Part 1
Part 2

Last time, we established several initial estimates for “thresholds” at
which point sacrificing becomes a good idea, either increasing raw run
scoring or increasing the probability of scoring at least one run. While
these estimates are a much more accurate way to evaluate the strategy of
sacrificing, they are lacking in several key areas.

First, BP’s resident Royals nut, Rany Jazayerli, pointed out that I
ignored one of Tony Pena’s favorite sacrifice situations: runners on
first and second and no outs. This situation is easily punched into the
equations developed last time and, jumping straight ahead to the
conclusions, this state–nicknamed Situation 4–falls somewhere in
between Situations 2 (a runner on first and no outs) and Situation 3 (a
runner on second and no outs). Here are the actual numbers:

Situation 4 - Runners on 1st & 2nd, 0 Out
Metric    Threshold    R-Squared
AVG         .201         .5204
OBP         .223         .7901
SLG         .211         .7055

Situation 4 - Runners on 1st & 2nd, 0 Out, Playing for 1 Run
Metric    Threshold    R-Squared
AVG         .277         .3875
OBP         .351         .5685
SLG         .452         .3712

As you can see, when playing for multiple runs, sacrificing in
Situation 4 makes sense only for pitchers. The threshold is low enough
that even the two more extreme hitters–one with terrible hitting
statistics followed by one with a high propensity for singles and doubles–cannot make this situation favorable for sacrificing. Playing for one
run, it rises to the levels of validity, but not nearly as much as
Situation 3 (.351/.436/.619). On a macro level, we can broadly say that
it makes sense to sacrifice in that spot a little more than half of the
time. Of all the situations encountered so far, this is the one in which
the conventional strategy is most in line with the equations presented
here: The best players will not sacrifice, but the average player will be
called upon by his manager when the game is close and one run is

The second major shortfall of the equations is found in the assumptions
presented at the beginning of Part 1. Primarily, assuming a 100% success
rate for sacrificing is not an accurate reflection of the events on the
field, a fact pointed out by more than a few readers. Therefore, the next
improvement involves trying to estimate the outcomes of a sacrifice based
on empirical data.

Rather than look at the batter’s results in various sacrifice
situations, we’ll look at the resultant base/out situation. The reason
for this is because the sacrifice is a play that both gives the defense a
choice and places it under a great deal of stress. Trying to cut down the
lead runner on a sacrifice is a high-risk, high-reward strategy and
results in a variety of scoring decisions (errors, fielder’s choices, etc.)
that don’t map absolutely to the resultant base/out situation. Further,
the results of a sacrifice can be thought of as falling into three
categories: success, failure, and overachievement. Obviously, when
sacrificing, the batter is attempting to concede himself for the
advancement of the runner. In “success,” the batter is out, but the
runner advances. In “failure,” the runner is out and the batter is safe
at first. In “overachievement,” the runner advances and the batter is
safe. (There is also the possibility of “miserable failure”–a double
play–and a few other rare ending states after errors, etc.) Looking at
the data for 2003 in three baserunner situations, the data yield the
following results:

Situation                     Success   Failure  Overachievement
Runner on first                 61.7      23.5        14.8
Runner on second                60.4      21.2        18.4
Runners on first and second     59.3      25.7        15.0

There are some more detailed breakdowns within those measurements that
I will include in the equations, but we can see from the numbers above
that sacrifices are successful about 60% of the time. The question now is
whether the overachievement and failure cancel each other out when looking
at run expectation–verifying the original threshold estimates–or if
our conclusions have changed significantly based on these new estimations
for success rates.

To incorporate this information into the existing equations, we will
simply enhance our estimation of run expectation for sacrificing, much
like when Batter One was upgraded from a singles hitter to a full hitter.
These outcome estimations will be added uniformly over all hitters; there
will be no adjustment for “good bunters” versus “bad bunters.” The
reasons for this are many, but the primary one is that there just isn’t
enough data out there to qualify each player’s sacrificing abilities. How
good of a bunter is Barry Bonds? I have no idea, and we
have no data on which to base assumptions. Observed data would certainly
lead us to believe that there are certain players more adept at succeeding
in a sacrifice situation than others, but the impossibility of accurately
gauging the differences combined with the likely marginal increase in
accuracy makes including them in the equations foolhardy. (Most of us do
our best not to be foolhardy around here, so we won’t add them.)

Having taken these adjustments into account, the updated threshold
estimates when attempting to maximize run scoring are:

Situation 1 - Runner on 1st, 1 Out
Metric    Threshold    R-Squared
AVG         .195         .5788
OBP         .221         .7913
SLG         .178         .8893

Situation 2 - Runner on 1st, 0 Out
Metric    Threshold    R-Squared
AVG         .191         .5916
OBP         .206         .9086
SLG         .182         .7891

Situation 3 - Runner on 2nd, 0 Out
Metric    Threshold    R-Squared
AVG         .249         .7195
OBP         .305         .8511
SLG         .363         .8074

Situation 4 - Runners on 1st & 2nd, 0 Out
Metric    Threshold    R-Squared
AVG         .218         .5810
OBP         .253         .8786
SLG         .266         .7870

And the data when the primary objective is one run:

Situation 1 - Runner on 1st, 1 Out
Metric    Threshold    R-Squared
AVG         .199         .4532
OBP         .224         .6506
SLG         .174         .7928

Situation 2 - Runner on 1st, 0 Out
Metric    Threshold    R-Squared
AVG         .233         .6333
OBP         .282         .8688
SLG         .322         .7677

Situation 3 - Runner on 2nd, 0 Out
Metric    Threshold    R-Squared
AVG         .364         .7390
OBP         .450         .5197
SLG         .646         .4976

Situation 4 - Runners on 1st & 2nd, 0 Out
Metric    Threshold    R-Squared
AVG         .268         .5323
OBP         .338         .7738
SLG         .430         .5070

The first thing to note is that most of the numbers have moved in from
the extremes. On the lower end of the spectrum, the threshold in
Situations 1 and 2 have come up from the extremely low levels, sometimes
under .100, to numbers slightly under and around .200. While this doesn’t
really change the conclusion about these situations, it does add a small
degree of validity to the idea of pitchers bunting in these situations.
Additionally, across the board, adding the probabilities for actual
sacrifice outcomes–instead of using an assumption of 100% success rate–actually increased run expectation for sacrificing. While sacrifices
“overachieve” less often than they “fail”–as noted above–the cost of
the failure is much less than the gains of the overachievement. These
calculations had a greater difference on Situations 1 and 2 than they did
on Situations 3 and 4.

For a final update, we’ll use the opportunity to take the opposing
strategy into account to some extent. As reader J.P. pointed out, the
opposing manager would likely intentionally walk the next batter or two
after a successful sacrifice in a late-game situation where one run is
paramount. To take this into account, the equations that compute the
percentages for scoring at least one run now assume the same double play
rates even after a successful sacrifice. This update will obviously not
affect Situation 1 (a runner on first and one out) since after a sacrifice
there are already two outs, but the other three situations are updated.

Playing for One Run (IBB)

Situation 1 - Runner on 1st, 1 Out
Metric    Threshold    R-Squared
AVG         .199         .4532
OBP         .224         .6506
SLG         .174         .7928

Situation 2 - Runner on 1st, 0 Out
Metric    Threshold    R-Squared
AVG         .177         .6314
OBP         .192         .8686
SLG         .153         .7636

Situation 3 - Runner on 2nd, 0 Out
Metric    Threshold    R-Squared
AVG         .277         .7823
OBP         .350         .5505
SLG         .451         .5240

Situation 4 - Runners on 1st & 2nd, 0 Out
Metric    Threshold    R-Squared
AVG         .206         .5521
OBP         .235         .8028
SLG         .263         .5234

Thus, having eliminated some of the key inefficiencies of the equations
from their initial iteration, the following conclusions can be drawn about
the data.

When run maximization is paramount (early in the game, in high run-scoring environments, etc):

  • Only pitchers should sacrifice a man from first to second in any
    circumstances. Even then, certain pitchers who are decent hitters
    should swing away.

  • With a runner on second and no one out, sacrificing makes sense when
    some of the league’s worst hitters are due up, with a hitter with a high
    propensity for singles and doubles following. The most likely instance
    of this is as a lineup in the AL turns over from the ninth spot to the
    first spot. Even then, instances where sacrificing increases run
    expectation are rare.

  • Sacrificing with men on first and second is only a good idea when
    pitchers are due up. While the thresholds here are higher than in
    Situations 1 and 2, they still remain far too low for even the worst
    regular position players.

When the probability of scoring at least one run is paramount (late in
a close game, in a low run-scoring environment, or facing a dominating
pitcher, etc):

  • Similar to the run maximization situation, only pitchers should
    sacrifice a man from first. Given that a pitcher would likely rarely be
    batting in this situation where runs are at a premium, this situation is
    likely to never occur.

  • Most of the league should sacrifice a man from second with no one out.
    While a line of .277/.350/.451 is slightly above average, recall that the
    skill set of the second batter due up should also be taken into account.
    On the whole, this finding is in the greatest agreement with conventional

  • When runners are on first and second, sacrificing is, again, not a
    good idea, a finding that is due almost entirely to the opposing manager’s
    propensity to intentionally walk the next batter to keep the double play
    in order. This 10% decrease (approximately) in the scoring probability of
    the situation is enough to reduce the threshold across a great deal of
    current hitters.

  • If a manager is certain that the opposition will not intentionally
    walk Batter Two, the validity of the sacrifice is increased in these

Therefore, in the broadest conclusion possible, we can say that
sacrificing is a good idea when pitchers are batting and, for most of the
hitters in the league, when there is a man on second, no one out, and a
single run is the goal. Even then, there is a set of the league’s best
hitters who should never lay down a bunt; which is too bad, because it
would be fun to see Bonds square around, just once.