*"The grass has always been tall here. I love it."*

—Willy Taveras, in anticipation of his recent visit to Wrigley Field

—

Last week we took a look at the bunting feats of Rockies center fielder Willy Taveras, and how that stacked up from a historical perspective. Since that column ran, Taveras continues to be sidelined with a right quadriceps injury, so he hasn't added to his total of 27 bunt hits. However, that won't stop us from a taking a closer look at some of the strategic considerations behind bunting for hits and answering a few reader inquiries.

**Decisions, Decisions . . .**

Simply put, batters should bunt for hits when their expected success rate is equal to or greater than the break-even success rate. The break-even success rate is calculated by taking the expected outcome assuming the batter swings away, and combining it with the outcomes for both success and failure if the batter attempts to bunt for a hit. That simple formula, where BE is the break even success rate, P is the present value (the outcome assuming the batter swings away), F is the failure outcome, and S is the success outcome, is as follows:

So, in order to determine whether a particular player should bunt for a hit, all we need to know are the three values P, F, and S. Fortunately, those values are available for 2007 in the form of the Run Expectations report here at BP. Using that report, for example, you'll notice that with no runners on and nobody out, the average number of runs scored in the remainder of the inning is around .522. So, for a leadoff hitter like Taveras, that value can serve as our P. The F value will then be the run expectation if Taveras were to fail in bunting for a hit, resulting in nobody on and one out (.277). The success value S is then a runner on first with nobody out (.910). Plug these numbers into our simple formula and we get a value of .387 or 38.7 percent.

In other words, if it were the case that Taveras thought he had a better than 38.7 percent chance of beating out the bunt when leading off an inning, then he should lay one down. It turns out that in those situations Taveras is actually 7-for-9 in 2007. Given just this as our inputs, what this indicates is that Taveras is not bunting for hits nearly often enough in this situation—he's had a total of 127 plate appearances with nobody out and no runners on—and that the defense is not adjusting enough to take away his ability to drop one down. Risking more attempts on Taveras' part, or more aggressive defense—say, instructing the third baseman to charge hard on the pitch—would both be factors that would drive down his success rate. In reality, however, things are a bit more complex than that. A small sample of nine attempts can be heavily influenced by chance, so it simply could be the case that he's been fortunate thus far and that both he and the defenses he plays against know this. This is supported by the fact that prior to this season he was 11 for 32 in these situations (34.4 percent), just a shade under the 2007 break-even success rate.

Juxtaposed against this are several important considerations. First, keep in mind that our calculation of the break-even success rate was predicated on an average 2007 hitter swinging away in the context of an average lineup. Since Taveras is much less likely to drive himself in with a home run or put himself into scoring position with a hit (despite playing in Houston and Denver, the man boasts a career .350 slugging percentage), the break-even rate would actually go *down*, making it more inviting to try and bunt. The same consideration applies to the hitters following Taveras, since hitters with above-average slugging percentages would lower the break-even rate, since it would not be as important for Taveras to slug an extra-base hit in order to eventually come around and score.

Second, it's also true that a speedy player, once aboard with a bunt hit, can get himself into scoring position via the stolen base. The higher his stolen base percentage, the lower the break-even success rate would have to be. Taveras is a 73.3 percent base stealer in 2007 (22 for 30), so the break-even rate only goes down a hair since the overall break even rate for stealing second base is 72 percent (calculated as (.910 – .277) / (1.156 – .277)). To see how you would combine a player's stolen base rate into the overall calculation, you simply substitute the original S value with a value representing the weighted average of the outcomes for that player attempting to steal second. In Taveras' case, we multiply his success rate with the run expectation of that outcome, and add the result to his probability of failure multiplied by the run expectation of that outcome, or do this:

.921 = (0.733 x 1.156) + (0.267 x 0.277)

Since the run value of .921 is greater than .910, when we substitute that value into the initial calculation the break even success rate goes down to 38.0 percent.

Finally, there is certainly value from the hitter's perspective in allowing the defense to think you *may* bunt, since this pulls infielders out of position, effectively increasing batting average and slugging percentage since ground balls down the lines can more easily sneak past defenders for extra bases. For someone like Taveras, who has historically hit over 50% of his groundballs to the left side of the infield, the tactic can be even more effective. We see hitters do this every day by bluffing the bunt at the cost of occasionally putting themselves down in the count. The more difficult variant of faking the bunt and still taking a swing at the ball is also used occasionally (mostly by pitchers) for the same reason. As you can imagine, this is a difficult aspect of bunting as a tactic to quantify, since defensive positioning is not available in today's play-by-play data, nor are bluff and fake bunt attempts.

When we look at the league as a whole, we find that with no outs and no runners on base batters are 95 for 229 (41.5 percent), with the added bonus that four of those attempts resulted in errors with the batter actually being safe, and five more included errors that allowed the batter to advance past first base. From this we again conclude that to a small degree batters may not be taking enough risks, and defenses are probably not doing enough to try and stop them.

Thus far we've only looked a single situation, namely nobody out and nobody on. We can apply the same procedure to all 24 base/out situations to come up with the following break-even success rates for bunting for hits in 2007:

Outs Bases 0 1 2xxx .387 .406 .468 1xx .262 .355 .512 x2x .230 .377 .690 xx3 .496 .488 .338 12x .021 .264 .549 1x3 .348 .465 .346 x23 .623 .637 .426 123 .435 .435 .446

Now, there are several assumptions built into this table that can be quibbled with. For example, I've assumed that when a runner is not on third, a failed bunt hit attempt still results in a successful sacrifice with the runners advancing (which explains the incredibly low break-even rate for runners on first and second). Similarly, a successful bunt hit attempt advances all runners including runners from third (explaining the relatively low break-even rates with runners on third and less than two outs). Attentive readers will also note that these values, calculated as they are from the empirical run expectancy chart, reflect what has actually happened during the 2007 season, and are therefore prone to the randomness inherent in any statistical sample. Theoretical run expectancy tables can also be generated using techniques such as a Markov Chain, or simulations that tend to even out the values somewhat.

Even given those caveats we can see that the break-even rate increases with the number of outs when a runner is not on third because of the trade-off the batter is explicitly making in terms of runner advancement (including his own). When comparing this table to the table presented last week titled "Success Rate by Outs," we can see that with the bases empty the success rate does indeed go up as the number of outs increases to 45 percent with one out, and 48.8 percent with two outs. The obvious reason that both the break-even and actual success rates correlate is that because the value of reaching first via a bunt hit *decreases* with the number of outs, the defense reacts accordingly by allocating their resources to other important tasks, such as preventing extra-base hits by guarding the lines. At the same time, hitters recognize the diminishing returns by bunting for hits less frequently. In the table labeled "Frequency By Outs" from last week we see that whereas 27.4 percent of all bunt hit attempts occurred with nobody on and nobody out, just 15.3 percent and 7.0 percent occurred with the bases empty and one or two outs, respectively.

Unfortunately, it's more difficult to compare the actual success rates from last week's table to the table above for any of the situations where runners are on base. As noted last week, the problem is that amongst our tabulation of bunt hit attempts will be included actual sacrifice attempts where the batter was credited for a hit as well as sacrifice attempts that resulted in a forc-out (and therefore an at-bat for the batter).

This entire section was based on average run expectancies in the 24 base/out situations but you'll notice we did not consider the score, the inning, or the specific run environment. All of this can be done using Win Expectancy instead of run expectancy, but not this week.

**Questions, Questions. . .**

Before we wrap things up, I'll take a crack at three reader questions. First, Larry Smith asks:

One thing that has had me frustrated the last few years is how announcers use the term 'drag bunt' for almost any bunt that is laid down for a hit. Correct me if I'm wrong but I always thought that a drag bunt, something that Mickey Mantle batting lefty often did, was a bunt laid down by a left-handed batter down the first base line. Bunts down the third base line should be more properly called 'drop bunts.' And whether a right-handed batter can drag a bunt at all is doubtful in my opinion. What are your thoughts about the terminology?

When I was writing last week's column I considered this briefly, and like Larry thought the term "drag" could only be used in the context of left-handed hitters, and only used it as such. That's why the title of the column was "Dropping One Down," and did not include a reference to dragging bunts.

In looking more closely, it appears that the technique of drag bunting made its way into the game in the years before 1925. It was then that F.C. Lane in his book *Batting* described the technique as one where "as the ball crosses the plate you hook the bat around it and drag it past the pitcher." It's fairly clear that he's using the word "hook," and the fact that he quotes John Tobin and Carson Bigbee (both left-handed hitters) suggests that early on drag bunting was a technique that applied only to lefties. However, I was able to locate several modern definitions of the term that include only the idea of taking a step towards first while the bunt is executed and one that explicitly notes that left-handers "perform this more often than right-handed hitters."

Even so, I still side with Laurence, since the mental picture of dragging the ball just doesn't jive with what a right-handed can do.

Next, reader Ryan Sullivan chimes in on the success rate for Taveras:

Great article. I can't believe Tavares has a success rate that high. I don't know how many of those bunt hits were at home and I have never been on that field but I know players talk about how high the grass is on the infield in Colorado. That could explain a little bit of an increase in success rate this year.

It is certainly true that the grass has been kept higher at Coors Field beginning last season in an attempt to depress run scoring somewhat. In 2007 Taveras has been slightly more successful at home than on the road, going 14 for 18 at home versus 13 for 18 on the road, but that difference is obviously not significant.

As far as parks go, there are some such as Wrigley Field that have a reputation for providing the long grass that makes bunting for hits easier, as Taveras indicates in his quote at the beginning of this article. However, since 2000 opponents at Wrigley have only been successful 28.4 percent of the time (25 for 88), and Taveras himself failed in his only attempt in the 2005 and 2006 seasons. On the other hand, in Anaheim opponents are 45 for 72, good for a 62.5 percent success rate. Playing surface is only one variable here, and the quality of the infield defense and pitchers also play a large role.

Finally, reader Marc Stone puts in his two cents on bunting for hits with two strikes.

As for bunting with two strikes. When a hitter is behind in the count 0-2 or 1-2, his batting average in that situation is likely to be south of .200 and his OPS under 500, it's not unreasonable to try something that has even a 30% chance of getting him on base—a batting average of .300 and an OPS of 600. On the other hand, bunting for a base hit when you're ahead in the count is just plain dumb, surprise or not—you're more likely to draw a walk and you might just get a cookie to hit.

Before responding to Marc's comment I should at this point correct an error in last week's column. Near the end I noted that 10 percent of two-strike bunt hit attempts ended in strikeouts. Other readers questioned this, and indeed my calculation was off—rather than 10 percent, fully **58 percent of two-strike bunt hit attempts in the 1970-2006 period ended in strikeouts**.

The high percentage of strikeouts on bunt attempts is what primarily accounts for the very low success rates (nine percent at 0-2, 11.6 percent at 1-2, 13.6 percent at 2-2, and 12.5 percent at 3-2) with two strikes. As you might have guessed, however, these percentages don't represent the normal case for two reasons. First, "bunt hit attempts" with two strikes as we've defined them are probably primarily first and foremost sacrifice attempts, as fully 81 percent of such attempts occurred with runners on base, and second, they are more likely to be tried by pitchers. When looking only at bunt hit attempts with the bases empty, we calculate a success rate of 27.7 percent, and as Marc points out, this is not bad considering that batting average and on-base percentage with two strikes drop precipitously.

On the other hand, I wouldn't be as quick as Marc to discount the advantages of bunting when ahead in the count, especially with nobody on base. Overall, the success rate is 44.8 percent when the batter is ahead, so unless the hitter has an on-base percentage of around .450 in these situations, it still may make sense to lay one down.