## Reading The Book: Bunt Me Baby One More Time

Before we get into details, let's do some background. What is The Book? It's a book about baseball titled The Book: Playing the Percentages in Baseball. It was published in 2007 by three statisticians and/or computer programmers named Tom Tango, Mitchel Lichtman, and Andrew Dolphin. It covers a range of topics including hot streaks, pitcher-batter matchups, and platoon splits. A lot of their analysis is based on something called Run Expectancy, which depends on the base-out state. What's a base-out state? It's one of the 24 possible combinations of runners on base (empty, 1st, 2nd, 3rd, 1st and 2nd, 1st and 3rd, 2nd and 3rd, loaded) and outs in the inning (0, 1, or 2). For example, runner on third with two outs is a base-out state; bases loaded with no outs is a base-out state. You can total up the average number of runs that real teams in real games scored from each base-out state until the end of the inning. That's Run Expectancy. A related concept is Run Frequency, which is the chance that real teams in real games scored at least one run from each base-out state. Check out this table, from Tom Tango's website.

 Base Runners Run Expectancy Run Frequency 1B 2B 3B 0 outs 1 outs 2 outs 0 outs 1 outs 2 outs __ __ __ 0.544 0.291 0.112 0.293 0.172 0.075 1B __ __ 0.941 0.562 0.245 0.441 0.284 0.135 __ 2B __ 1.170 0.721 0.348 0.637 0.418 0.230 1B 2B __ 1.556 0.963 0.471 0.643 0.429 0.237 __ __ 3B 1.433 0.989 0.385 0.853 0.674 0.270 1B __ 3B 1.853 1.211 0.530 0.868 0.652 0.288 __ 2B 3B 2.050 1.447 0.626 0.866 0.698 0.280 1B 2B 3B 2.390 1.631 0.814 0.877 0.679 0.334

These numbers are from charts created by Tom Tango at the site I linked above, and are based on Retrosheet data from 1993-2010. How do you read them? Well, let's say you've got a runner on second and two outs. This corresponds to the third line of the third column of each table. The numbers tell us that teams scored an average of .348 runs from this situation until the end of the inning. There was also a .230 chance of scoring in this situation. In other words, teams scored at least one run 23% of the time when they had a runner on second and two outs.

What does any of this have to do with bunting? A ton. First, let's analyze the results of a "successful" sacrifice bunt using Run Expectancy (from now on: RE). Take a look at the RE with a runner on first and no outs: .941 runs. If the batter just swings away, you can expect to score an average of .941 runs. Sacrifice and you've now got a runner at second with one out, for an RE of .721 runs. Even though the runner advanced, by giving up that out the team has cost itself runs! How about a runner on second and no outs? Before: 1.170 runs; after (third base with one out): 0.989 runs. Again, a decrease in runs scored. Look at the before and after state for any other bunting situation you can imagine, and you'll see that giving up the out for a base always costs the team runs.

There are some situations, however, where we're not really concerned about how many runs the teams scores. All we care about is whether the team can score one run. This can happen late in close games, particularly when the score is tied in the bottom half of the 9th or extra innings. That's where Run Frequency (from now on: RF) comes in. Take a look at the chance of scoring at least one run with a runner on first and no outs: .441. Sacrifice the runner to second and the RF is down to .418, meaning there is now a lower chance of getting the one run. But with a runner on second with no outs, you can actually increase your chances of scoring by sacrificing. The RF jumps from .637 to .674 when the runner moves to third with one out, for an increase of a little less than 4% in the chance of getting at least one run. This is the only place I could find a benefit to the sacrifice in the numbers: close and late with a runner on second and no outs.*

Ok, so the sacrifice bunt is a bad play, right? In all but a tiny handful of cases, bunting hurts the team by decreasing, not increasing, the number of runs scored. This probably isn't news to most sabermetrically-inclined baseball fans. I was a strict adherent to this mode of thinking until The Book blew my mind by explaining that a sacrifice hit is not remotely equivalent to a sacrifice attempt.

How often do you think that a batter is successful when he attempts to sacrifice? If you first asked, "What is your definition of success?" then you are following this chapter very nicely so far. Let's say that a successful sacrifice bunt attempt is when the runner advances to second or beyond, whether or not the batter made an out, and whether or not the batter actually bunted the ball fair (i.e., it includes when the batter first attempts a bunt and then switches to swinging away, or walks or strikes out while still attempting to bunt). If you answered, "69.3% of the time," you were exactly correct! (The Book, page 247)

In fact, the authors go on to show that of all plate appearances from 2000-2004 where non-pitchers squared to bunt on the last pitch they saw, only about 65% ended as a typical sacrifice hit! That is, in 35% of cases, something other than "batter out, runner advances" occurred. I don't know about you, but I found this number shockingly high. Sometimes bad things happened: the batter struck out 1.5% of the time, got out without advancing the runner 6.8% of the time, caused a force out 7.8% of the time, and bunted into a double play 1.6% of the time. However, sometimes very good things happened: in about 16% of cases everyone was safe on a bunt single or an error, and another 1.2% of the time the batter pulled back the bat and received a walk or HBP. Note that this is for all bunt attempts, including situations where the defense was playing bunt all the way and charging hard. One would expect a higher proportion of good outcomes when the bunt is a surprise or perhaps even in traditional bunting situations when the defense isn't charging very hard.

The authors continue with a lot of interesting analysis (50 pages' worth, or about 1/7 of the book). I won't get into the details here, but their general conclusion is that it is almost never completely "right" or completely "wrong" to bunt in any situation. The hitting skill and bunting skill of the batter, the speed of the batter and runner(s), and most importantly the positioning of the defense all influence how often teams should attempt a bunt. Bad hitters should bunt more often, but even in late and close situations should not do it every time. Good hitters should bunt less often, and should try it both early in the game and late. Pretty much everyone should drop one at least every once in a while to keep the defense honest, except maybe someone like David Ortiz (a great hitter who is very slow and probably terrible at bunting). Although, with the crazy infield shift they put on Ortiz, perhaps even he should learn how to bunt.

To get an idea of how to think about this while watching a game, let's look at one more hypothetical. Suppose we've got a runner on first with no outs and an average batter at the plate. For simplicity, also suppose that a bunt attempt will result in one of only two outcomes: either the batter will be out and the runner will advance, or the batter will reach safely, whether on a bunt single or an error. How good do the chances of the latter outcome have to be in order to justify dropping a bunt? We can figure that out with the Run Expectancy chart and some simple algebra.

RE(1B, 0 out) = RE(2B, 1 out) * P(batter out) + RE(1B, 2B, 0 out) * P(batter safe)

.941 = .721(1 - x) + 1.556(x)

Remember, we're only considering two possible outcomes. Either the batter is out or the batter is safe at first. That's why the sum of those two possibilities adds up to 1. We set it equal to the RE with a batter on first with no outs because that's the starting RE of swinging away, and we want to find the break-even point. Solve for x and you get about 26%. In other words, in this hypothetical situation the batter needs a 26% chance of reaching safely to make the bunt as good a play as swinging away. Any higher and it's actually a better play. That number seems pretty high, especially when you compare it to the 16% rate of good outcomes that they found in The Book. And that's not including all the bad outcomes, which we didn't even consider. However, that 16% rate of good outcomes included all non-pitcher bunts in all situations, and one can imagine that in cases where the defense is playing back, the chances of bad things happening go down and the chances of reaching safely go up. I would guess that in quite a few cases, even decent hitters might reach this breakeven point if they surprise the defense every once in a while.

The one really important conclusion from all of this is that treating the bunt as a straight up sacrifice, where the batter is out and the runner advances one base, is simply incorrect. The prevalence of other outcomes is surprisingly high. Bunting as much as Barton did in 2010 probably hurt the team, but he may have been onto something. Who knows, adding the bunt as a regular part of his game may have pulled in the infield defense a bit and helped cause his career year. Rather than simply dismissing the bunt as a bad play in all but a handful of late-game cases, we should acknowledge it as an important, though perhaps still overused, part of any manager's arsenal.

*One caveat: this assumes average batters at the plate, on deck, etc. If you have a bad batter at the plate and a good batter on deck, some of these gaps could get smaller. This is because, compared to the numbers in the chart, the bad batter will have a lower RE from swinging away (the "before bunt" state) and the good batter will increase the RE in the "after bunt" state. The gaps are probably far too large to ever make sacrificing worthwhile from a Run Expectancy point of view, but perhaps in some late and close situations, where you're playing for one run, it could make sense to sacrifice runners from first to second as well as second to third.

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