... .268
3rd ... 313880 ... .272
4th .... 90994 ... .276

So it does appear that batters “learn” to hit pitchers better as the game goes on. However, because of a bias in the data, the effect is probably stronger than the statistics show.

To stay in the game long enough to face a hitter for the fourth time, the pitcher has to have been pitching a very good game. More likely then, the pitchers in the fourth group are better than average pitchers. So for the batters, in the aggregate, to still improve the fourth time, they have to overcome an extra obstacle – that the fourth group of pitchers is actually of better quality than the other three groups. The .259 in the first group is obtained against all starters – the .276 in the fourth group is obtained against better starters. The value of “learning,” then, is actually greater than the 17-point difference in the data.

Notice that the PA barely change between the first and second group, indicating this is still largely the same group of pitchers. The difference there is nine points. But the second difference is four points when the PAs drop (meaning the pitchers have improved), and the third difference is again only four points when the pitchers have improved a lot. This is consistent with the hypothesis that the batters’ learning is constant, but the pitchers are getting better.

You could get a more accurate estimate of batter learning by looking at the caliber of pitcher. That’s what Tom Tango, Mitchel Lichtman, and Andy Dolphin did in “The Book.”

They did it a little differently. David Smith’s study included only PAs where the batter was in the starting lineup. The Book’s study included all batters. It found that the quality of batters increased as the game progressed (presumably because of pinch hitting or platooning), so you can’t go by the raw batting stats – you have to adjust for the batters. They did.

Here are their results. The numbers are in “wOBA,” which is their measure of offensive effectiveness, and are the difference between what you’d expect the batter to do against that pitcher, and what actually happened:

____________PA______wOBA diff
1st ... 163,900 ... -.008
2nd ... 158,872 ... +.001
3rd ... 124,603 ... +.008
4th .... 22,221 ... +.001

So, the first time through the order, the pitcher was eight points better than you’d expect. The batters then take the advantage for the rest of the game.

To convert wOBA to runs per game, you divide by 1.15 and multiply by the number of PA in a game (call it 40). That means that the first time through the order, the pitcher’s ERA is about 0.25 lower than average. By the third time through, it’s about 0.25 higher than average. That’s a pretty big jump, from (say) 4.25 to 4.75.

You could save a tenth of a run per game by taking out your starter after 18 PA and bringing in someone new. That’s assuming that the increase is caused by the pitcher, and not by the batters just being warmed up and better against any pitcher. But the strategy probably wouldn’t work in real life – even with an 0.5 disadvantage, your ace starter is probably still better than your middle reliever. And the ensuing revolt by the starters would probably negate that tenth of a run pretty quick.