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June 2, 2005

Crooked Numbers

Eight Is Enough

by James Click

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Any Dean Chance fan will tell you: Pitchers stink at the plate. Perhaps it's a mental thing: they spend so much time trying to prevent that crack of the bat that they just can't make it themselves or they'll break out in hives. Regardless, in 2004, NL pitchers notched a line more reminiscent of my Little League career than of major-league players: .146/.179/.187. That's an MLVr of -0.653; pitchers cost their team well over half a run over the course of a full game of plate appearances. That they rarely see more than two or three PAs in a game means they're probably only costing the team 0.3 to 0.4 runs. This estimate is borne out by the league run scoring averages: the NL averaged 9.26 R/G in 2004 while the AL managed 10.04, so the two pitchers managed to sap 0.78 R/G compared to their DH counterparts last year.

It's this discrepancy that leads to some interesting strategic situations when the pitcher is due to bat. And these situations are what prompted the following e-mail regarding one of the more common techniques employed in the National League: intentionally (or "unintentionally") walking the eighth-place batter to face the pitcher when there are already men in scoring position and two outs:

Watching Mike Matheny being intentionally walked to get to Jason Schmidt the other day, I wondered if you (or anyone else) had ever thought about whether the conventional wisdom about walking the number-eight hitter to get to the pitcher when there are men on base actually makes sense. Obviously, it should reduce run expectancy for that inning given average number-eight hitters and average hitting pitchers. But avoiding the pitcher leading off the next inning must increase run expectancy significantly as well. Is the short term gain worth intentionally avoiding getting an out from the second worst hitter on a team and facing better batters later in the game?

-- Adam S.

Thanks for the question, Adam. This strategy is almost exactly the same as the idea of walking Barry Bonds to face whomever the Giants have dragged back from the Shady Acres retirement home to bat fifth: Is the difference in the resultant run expectation after a walk greater than the difference in the performance of the current and following batter?

As a first pass, let's look at the various run expectation situations and how they change if a batter is walked. Here's the run expectation chart from 2004:


        0 Outs  1 Out   2 Outs
        ------  -----   ------
Empty    .5379   .2866  .1135
1st      .9259   .5496  .2460
2nd     1.1596   .7104  .3359
1st&2nd 1.4669   .9577  .4605
3rd     1.4535   .9722  .3623
1st&3rd 1.8540  1.2236  .5219
2nd&3rd 2.1343  1.4717  .6179
Loaded  2.2548  1.5946  .8082

Putting an additional runner on base increases the run expectation in every situation, so intentionally walking the eighth-place batter is a bad idea most of the time. But teams still employ this strategy. In 2004, here's how often they did so (looking at all NL games in innings 1-6 as a rough proxy for situations in which the pitcher is likely batting next):


Runners   Outs  Instances    IBB     UBB
-------   ----  ---------    ---     ---
2nd&3rd     2        62     .484    .081
3rd         2       114     .360    .096
2nd         2       311     .302    .074
2nd&3rd     1        71     .239    .071
1st&3rd     2       132     .023    .121
3rd         1        67     .015    .060
2nd         1       192     .010    .178
1st&2nd     2       231     .009    .117
Empty       0      1428     .000    .050
Empty       1      1169     .000    .061
Empty       2       896     .000    .070
3rd         0        20     .000    .000
2nd         0       124     .000    .089
2nd&3rd     0        14     .000    .071
1st         0       352     .000    .031
1st         1       438     .000    .050
1st         2       466     .000    .047
1st&3rd     0        41     .000    .073
1st&3rd     1       101     .000    .040
1st&2nd     0        96     .000    .083
1st&2nd     1       200     .000    .075
Loaded      0        41     .000    .000
Loaded      1        81     .000    .086
Loaded      2        76     .000    .066

IBB is the percentage of PAs that resulted in an IBB; UBB is the same but for unintentional walks. In the extreme cases (like second and third with two outs), intentional walk rates can approach 50%, but the strategic employment declines precipitously as the outs and baserunners decrease. In general, it appears teams only employ the strategy with any frequency in four situations: second and third with two outs, third with two outs, second with two outs, and second and third with one out.

As such, let's look at those situations. From the run expectation chart in 2004, we can see how much the run expectation changes by walking that batter:


Runners Outs   RE   RE w/BB  Diff
------- ---- ------ ------  ------
2nd&3rd 2   0.6179  0.8082  0.1903
3rd     2   0.3623  0.5219  0.1596
2nd     2   0.3359  0.4605  0.1246
2nd&3rd 1   1.4717  1.5946  0.1229

As a general rule, walking the eighth man is going to cost a team 0.15 runs in that inning. So if the difference between the pitcher and the eighth man is more than 0.15 R/PA, then it's a good idea to walk the eighth man in these situations.

To check it out, Marginal Lineup Value Rate (MLVr) comes in quite handy because it's based in runs and it's a rate stat. However, MLVr is R/G, not R/PA and because MLVr is based on inserting a player into a lineup of league average players, simply dividing MLVr by PA/G would give us a slightly incorrect answer. With an assist from the great Keith Woolner who saves little lost children and BP columnists in his spare time, the estimated R/PA of a batter can be estimated by dividing MLVr by the expected PA/G of that specific player in an average lineup. Thus:

MLV / PA = MLVr * (9-OBP-8*LgOBP) / 27

Where OBP is the player's OBP and LgOBP is the OBP of the league. In the NL in 2004, LgOBP was .329 and pitcher OBP was .179, so MLV / PA for pitchers is:

-.653 * (9--.179--8*.329) / 27 = -0.150

And for eighth-place batters (who batted .251/.323/.381 with a -.075 MLVr):

-.075 * (9--.323--8*.329) /27 = -0.017

On average, the eighth-place batter is 0.133 runs better than the pitcher--right in the middle of the run differentials in the four situations in which eighth place batters were typically walked last year. This means that in the average situation of eighth-place hitter and pitcher, walking the eighth-place batter makes the slightest bit of sense with men on second and third with two out and a man on third and two out, but not so with a man on second with two out or men on second and third with one out. Again, these differences are very, very slight, meaning teams are likely not doing wrong either way and the individual batters involved deserve high consideration. If a good hitting pitcher is coming up or the eighth-place hitter is someone like Matheny, walking the eighth-place hitter is probably not a good idea, but the converse is also true.

However, those run expectation numbers are based on the entire league. We have more information about the situation in which we're interested, namely the fact that the eighth-place hitter and the pitcher are not nearly as good as the rest of the league. Rather than looking at MLVr, let's look at potential outcomes for each PA and the resultant runs scored and baserunners--outs situation (and the expected runs from that situation). Essentially, we'll add the probabilities of the various outcomes of the pitcher's PA multiplied by the runs scored plus expected runs for the rest of the inning after each event, giving up a much more specific run expectation based on the situation and the quality of batters involved. (For more details on this kind of analysis, check out the description of the methodology involved in this article on sacrifices.)

Let's see how our four situations come out:


Runners Outs  Eighth Pitcher  Diff
------- ----  ------ ------  ------
2nd&3rd 2    .5783   .4213   .1571
3rd     2    .4518   .4254   .0264
2nd     2    .3199   .2419   .0780
2nd&3rd 1   1.2381  1.3731  -.1350

Note that in all eight situations (four for the pitcher, four for the eighth-place hitter), the run expectation is lower than in the league-average tables because the eighth-place hitter and pitchers are significantly worse hitters than average, despite my Little League coach telling me batting eighth was good because it was the "second cleanup hitter." However, as opposed to the rather steady and even distribution above among the four situations, here the differences are more varied. Given an average eighth-place hitter and pitcher, walking the eighth-place hitter with runners on second and third with two outs saves a run about every six or seven times, but doing so with only one out actually gives up an additional run nearly as often. When considering walking the eighth place hitter, the actual batters involved is important, but the differences between them and the league average will have to be much larger with men on second and third to change the strategy from the above conclusions.

Finally, let's try to take into account the second point of the e-mail: Does pitching to the eighth-place man decrease run scoring later in the game? This part of the question is much more difficult to answer because there are so many factors at play, most notably pinch hitters and leverage. For example, if the eighth place hitter gets out and it's the sixth inning, the chances of the pitcher batting in the seventh are significantly lower than of him batting in the second or third inning. But that doesn't mean we can't try to take a stab at things.

Let's assume that the pitcher is going to remain in the game the next time up no matter what. Given that, by pitching to the eighth-place batter and getting him out, the batting team is left with a certain number of plate appearances that must be doled out to a lineup leading off with their worst hitter: the pitcher. Thus, in essence, the pitcher's next PA takes the place of 1/8th of a PA from everyone else in the lineup, assuming that the last PA of the game will fall equally among the nine men in the lineup.

Using the same method for converting MLVr to MLV/PA, the average NL lineup in 2004 looked something like this:


Year  Lineup Lg   MLV/PA
----  ------ --   ------
2004    1    NL   0.0007
2004    2    NL   0.0049
2004    3    NL   0.0367
2004    4    NL   0.0488
2004    5    NL   0.0303
2004    6    NL   0.0067
2004    7    NL  -0.0090
2004    8    NL  -0.0170
2004    9    NL  -0.1030

If the pitcher gets an extra PA and everyone else 1/8th fewer, the lineup adds -0.103 (the pitcher) and removes 0.013 (the average of everyone else) for a net loss of -0.116. Now we must consider the probability that the eighth place hitter gets on base on his own if pitched to: his OBP. (While it's interesting to note that the OBP of eighth-place NL hitters is artificially inflated by the intentional walks they draw batting in front of the pitcher, we'll assume for the time being that their OBP is a fair approximation of their ability to get on base regardless.) As noted above, eighth-place batters reached base at a .323 clip in 2004 in the NL, so there's just a 67.7% chance that the above strategy will work, meaning the expected difference for the rest of the game is -0.078, again assuming that the pitcher is not replaced by a more talented pinch-hitter.

Let's sum things up: In our four situations in which the eighth-place batter is intentionally walked the most often, the strategy is effective when there are two outs, but in the case when there is one out, walking to get to the pitcher carries a negative run expectation based on average eighth-place hitters and pitchers. Further, attempting to get the eighth place hitter out as opposed to walking him generally gains a team 0.078 runs per game by removing that one plate appearance from the end of the game. Going back to run expectation differentials, suddenly only with runners on second and third with two outs looks like a viable situation in which to employ this strategy. In that case, the run expectation in the inning in question drops from .5783 to .4213 runs if the eighth place man is walked for a gain of .1571. Subtract from that .078 (the expected runs lost by granting the extra PA to the other team) and the net gain is .0791. Walking with a runner on third now costs a team .0516 runs, with a runner on second it's dead even, and with only one out and runners on second and third, it's .213--more than a fifth of a run.

In general, with average batters in the eighth and pitcher's spots in the order, walking the eighth-place batter to get to the supremely weaker hitter appears to only make sense with runners on second and third and two outs, of the situations in which the strategy is typically employed. There may very well be other situations in which this strategy is positive, but the high cost of granting the opposing team another PA makes the likelihood of finding immediate returns that outweigh that cost small.

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