**Part 1:** Taking One for the Team: When Does it Make Sense to Sacrifice?

**Batter One Hits for More Power**

Since Batter One, who hits nothing but singles, does not exist, we obviously have to expand our information on Batter One to more realistically simulate the choices facing the manager. Thus, we’ll add in the full range of result possibilities for Batter One, allowing him to do more than just single. Of course, that results in an immense formula that reveals little about the efficacy of sacrifices. Instead, viewing how the formula handles actual players can yield some more useful insights.

The data for all regular players from 2000-2003 still shows that sacrifices are almost never a good idea. Putting the 2001 version of Ichiro–the player with the highest breakeven point for Batter One’s AVG–in front of every batter, the minimum expected runs lost by sacrificing over swinging away is 0.018, when Ichiro bats in front of **Chris Truby** in 2002 and his massive .199/.215/.282 line while he was in Detroit. Using other batters who are also highly adept at taking advantage of a sacrifice for Batter Two yield no situations in which run expectation increases by sacrificing, at least when there’s a runner on first and one out.

Expanding the results to look at other sacrifice situations does not change these conclusions. Looking at the second situation–a runner on first and no outs–and using the same plan of attack, the smallest difference between sacrificing and swinging away is again Truby and Suzuki, but this time the difference is .085 runs. Other players who come close are **Craig Paquette** in 2002, **Alex Gonzalez** in 2000, and **Pat Meares** in 2001 with .100, .107, and .114, respectively. (Not surprisingly, the three players who should never sacrifice as Batter One are Barry Bonds 2003, Barry Bonds 2001, and Barry Bonds 2002, costing the team .466, .481, and .518 runs respectively.)

**Equation Enhancements**

Before tackling the third sacrifice situation–a runner on second and no outs–a few adjustments need to be made to our equations. In this situation, the objective of sacrificing is no longer to get the runner into scoring position, but rather to get him to third with fewer than two outs so that he can score on a sacrifice fly. As our equations do not allow that, judging this situation would not be fair using the current numbers.

Referring back to the 2003 base-running numbers above, runners score from third with fewer than two outs on a ball in play 61.5% of the time. Rather than simply letting the runner from third score 61.5% of the time, we’ll add in strikeouts for Batter Two so that the runner only scores when the ball is in play. (This percentage includes hits, so to get the percentage of times the runner scores on an in play out, we’ll use .615-AVG / OUT – K.) This, again, will mean that sluggers–who tend to have higher strikeout rates–will not make for good batters to follow a sacrifice. Certainly, the strikeout is as much due to the pitcher on the mound as it is due to the batter, but to keep things simple, we’ll leave the pitcher out of it.

Additionally, we can now add in one of the main benefits of the sacrifice: staying out of the double play. This sacrifice situation won’t take advantage of this benefit as much as the other two would, but we’ll add it in to increase the robustness of the model. As stated in the initial assumptions, batters will all GIDP at the league average rate: 13.0% of GIDP opportunities.

Having added these two new adjustments and running the model again, this time some positive results are generated. Looking once more to our Truby-Suzuki combination, in this case, sacrificing increases run expectation by .085. Therefore, in this situation, we have finally passed the threshold and found a reason to justify sacrifices. However, any enthusiasm should be tempered, as only the very worst hitters in the league, batting in a particular spot in the order, see any benefit. In fact, only 49 hitters in 2003 qualify in this situation. If we change Batter Two from Ichiro to another player, the numbers decline precipitously.

It would be remiss to add sacrifice flies and GIDP and not rerun the first two situations to see if the results change. In fact, sacrificing looks even worse when these adjustments are made. The minimum benefit of swinging away instead of sacrificing is increased to .056 and .103, respectively. Thus, we can more confidently dismiss these two situations, as there appears to never be a benefit to sacrificing, no matter who is batting in any particular part of the lineup.

**Simplification and Extrapolation**

Having done what we can to better estimate the sacrifice threshold based on the two batters due up in the lineup and the situation, we can take a step back and try to draw some conclusions based on the large data sample generated. Rather than generate the total number of combinations between every pair of batters, we’ll use a random sampling of about 2000 combinations, with every player included twice as Batter One and twice as Batter Two. The results should be extremely close to what we’d get if we did use the full set of over a million combinations.

In each situation, we can map various batting metrics to the expected runs scored by swinging away (RE_{SWING}) minus the expected runs for sacrificing (RE_{SAC}). Then, by setting (RE_{SWING} – RE_{SAC}) to zero, we can an estimated “average” threshold for that metric. Taking a quick look at the data, the metrics for Batter One map very well to the data while Batter Two does not, a discrepancy rooted in the fact that it is Batter One who is deciding between swinging and sacrificing. Looking at our three situations and mapping to AVG, OBP, and SLG for Batter One, here are the thresholds and Coefficients of Determination for each metric.

Situation 1 - Runner on 1st, 1 Out Metric Threshold R-Squared AVG .167 .5481 OBP .176 .7495 SLG .082 .8421 Situation 2 - Runner on 1st, 0 Out Metric Threshold R-Squared AVG .157 .5644 OBP .150 .8678 SLG .078 .7509 Situation 3 - Runner on 2nd, 0 Out Metric Threshold R-Squared AVG .244 .5394 OBP .303 .6481 SLG .359 .6076

Some bells should start to go off. Specifically, it’s impossible for one hitter to have a .157/.150/.078 line–the threshold in Situation 2. Since the threshold for each metric has been computed separately, discrepancies like this are going to arise, most notably because the slope of the line of best fit is steeper when mapping SLG than when mapping AVG or OBP. However, some general conclusions can be drawn. First and foremost, sacrificing almost never increases run expectation in the first two situations. No position players putting up lines like that are likely to keep their jobs, so we can conclude that sacrificing should always be restricted to some pitchers. Those lines are weak even for hurlers, so the better hitting pitchers, such as **Mike Hampton** or **Russ Ortiz**, should probably be restrained from sacrificing in those situations.

Also note that the threshold is significantly higher for Situation 3 than for situations 1 or 2. There are a significant number of everyday players, mostly on last year’s Tigers, that fall under some of those numbers. Based on the respectable r-squareds, we can comfortably set the threshold for sacrificing in this situation at .244/.303/.359.

There are a few notes of caution. Obviously, a player who is under all three numbers should sacrifice in these situations, but players who are under one but over two probably should not. Furthermore, these are based on an average lineup coming up behind Batter One. If Ichiro is due up, the threshold moves to .249/.310/.374; for Bonds, .203/.238/.241. This is certainly a wide range, but even the upper extreme for Ichiro barely enters the lower range of regular lines. This fact again reduces the instances when sacrificing increases run scoring. Thus, we can conclude that sacrificing increases run expectation in an extremely limited number of situations.

**Changing Goals**

Finally, we need to consider that sacrificing is a strategy often employed when getting one run is more important to winning than scoring many. To use the current set of equations to check that, we simply replace the run expectation values with the probability that the team will score at least one run. For 2003, the probabilities are:

Runners Outs None 1st 2nd 3rd 1st&2nd 1st&3rd 2nd&3rd Loaded 0 0.284 0.433 0.635 0.835 0.644 0.863 0.869 0.888 1 0.167 0.273 0.411 0.693 0.420 0.642 0.689 0.660 2 0.073 0.132 0.224 0.274 0.227 0.280 0.254 0.327

Running the same equations yields the following threshold estimates:

Situation 1 - Runner on 1st, 1 Out Metric Threshold R-Squared AVG .162 .4453 OBP .165 .6366 SLG .041 .7772 Situation 2 - Runner on 1st, 0 Out Metric Threshold R-Squared AVG .236 .5708 OBP .287 .7822 SLG .332 .6917 Situation 3 - Runner on 2nd, 0 Out Metric Threshold R-Squared AVG .351 .5161 OBP .436 .3660 SLG .619 .3507

The values in Situation 3 immediately stand out. Virtually every major leaguer is under those numbers; so, according to this model, it is almost ALWAYS a good idea to sacrifice in that situation if only one run is needed. This conclusion flies in the face of many of the conventions usually espoused by performance analysts, but it is supported by the raw numbers from the run probability table. Notice that the probability of scoring at least one run increases from .635 to .693 when sacrificing in Situation 3. Out of every combination tested, only one–**Brian Giles** and **Russ Branyan**–decreased the probability of scoring at least one run.

On the other hand, in Situation 1 it is still almost never useful to sacrifice, despite having reduced the requirement for a successful sacrifice. Situation 2 makes a significant move up the charts, but the numbers are still too low to make sacrificing beneficial in that situation in all but the rarest of circumstances.

**Conclusions**

These findings cannot be directly applied to real-life situations without first making some adjustments. First and foremost, the model assumes 100% chance of success when sacrificing. This is clearly not a realistic reflection of the choices facing the manager and thus, all conclusions that show sacrificing to be a good idea should be tempered. Second, the effect of steals has not been accounted for. Third, we assumed that all batters hit into double plays at the league average rate, and while that assumption should have a marginal effect on the results, it is not a true reflection of the events on the field. Fourth, the strengths and weaknesses of the opponent have not been quantified. Players seem to feel that certain pitchers are easier to bunt against than others and, if that is true, the quality of the opposing pitcher should certainly be taken into account, both with regard to the ability to sacrifice successfully and the likelihood of striking out.

Finally, team-specific information has not been included. By using league-average run expectation numbers, the model fails to account for the fact that teams with better offenses likely cost themselves more runs by sacrificing than teams with poorer offenses. However, by computing the individual numbers for Batter One and Batter Two, much of the rest of the team is removed from the equation. Also, team-specific run expectation charts do not contain enough information to instill confidence in their results. Teams don’t face all 24 base-out situations often enough over the course of a season to reduce the amount of noise in the data collected. Universally adjusting all run expectations for each team doesn’t affect the results either, as both RE_{SWING} and RE_{SAC} are adjusted. In the end, the effect is probably minimal, but it’s another improvement that could be made.

While it’s still true that sacrificing is an archaic, outdated strategy, there appear to be a few select game situations in which it remains a better option than swinging away. However, those situations are almost entirely limited to when there is a runner on second and no outs. Even in such a situation, it is only beneficial to sacrifice in certain parts of the lineup or when the quest of a single run is more important than maximizing run scoring. Hopefully, in those select situations, those of us who think that sacrificing is always a bad idea will remember that it’s likely helping our team out. This does not excuse the abuse of sacrificing, but the old school strategy does hold water on a few occasions.

#### Thank you for reading

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