Designated Hitter July 13, 2006
Empirical Analysis of Bunting

Baseball analysts have been near universal in their condemnation of the overuse of the sacrifice bunt. While acknowledging it as the correct strategy in a small number of cases, most feel that any gain in moving players around the bases is more than offset by giving up an out, the "clock" in baseball. Much of this disparagement of the sacrifice bunt derives from analysis based on expected runs tables (ERT). In this essay, I will introduce more detailed and targeted expected runs tables along with an empirical comparison with what happens when teams actually bunt.

In their seminal book, The Hidden Game of Baseball, John Thorn and Pete Palmer popularized the ERT. Although introduced in concept a number of years earlier, it was Palmer's baseball credentials, the authors' lucid highlighting of its numerous applications, and the book's popularity that led to the ERT gaining a more widespread usage. Essentially, an expected runs table provides the average runs scored over the remainder of the half-inning (i.e. the batting team) from any of the 24 possible base/out states (i.e. the number runners on base and outs).

By necessity, the original expected runs tables were mainly derived from computer simulation of baseball games. The more recent availability of game data through Retrosheet, however, allows for the ERT to be calculated from actual play-by-play information. The overall expected runs table derived from this play-by-play data (for the years 1977 through 1992) is shown in table 1. Because the designated hitter materially impacts scoring, separate tables are necessary for each league. As a technical note, although I refer to these tables as expected runs tables to conform to common terminology, they technically reflect averages. That is, the tables represent the total runs scored over the remainder of the half-inning starting from a particular base/out situation divided by the number of such situations.

TABLE 1 - Expected Run Table (1977-1992)

```AL         0        1        2      NL        0        1        2
-----------------------------------------------------------------
---     .498     .266     .099     ---     .455     .239     .090
x--     .877     .522     .224     x--     .820     .490     .210
-x-    1.147     .693     .330     -x-    1.054     .650     .314
xx-    1.504     .922     .446     xx-    1.402     .863     .407
--x    1.373     .967     .385     --x    1.285     .907     .358
x-x    1.758    1.187     .507     x-x    1.650    1.123     .466
-xx    2.009    1.410     .592     -xx    1.864    1.320     .566
xxx    2.345    1.568     .775     xxx    2.188    1.487     .715
```

Where, for example, "xx-" means runners on first and second, third base empty. Thus if a team has runners on first and second with no outs, on average they will score 1.504 runs over the remainder of the half-inning; with one out, .922; and with two outs, .446.

Because it is always the first event of an inning, the no runners/no outs state (the top left corner) reflects the average number of runs that a team scores in an inning. As table 1 indicates, on average over the 16 year period, the AL scored .043 runs per inning more than the NL (almost entirely due to the DH). This translates to about 4/10 of a run per game.

One fairly common application of the ERT is the evaluation of various in-game strategies, such as the sacrifice bunt. For example, if an AL team has a runner on first base with no outs, the team can be expected to score .877 runs before the end of the inning. If the batter successfully executes a sacrifice, the team would find itself with a runner on second and one out. Based on the table, the expected runs in this latter situation is .693. In other words, according to the ERT executing a successful sacrifice bunt actually lowers the run expectation over the remainder of the inning by .184 runs (.877-.693), while a failed sacrifice lowers the expected runs by .255 (i.e. to a runner on first, one out).

In large part because of these expected runs tables, most baseball analysts have concluded that except in very rare instances, the sacrifice bunt is a poor strategic decision, and that managers use the bunt much more often than optimal. In fact, overuse of the bunt is one of the main criticisms baseball analysts level at the conventional baseball wisdom.

Of course there are many caveats that apply to conclusions based on an ERT. Most important, the table reflects an overall average; in many situations the actual run expectation may differ significantly than that identified by the table. For example, with a pitcher coming up, the expected runs are almost surely less than reflected in the table. On the other hand, with the heart of the order due up, the run expectation may materially exceed that indicated by the table.

Furthermore, late in games teams may be playing for one run, and increasing the overall run expectation may be secondary to simply scoring one run. To examine this topic, one really needs run probability tables to evaluate the probability of scoring at least one run in the various base-out situations. In this essay, I will use run probability tables as well as the ERT to evaluate bunting.

Recently several baseball researchers have further dug into the advisability of the sacrifice bunt by evaluating run potential based on a detailed probabilistic model of a specific sequence of batters and all possible outcomes. This research is valuable and suggests that bunting may not always be such a flawed strategy. But like the original expected runs tables, they are based on modeling outcomes, not on the outcomes themselves. With the availability of the Retrosheet files which include play-by-play data from games, one can begin to evaluate bunting strategies, not only from probabilistic models but from the results themselves.

A key drawback of the above expected runs tables is that they represent only overall league averages. Using the Retrosheet play-by-play output, though, allows for more finely parsing the data. As noted above, one of the problems with the overall ERT is that it takes no account of the ability of the hitter or the actual string of batters following him. One proxy for the quality of the batter and the following hitters is the batting order.

As readers of this essay likely know, the typical batting order follows rather orthodox principles. The leadoff hitter is usually good at getting on base and has some speed. The second place hitter has good "bat control", i.e. the ability to bunt or hit to the right side so as to move the runner along. A team often places its best overall hitter third, and its top power hitter in the cleanup position. The best remaining hitter with power typical hits fifth. The specific positioning of the remaining four hitters often depends on specific player abilities and managing philosophies, but very generally, these final four hitters typically bat in descending order of ability with the pitcher batting ninth in the National League.

By subdividing the data by lineup position, one can evaluate expected runs based on subsets that have different run potentials due to the average ability of the batter and immediately following sequence of hitters. Table 2 shows the expected runs based on lineup position for each league. For example, what does the expected runs table look like if the cleanup hitter is at bat? In other words, the ERT in table 2 reflects the expected runs over the remainder of the inning from each of the 24 base-out situations broken down by the nine lineup positions. To keep the data as pure as possible to reflect lineup position, appearances by pinch hitters in the identified lineup position are not included.

TABLE 2 - ERT by Lineup Position

```AL					NL
1	    0	    1	   2		  1	    0	    1	   2
---	 .553	 .291	.100		---	 .542	 .294	.102
x--	 .951	 .567	.210		x--	 .911	 .530	.213
-x-	1.263	 .753	.323		-x-	1.130	 .720	.342
xx-	1.614	 .966	.428		xx-	1.526	 .868	.418
--x	1.395	 .976	.399		--x	1.319	1.003	.399
x-x	1.840	1.242	.527		x-x	1.786	1.107	.506
-xx	2.182	1.456	.623		-xx	1.978	1.336	.621
xxx	2.365	1.621	.773		xxx	2.081	1.480	.722
```
```  2	   0	    1	   2		  2	    0	    1	   2
---	 .543	 .297	.113		---	 .530	 .286	.104
x--	 .966	 .576	.253		x--	 .977	 .611	.251
-x-	1.214	 .752	.346		-x-	1.180	 .723	.333
xx-	1.599	1.028	.453		xx-	1.583	 .979	.450
--x	1.435	1.012	.432		--x	1.368	 .971	.394
x-x	1.865	1.286	.531		x-x	1.778	1.211	.523
-xx	2.100	1.487	.609		-xx	2.068	1.375	.570
xxx	2.434	1.685	.822		xxx	2.398	1.473	.732
```
```  3	    0	    1	   2		  3	    0	    1	   2
---	 .536	 .305	.117		---	 .517	 .297	.118
x--	 .945	 .581	.268		x--	 .928	 .582	.278
-x-	1.192	 .740	.385		-x-	1.129	 .735	.395
xx-	1.609	1.002	.522		xx-	1.607	1.007	.518
--x	1.422	1.017	.400		--x	1.337	 .993	.401
x-x	1.820	1.249	.574		x-x	1.831	1.266	.562
-xx	2.052	1.534	.674		-xx	2.031	1.518	.715
xxx	2.468	1.699	.867		xxx	2.402	1.720	.817
```
```  4	    0	    1	   2		  4	    0	    1	   2
---	 .488	 .293	.118		---	 .442	 .274	.115
x--	 .885	 .567	.252		x--	 .849	 .553	.261
-x-	1.160	 .711	.343		-x-	1.098	 .719	.350
xx-	1.501	 .962	.488		xx-	1.488	 .961	.532
--x	1.318	 .972	.412		--x	1.308	 .958	.390
x-x	1.816	1.230	.530		x-x	1.741	1.247	.559
-xx	1.950	1.445	.644		-xx	1.864	1.426	.596
xxx	2.345	1.616	.863		xxx	2.457	1.615	.867
```
```  5	    0	    1	   2		  5	    0	    1	   2
---	 .452	 .254	.107		---	 .403	 .224	.103
x--	 .835	 .537	.245		x--	 .757	 .494	.220
-x-	1.110	 .706	.339		-x-	 .925	 .648	.340
xx-	1.453	 .930	.463		xx-	1.336	 .913	.452
--x	1.223	 .946	.373		--x	1.159	 .942	.389
x-x	1.674	1.200	.529		x-x	1.579	1.163	.496
-xx	1.900	1.353	.550		-xx	1.881	1.356	.607
xxx	2.301	1.601	.795		xxx	2.284	1.588	.775
```
```  6	    0	    1	   2		  6	    0	    1	   2
---	 .446	 .231	.094		---	 .370	 .191	.079
x--	 .791	 .464	.220		x--	 .725	 .430	.210
-x-	1.059	 .646	.336		-x-	 .941	 .585	.309
xx-	1.415	 .905	.459		xx-	1.311	 .851	.404
--x	1.328	 .951	.367		--x	1.095	 .829	.342
x-x	1.712	1.129	.518		x-x	1.435	1.106	.452
-xx	2.016	1.340	.581		-xx	1.764	1.336	.531
xxx	2.200	1.532	.755		xxx	1.997	1.536	.726
```
```  7	    0	    1	   2		  7	    0	    1	   2
---	 .439	 .225	.083		---	 .363	 .183	.061
x--	 .800	 .438	.201		x--	 .652	 .388	.176
-x-	1.076	 .617	.310		-x-	 .913	 .540	.261
xx-	1.408	 .836	.419		xx-	1.293	 .756	.385
--x	1.230	 .888	.354		--x	1.242	 .749	.327
x-x	1.625	1.107	.453		x-x	1.507	1.036	.419
-xx	1.852	1.360	.570		-xx	1.718	1.220	.469
xxx	2.337	1.480	.753		xxx	2.062	1.450	.717
```
```  8	    0	    1	   2		  8	    0	    1	   2
---	 .474	 .226	.077		---	 .397	 .172	.054
x--	 .798	 .461	.179		x--	 .678	 .375	.127
-x-	1.039	 .609	.283		-x-	 .923	 .485	.230
xx-	1.431	 .804	.410		xx-	1.179	 .694	.321
--x	1.419	 .919	.347		--x	1.212	 .782	.274
x-x	1.674	1.105	.444		x-x	1.514	 .945	.429
-xx	1.962	1.322	.561		-xx	1.620	1.157	.495
xxx	2.289	1.465	.686		xxx	1.994	1.315	.661
```
```  9	    0	    1	   2		  9	    0	    1	   2
---	 .519	 .263	.081		---	 .450	 .194	.050
x--	 .852	 .480	.182		x--	 .739	 .362	.125
-x-	1.128	 .641	.293		-x-	1.022	 .542	.181
xx-	1.475	 .927	.382		xx-	1.238	 .705	.230
--x	1.423	 .947	.341		--x	1.281	 .753	.236
x-x	1.725	1.145	.457		x-x	1.466	 .891	.269
-xx	2.108	1.396	.513		-xx	1.730	1.048	.387
xxx	2.386	1.533	.709		xxx	1.930	1.219	.470
```
Table 2 clearly illustrates the impact of batting order position on expected runs. In the AL with the leadoff hitter starting an inning, one can expect .553 runs to score. On the other hand, with the seventh place hitter leading off an inning, this falls to .439 runs. In the NL where the pitcher hits, the fall off is even more drastic, from .542 with the leadoff hitter starting an inning, to .363 for the seventh place hitter. This difference equates to over two runs per game.

One technical qualification to note is that the tables are derived from all events and include actual bunts in the calculation of their averages. A judgment was made, however, that this confounding effect was less significant than incorporating only those cases in which no bunt occurred. The latter involves significant self-selection: only the poorer hitters (and pitchers) bunt, resulting in a data set unrepresentative of the overall expectations.

Based on table 2, the sacrifice bunt still seems like a poor play in most runner on first, no out situations. The one exception, not surprisingly, is the NL pitcher spot, where a successful bunt reduces the expected runs by only .019 (.739 with the ninth place hitter up and a runner on first/no outs to .720 with a leadoff hitter up, a runner on second and one out). Given that these tables reflect overall averages of many teams over many seasons, it follows that a pitcher bunt would make sense in many specific instances.

However, one can now uncover an instance where a successful bunt actually increases the expected runs using the ERT. The run potential with runners on first and second with no outs and the pitcher hitting is 1.238. A successful sacrifice bunt brings up the leadoff hitter with runners on second and third and one out; a state with a run expectation of 1.336. Again, this needs caveats: not all bunts are successful, but clearly given that this is based on overall averages, one can imagine circumstances in which a bunt is the correct strategy.

One examination that researchers sometimes apply to the expected runs tables is that of breakeven percentages. That is, on what percentage of sacrifice attempts does one have to be successful to make the attempt at least a breakeven proposition with respect to the expected runs over the remainder of the inning. To take the example above: if the run potential without a bunt is 1.238 and with a bunt is 1.336, on what percentage of sacrifice attempts does one have to be successful to raise the run expectation above 1.238? Assuming an unsuccessful attempt results in no base runner advance and an out, the resulting run potential is .868. Thus the breakeven sacrifice percentage in this instance is 79% [(1.238-.868)/(1.336-.868)].

While mathematically these breakeven calculations appear helpful, I find them mostly irrelevant and avoid them for two reasons. First, more than two possible outcomes exist on any bunt attempt. For example an error would load the bases with no out: an increase to a run potential of 2.081. Other outcomes such as a double play could drastically reduce the run potential. While the probability of these and other potential outcomes remain small, they alter the run potential enough that any breakeven analysis that ignores them risks materially invalid conclusions.

But more basically, as I hope these tables begin to illustrate, the actual run potential in any situation is extremely dependent on the batter and hitter sequence following his at bat. A breakeven analysis offers value only after working out an accurate expected runs table. Until we have the ability to generate expected runs tables for each applicable batting sequence it does not really make sense to begin calculating a breakeven analysis, and only then if we can include probabilities for all the possible outcomes as discussed above.

The most interesting part of the analysis, however, is investigating the average results of actual bunts. While the ERT can help calculate the expected change in run potential given a successful or unsuccessful bunt, detailed analysis of the Retrosheet data provides an understanding of what actually happens on bunts. Managers want to win; therefore they may very well bunt in situations which offer a better run potential than average. To go back to the original example, a successful bunt with a runner on first and no outs in the AL appears to lower the run expectation by .184 runs; what happens in practice when teams bunt?

Table 3 provides the results of what actually happens when teams bunt. The analysis looks at all situations in which there were at least 200 bunt attempts and compares the results from bunting to all results. Unfortunately, Retrosheet does not make a specific notation for a sacrifice bunt attempt. From the data one can track either successful sacrifice bunts or all bunts (including those attempted for base hits). Fortunately, one can assume that few bunt attempts are for base hits and often occur with the bases empty.

As an example of how to interpret table 3, with no outs, a runner on second, and the eighth place hitter up in the American League, on average 1.039 runs will score over the remainder of the inning; this ties back to table 2. In those instances in which the batter executed a successful sacrifice bunt the expected runs over the remainder of the inning increased to 1.057. After any bunt, the expected runs over the remainder of the inning grew to 1.082.

Table 3 demonstrates that when teams actually bunt they sometimes do, in fact, increase the expected runs over the remainder of the inning, particularly late in the order with no outs and a runner on second or first and second. And this analysis aggregates all bunts: intelligent, ill-advised and those in between. The results imply that managers have at least some ability to recognize those situations in which bunts can increase run scoring. Bunting runners from first to second, except by the pitcher, still appears more problematic. But it must be remembered that the overall run potential of a base/out situation reflects the average of a large number of occurrences, and in many situations the expected runs are surely as low or lower than those that result from a bunt.

TABLE 3 - Results of Actual Bunts Compared to All Events

```Lg      BOP	Runners	  All   SH Only  All Bunts
No Outs
A	1	x--	 .951	 .848	 .899
A	1	-x-	1.263	1.062	1.203
A	1	xx-	1.614	1.635	1.676
A	2	x--	 .966	 .753	 .848
A	2	-x-	1.214	1.131	1.206
A	2	xx-	1.599	1.694	1.744
A	3	x--	 .945	 .769	 .818
A	5	x--	 .835	 .702	 .752
A	6	x--	 .791	 .642	 .643
A	6	xx-	1.415	1.416	1.388
A	7	x--	 .800	 .664	 .709
A	7	xx-	1.408	1.517	1.430
A	8	x--	 .798	 .714	 .715
A	8	-x-	1.039	1.057	1.082
A	8	xx-	1.431	1.575	1.496
A	9	x--	 .852	 .802	 .790
A	9	-x-	1.128	1.146	1.137
A	9	xx-	1.475	1.464	1.455
N	1	x--	 .911	 .878	 .909
N	2	x--	 .977	 .784	 .837
N	2	-x-	1.180	1.094	1.185
N	2	xx-	1.583	1.606	1.612
N	5	x--	 .757	 .800	 .714
N	6	x--	 .725	 .683	 .682
N	7	x--	 .652	 .575	 .587
N	8	x--	 .678	 .619	 .611
N	9	x--	 .739	 .769	 .724
N	9	-x-	1.022	1.159	1.137
N	9	xx-	1.238	1.404	1.325
1 Out
N	9	x--	 .362	 .380	 .354
N	9	xx-	 .705	 .732	 .724
```

(Technical note: the "All" column does not include pinch hitting appearances, while the two bunt columns do)

Overall, table 3 highlights that, in general, when managers elect to bunt they produce results superior than that assumed by the expected runs tables. In the example above--AL: eighth place hitter up, runner on second, no outs--a successful bunt ought to reduce the run scoring potential for the remainder of the inning from 1.147 to .967 according to the overall ERT in table 1. Even the batting order subsets generated in table 2 suggests that a successful sacrifice bunt reduces the expected runs over the remainder of the inning declines from 1.039 to .947, a smaller reduction but a reduction nonetheless. Using the game generated data, however, illustrates that on average, managers use the bunt strategically enough to actually increase the run expectation over the remainder of the inning from 1.039 to 1.057.

Of course, when bunting, teams are often not as concerned about the overall run potential of an inning, but the probability of simply scoring one run. One of the terrific things about the Retrosheet play-by-play data is that one can also generate tables that contain the probability of scoring at least one run. Table 4 resembles the overall expected runs table in table 1, but the numbers reflect the probability of scoring at least one run as opposed to the expected runs over the remainder of the inning. For example, over all the games in the AL from 1977 through 1992, the probability of scoring at least one run with no out and a runner on second is .634.

TABLE 4 - One Run Probability Table (1977-1992)

```AL	   0	   1	   2	NL	   0	   1	   2
---	.276	.161	.067	---	.261	.148	.061
x--	.432	.277	.129	x--	.424	.268	.124
-x-	.634	.414	.226	-x-	.609	.400	.216
xx-	.637	.430	.236	xx-	.622	.413	.220
--x	.839	.670	.279	--x	.814	.648	.267
x-x	.870	.656	.289	x-x	.847	.650	.275
-xx	.867	.689	.275	-xx	.838	.664	.267
xxx	.875	.679	.331	xxx	.860	.668	.315
```
Table 4 provides a little more evidence of why managers bunt. The probability of scoring a run decreases only from .432 with a runner on first and no outs to .414 with a runner on second and one out. As these values represent an overall average of all games, one can imagine that it many cases it surely increases the probability. Once again we can generate these tables by batting order position as a surrogate for the multiple batter sequences that can produce a huge variation in expected outcome.

TABLE 5 - One Run Probability Table by Lineup Position

```AL				NL
1	   0	   1	   2	  1	   0	   1	   2
---	.302	.170	.067	---	.301	.173	.066
x--	.458	.292	.121	x--	.426	.263	.120
-x-	.662	.436	.218	-x-	.606	.411	.232
xx-	.658	.427	.233	xx-	.653	.428	.228
--x	.827	.655	.295	--x	.794	.666	.284
x-x	.872	.672	.302	x-x	.864	.662	.305
-xx	.874	.691	.293	-xx	.852	.669	.290
xxx	.867	.696	.340	xxx	.829	.661	.338
```
```  2	   0	   1	   2	  2	   0	   1	   2
---	.298	.176	.073	---	.300	.171	.065
x--	.483	.306	.148	x--	.497	.320	.146
-x-	.665	.433	.236	-x-	.659	.429	.223
xx-	.678	.466	.233	xx-	.653	.433	.232
--x	.870	.687	.294	--x	.846	.653	.276
x-x	.880	.671	.296	x-x	.855	.651	.299
-xx	.887	.725	.278	-xx	.847	.696	.268
xxx	.889	.700	.347	xxx	.882	.663	.323
```
```  3	   0	   1	   2	  3	   0	   1	   2
---	.299	.185	.077	---	.301	.187	.078
x--	.457	.308	.150	x--	.470	.314	.156
-x-	.649	.439	.252	-x-	.644	.436	.254
xx-	.673	.460	.271	xx-	.668	.459	.262
--x	.842	.707	.288	--x	.854	.696	.291
x-x	.899	.688	.312	x-x	.889	.687	.302
-xx	.874	.731	.302	-xx	.888	.701	.315
xxx	.910	.702	.365	xxx	.879	.720	.350
```
```  4	   0	   1	   2	  4	   0	   1	   2
---	.280	.182	.081	---	.271	.176	.083
x--	.433	.294	.141	x--	.444	.304	.149
-x-	.635	.427	.231	-x-	.632	.437	.234
xx-	.645	.445	.252	xx-	.638	.443	.260
--x	.830	.667	.290	--x	.811	.680	.278
x-x	.868	.668	.299	x-x	.866	.680	.308
-xx	.867	.706	.288	-xx	.862	.694	.268
xxx	.886	.682	.351	xxx	.908	.692	.352
```
```  5	   0	   1	   2	  5	   0	   1	   2
---	.261	.163	.076	---	.245	.151	.073
x--	.405	.286	.136	x--	.396	.269	.132
-x-	.629	.416	.231	-x-	.581	.412	.235
xx-	.622	.435	.241	xx-	.630	.441	.236
--x	.842	.660	.263	--x	.794	.682	.288
x-x	.859	.654	.294	x-x	.855	.685	.284
-xx	.842	.672	.260	-xx	.853	.674	.278
xxx	.883	.691	.342	xxx	.878	.701	.326
```
```  6	   0	   1	   2	  6	   0	   1	   2
---	.252	.148	.066	---	.221	.127	.060
x--	.394	.253	.128	x--	.383	.249	.128
-x-	.599	.405	.233	-x-	.559	.380	.218
xx-	.604	.427	.239	xx-	.606	.418	.224
--x	.793	.668	.270	--x	.756	.640	.266
x-x	.869	.636	.291	x-x	.832	.651	.278
-xx	.876	.654	.278	-xx	.838	.668	.261
xxx	.844	.668	.315	xxx	.848	.684	.307
```
```  7	   0	   1	   2	  7	   0	   1	   2
---	.245	.140	.059	---	.211	.115	.046
x--	.394	.240	.117	x--	.350	.226	.113
-x-	.605	.380	.216	-x-	.556	.357	.200
xx-	.602	.403	.235	xx-	.592	.389	.222
--x	.814	.628	.266	--x	.784	.582	.267
x-x	.845	.643	.277	x-x	.830	.653	.257
-xx	.864	.690	.272	-xx	.815	.656	.236
xxx	.854	.668	.331	xxx	.831	.681	.322
```
```  8	   0	   1	   2	  8	   0	   1	   2
---	.259	.134	.053	---	.220	.104	.038
x--	.393	.247	.106	x--	.360	.204	.082
-x-	.593	.379	.207	-x-	.537	.324	.168
xx-	.608	.392	.216	xx-	.549	.346	.194
--x	.855	.652	.262	--x	.759	.583	.222
x-x	.847	.627	.264	x-x	.810	.600	.286
-xx	.843	.651	.266	-xx	.727	.625	.248
xxx	.864	.656	.302	xxx	.840	.610	.306
```
```  9	   0	   1	   2	  9	   0	   1	   2
---	.277	.154	.052	---	.240	.109	.030
x--	.423	.252	.108	x--	.397	.217	.072
-x-	.624	.386	.209	-x-	.585	.342	.133
xx-	.624	.425	.213	xx-	.580	.336	.136
--x	.822	.653	.266	--x	.781	.530	.194
x-x	.860	.646	.272	x-x	.734	.514	.165
-xx	.872	.678	.241	-xx	.770	.543	.197
xxx	.872	.662	.304	xxx	.808	.559	.218
```
Table 5 indicates a number of cases in which a successful bunt increases the probability of scoring a run. In the AL, when the ninth place batter bunts with a runner on first and no outs, the probability of scoring at least one run moves from .423 up to .441. The impact is even greater with a runner on second and no outs when playing for one run. For example, a successful bunt by the ninth place AL batter with runners on first and second increases the probability of scoring from .624 to .691. And this phenomenon is not limited to the bottom of the order. A successful sacrifice bunt by the second place hitter in this base/out situation raises the probability of scoring at least one run as well.

Again it needs to be emphasized that the batting order is simply a proxy for studying sequences of varying quality hitters. Even subdividing the data by the various lineup positions aggregates large amounts of data that mask many of the nuances in all the myriad possible sequences of batters. Thus, it certainly seems likely that many individual situations offer a much greater potential increase in the probability of scoring at least one run.

Table 6 compares the probability of scoring one run based on the overall run probability tables with what actually happens when teams bunt, using as reference the various lineup positions. As the table makes clear, when managers elect to bunt (on average) they typically increase the probability of scoring at least one run. In some cases the jump can be substantial. In the AL for example, if the number two hitter successfully sacrifices with a runner on second and no outs, the probability of scoring jumps from .665 to .736.

Once again the data underscores that managers employ the bunt much more advantageously than an arbitrary reading of the run probability table would suggest. Using the example above--AL: number two hitter at bat, runner on second, no out--a successful bunt should increase the probability of scoring from .634 to .670, based on the overall run probabilities in table 4. According to the lineup derived table 5, a successful sacrifice should increase the probability of scoring in the base/out example from .665 to .707 based. In fact, in those instances when managers chose to bunt, a successful sacrifice increased the probability of scoring to from .665 to .736.

TABLE 6 - Probability Results of Actual Bunts Compared to All Events

```Lg      BOP	Runners	 All   SH Only  All Bunts
No Outs
A	1	x--	.458	.476	.478
A	1	-x-	.662	.681	.692
A	1	xx-	.658	.766	.729
A	2	x--	.483	.455	.474
A	2	-x-	.665	.736	.726
A	2	xx-	.678	.757	.738
A	3	x--	.457	.448	.451
A	5	x--	.405	.405	.400
A	6	x--	.394	.386	.372
A	6	xx-	.604	.686	.667
A	7	x--	.394	.386	.395
A	7	xx-	.602	.715	.654
A	8	x--	.393	.421	.404
A	8	-x-	.593	.664	.646
A	8	xx-	.608	.714	.681
A	9	x--	.423	.454	.440
A	9	-x-	.624	.708	.693
A	9	xx-	.624	.703	.656
N	1	x--	.426	.457	.459
N	2	x--	.497	.461	.472
N	2	-x-	.659	.741	.735
N	2	xx-	.653	.686	.670
N	5	x--	.396	.454	.403
N	6	x--	.383	.417	.419
N	7	x--	.350	.401	.387
N	8	x--	.360	.384	.376
N	9	x--	.397	.432	.405
N	9	-x-	.585	.723	.663
N	9	xx-	.580	.678	.616
1 Out
N	9	x--	.217	.247	.226
N	9	xx-	.336	.339	.342
```

(Technical note: the "All" column does not include pinch hitting appearances, while the two bunt columns do)

Over the past couple of decades baseball analysts have seemingly discredited the bunt in all but the most obvious situations. Much of their evidence is based on the use of an overall run expectation table that reveals a loss of run potential even with a successful sacrifice. These overall expected runs tables, however, fail to differentiate between the innumerable possible scenarios of the ability of the hitter at bat and those following in sequence. Subdividing the data by batting order position allows a look at more finely dissected sequences of player ability. Although most of this analysis still indicates a successful bunt does not increase the run potential, it certainly shows that it in certain base/out situations it is not as detrimental as commonly believed. In fact, disaggregating by batting order still averages over many different player ability sequences, suggesting that in a number of instances a bunt may actually increase the run potential.

The Retrosheet play-by-play data allows us to partially test this hypothesis that managers can outperform the run expectation tables by a selective employment of the bunt. While not conclusive, the data here is clearly suggestive: in some base/out situations teams do increase the run expectation with a sacrifice bunt beyond the overall run potential implied by the ERT. And furthermore, even in those cases in which the runs expected over the remainder of an inning after a sacrifice bunt are less than that derived from the ERT, the decrease is typically less than the derived value. In addition we can assume that a manager typically bunts in those situations in which the specific sequence of batters is inferior to the average reflected by the ERT.

It is in the case of playing for one run, however, that the overall aptitude of managerial decisions shows up most clearly. As table 6 reveals, when managers bunt they usually increase the likelihood of scoring at least one run in the inning. And this increase is materially greater than that suggested by simply looking at run probability tables. While the bunt should and will remain a controversial managerial decision, it is clear that managers use it more judiciously than a cursory analysis based on the run expectation and probability tables would suggest.

Dan Levitt is the co-author of Paths to Glory, winner of the 2004 Sporting News-SABR Baseball Research Award. He manages the capital markets for a national commercial real estate firm.

Wow, that's enough to make your head spin. I am a raging anti-bunt activist so I found that interesting reading. I noticed your caveats, but I think if possible, more information needs to be gathered on what percentage of sacrifices are actually successful. There should be some sort of system of demerits for bunting into a double play, fouling out, taking out the lead runner, etc. This is in no way scientific, but just from observation I'd say one of every 12 sacrifice bunts leads to a double play, to say nothing of fouling out or the lead runner getting nabbed (giving away an out for nothing).

I'm surprised there's no mention of The Book, which contains a very detailed analysis of when to bunt. Highly recommended if you're interested in the subject.

Sorry, I see that you've mentioned "other analysts" and that you're complementing their analyses with "Retrosheet era" data. Very nice job.

The last table is fascinating, but I'm not sure I'm interpreting it correctly. First of all, not all lineup places/basesits are included. I assume that's because there were no bunts, or too few bunts? Did you have a minimum sample size for each cell? Or are these selected for a reason?

And the first column is the average number of times one run scored before the play, while the last two columns are average number of times one run scored after the play, I assume? A plus/minus in the right-hand column would be very useful for interpreting that data.

If I'm interpreting it correctly, that's tremendous info. Thanks.

Just to clarify, I meant that adding two columns to the last table, showing the +/- of each event (SH and bunt) would be very helpful.

I don't know how much detail you can input into a simulation game, but if you had all your team's batter's projected probabilities of hitting against each of the other team's pitchers in any situation (runners on, count), then apply a factor for each opposing reliever (e.g. closer 95% as effective in 2nd inning or after 20 pitches in same inning), and have some pre-arranged situations you would pinch hit or opponent would swap pitchers, you could set up a situation such as:
top of 7th, tie, 0 out, #6 hitter reaches base. Opposing pitcher has thrown 90 and will get pulled at 110 or against a LH batter, 2 runners on w/ under 2 out, 2 runs down, etc.
Bunt, swing away, steal, hit & run? Run many games and see what wins the most rather than what scores the most, which also includes your ability to prevent runs (just use a table?)
Bonus points for getting their closer to pitch more than one inning (maybe about .02 win), deduction for allowing a complete game (maybe about .05 win) or other reliever rest/use apart from extra innings (which wears your pen equally).

wow. pretty comprehensive stuff. I cant help but think that we might see significant changes when you plug in the more recent numbers from 1993-present. Theres several smaller parks, the recent homerun binge years, etc. that would seem to have some bearing on the figures. Baseball just feels like a different animal than it was 20 years ago.
Regardless, thats a good lunch time read. I never would have looked into that myself. thanks.