Are pitchers chicken?

The issue of retaliation for hit batsman is a popular one among economists, due to its testing of the standard “people respond to incentives” theory.

The theory predicts that a pitcher will plunk more batters if he doesn’t have to come to the plate and risk being hit himself. That is, in the National League, when the pitcher faces potential retaliation, he has an incentive to refrain from deliberate plunkings – or, at least, to pitch more carefully to prevent the accidental hit batsman. But with the DH, the pitcher doesn’t have to bat, and he has less incentive to avoid the HBP.

(This is a hot topic, and there are more studies than I’m summarizing here, but I’ll eventually get to the only two Retrosheet-data-based ones I know of, which are probably the best.)

In 1997, Brian Goff, William Shughart, and Robert Tollison started the ball rolling – er, beaning – with a study showing that hit batsmen were substantially more common in the AL than NL, seemingly confirming the “moral hazard” hypothesis. (The paper doesn’t seem to be online and I haven’t actually read it, but it’s cited a lot.)

Steven Levitt responded in 1998 with this nice article, pointing out that almost the entire difference is explained by the fact that pitchers, being sucky hitters (I am paraphrasing), don’t get hit very much. Non-pitchers are hit at almost the same rate in both leagues. Furthermore, pitchers who hit more batters are no more likely to be hit themselves, which means that self-preservation is unlikely to be a motive.

More recently, this study by J. C. Bradbury (of sabernomics.com) and Douglas Drinen used Retrosheet data to analyze the question in more detail. For all interleague games 1997-2003, they ran a regression on a game-by-game basis instead of team by team, attempting to predict HBP (received) based on a bunch of season and game team factors.

They find some serious significance. Walks, HR, and game score difference were significant at 1% or better.

Most importantly, the DH was significant at the 5% level (the DH meant more HBP, as expected). Retaliation, as measured by the number of batters the team itself hit, was extremely significant at 1% -- in fact, the observed value was seven standard deviations from the mean.

The authors write that this means the DH is associated with an 11% increase in HBP, and each batter hit increases the number of its own HBP by 10% to 15%.

I don’t know if the authors accounted for this, but if one HBP in the game increases the frequency of opponent HBPs by 10% in that game, it follows that it must have increased the frequency of subsquent retaliatory HBPs by more – since only half of HBPs can be retaliatory. (Assuming a team won’t retaliate after a retaliation.) And that’s the number that we’re most concerned with: how the rate of plunking increases after an HBP, not before.

Here’s an example, which you can ignore. Consider two teams, A and B. A hits B unprovoked in 1/3 of games. B hits A unprovoked in 1/3 of games. Nobody gets hit in the last 1/3 of games. Finally, teams retaliate 50% of the time they are hit unprovoked.

On average, every six games will look like this:

1. No HBP
2. No HBP
3. A hits
4. A hits then B hits
5. B hits
6. B hits than A hits

On average, A and B hit 1/2 batter per game each. But in games when a team is hit, it hits 2/3 batters per game. That looks like an increase of 33% due to retaliation. But we’ve seen that there is actually 50% retaliation. That comes out only if you look at the order in which the plunkings occurred.

So when Bradbury and Drinen find an increase of 10% to 15%, that’s probably understating the actual retaliation effect, because they’re including “pre-taliations” – and those presumably are less frequent, which brings the average down.

The same would be true of the home run and score situation. It seems to me that because this study doesn’t account for what came first, the effects it finds are lower bounds, and real life probably has an even stronger cause and effect relationship than what the study found.

But anyway, the main purpose of the study was to find a DH effect. And it did find one, significant at the 5% level.

Finally, Bradbury and Drinen have one more paper, this time analyzing the question in even more detail – by plate appearance rather than by game. For each plate appearance from several seasons in both leagues, they regressed HBP on about 15 variables, including score, batter and pitcher quality, whether the previous batter hit a home run, whether there was an opposing HBP in the previous half-inning, and so on.

Their findings: the DH increases the probability of an HBP by 11 to 17 percent. Also, the study finds that the pitcher has four times the chance of being personally hit after hitting an opponent in the previous half-inning. It appears that pitchers do reduce their HBP out of fear of getting hit themselves.

We can even estimate the fear. I’ll run through the calculation – let me know if I’ve made any mistakes.

Levitt’s study shows that pitchers are hit once per 335 at-bats. The increase in risk is three times that (the difference between average and four times average) or, say, about once per 100 at-bats. The chance of a pitcher coming to bat the inning following an HBP is, say, 40%. So after an HBP, the increase in pitcher plunkings is about 1 in 250 at-bats. That is, each 250 HBP committed by the pitcher cause him to get hit once himself.

So since pitchers reduce their HBP in response to this incentive, they marginally value their own safety about 250 times as much as they value the safety of the other team’s batters.

That might not be fair … Bradbury and Drinen’s study assumes that pitchers are out of danger if they don’t bat the next inning. Suppose that they always bear the four times risk in their next at-bat, even if it comes in a later inning or game. Then the 40% factor disappears, and instead of valuing their own safety 250 times as much, it becomes only 100 times as much.

So: introduce the DH, and a pitcher feels emboldened enough to hit 100 batters – just because it’ll save him being hit once. Is 100 to 1 the normal level of human self-interest? Or are pitchers particularly chicken?