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Here's a question. Which late-inning situation is more important: When your team is up by three runs, or down by one? In the past, I've argued that closers should come into tie games far more often than they do now and that teams might use their closer for two innings to protect a one-run lead in the eighth, rather than a three-run lead in the ninth. Should a team actually use its closer/best reliever sometimes when it is behind? If a visiting team finds itself behind by a run heading into the bottom of the eighth, it will have one last chance to either tie or go ahead in the top of the ninth, but it has to get through the eighth inning first. A home team trailing going into the top of the ninth might have the same conversation. Should they use the best that they have to keep the score close in the hope of pulling off a comeback?

We know that because closers are rarely brought into non-save situations, someone else will generally come in when the team is behind. That someone probably won't be as good. In fact, teams seem to bring in their third- or fourth-best reliever in these situations. Standard sabermetric orthodoxy suggests that the third-best reliever should handle the three-run save while the closer might handle more delicate situations. Well, what would happen if they did switch?

Warning! Gory Mathematical Details Ahead!
For this study, I relied on Retrosheet game logs from 1993-2012. Retrosheet for the Hall of Fame!

Let's start with the beginning-of-the-ninth-inning, three-run save. For the home team faced with that situation, this is what has historically happened. Most likely, the closer was on the mound.

Runs Scored

Percentage

Win Probability After

0-2

96.4% (77.2%, 13.2%, 5.9%)

100%

3

2.3%

62.6%

4

0.9%

18.1%

5

0.3%

7.8%

6+

0.1%

3.4%

Now, let's see what has historically happened during the "down a run" situation, likely with the third-best reliever on the mound.

Runs Scored

Percentage

0-2

94.6% (74.1%, 14.2%, 6.3%)

3

3.1%

4

1.2%

5

0.5%

6+

0.6%

We can see that the crew that normally handles this situation isn't quite as good as the pitcher who handles the three-run save. No surprise there. Let's set those numbers side by side and pretend that the guy who handles "down a run" takes care of "up three runs":

Runs Scored

Percentage with Closer

Percentage with Third-Best Reliever

Win Probability After if this were a save situation

0-2

96.4%

94.6%

100%

3

2.3%

3.1%

62.6%

4

0.9%

1.2%

18.1%

5

0.3%

0.5%

7.8%

6+

0.1%

0.6%

3.4%

If you calculate that out, the expected win percentage with the closer on the mound is about 97.9 percent. With the third-best reliever (who usually gets the "down one run" assignment), it's 96.6 percent. So, a team that lets the non-closer take the three-run save will lose about 1.3 percent of an expected win. However, it would probably be less than that in practice, because if things really got that hairy in the ninth, it's likely that the closer could still be summoned.

The above numbers consider only what has happened when the home team finds itself in this situation. If it were the visiting team trying to guard a three-run lead in the bottom of the ninth inning (vs. trying to guard a one-run deficit in the bottom of the eighth), the chart would look like this:

Runs Scored

Percentage with Closer

Percentage with Third-Best Reliever

Win Probability After if this was a save situation

0-2

95.5%

93.8%

100%

3

2.6%

3.1%

50%

4+

1.9%

3.1%

0%

In this case, our closer has a win expectancy of 96.8 percent and our other bullpen guy has an expectancy of 95.3 percent, which is a difference of 1.5 percent. Let's call it a net loss of about 1.4 percent in win probability in save situations for having simply switched the two job descriptions.

Now, let's look at what has happened in reality when a home team tries to prevent a one-run deficit from getting bigger in the top of the ninth, usually with that third-best reliever. Again, we juxtapose the closer's performance in a three-run save next to it.

Runs Scored

Percentage with Third-Best Reliever

Percentage with Closer

Win Probability After if this had been a "one run down" situation

0

74.1%

77.2%

18.1%

1

14.2%

13.2%

7.8%

2

6.3%

5.9%

3.4%

3

3.1%

2.3%

1.5%

4

1.2%

0.9%

0.6%

5+

1.1%

0.4%

0.2%

With the third-best reliever in, our team has an 11.7 percent chance of winning after this inning, because remember, we're already down a run and it's late. With the closer, it's 12.3 percent. Bringing in the closer adds about 0.6 percent of a win.

What about the visiting team trying to prevent a one-run deficit from getting bigger in the bottom of the eighth? Let's again look at the performance of our third-best reliever vs. our closer and how often they give up different numbers of runs.

Runs Scored

Percentage with Third-Best Reliever

Percentage with Closer

Win Probability After if this had been a "one run down" situation

0

69.4%

75.0%

14.9%

1

17.0%

13.9%

6.5%

2

7.4%

6.5%

2.8%

3

3.0%

2.6%

1.2%

4

1.8%

1.6%

0.5%

5+

0.8%

0.3%

0.2%

With the third-best reliever in, our team has a 14.8 percent chance of winning. With the closer, it's 15.2 percent. There is a benefit to bringing in the better pitcher: summoning the closer here saves about 0.4 wins for a visiting team and 0.6 for a home team. Let's call it half of a percent of win probability in general.

The conclusion: You lose more by taking the closer out of the easy save situation than you gain from putting him in to keep a small deficit from getting bigger. Even when you factor in that teams face the "down by one" situation more often than "up by three" (about 9000 times from 1993-2012 vs. just shy of 7000) it's actually still a losing bet.

Of course, there are a couple of conceits that I've allowed myself. One is that pitchers might perform differently in different situations. Here I've assumed that the closer will simply pitch with the same quality in both the three-run save situation and the "make sure we don't fall any further behind" scenario. The other is that if a manager really were going to pull a closer away from his three-run save duties, he'd probably have his second-best reliever take over (and then that reliever would be pulled away from something else). If you squeeze real hard, you might be able to make the case that you could pull the two situations even, but using the closer for a three-run save has the head start. It's actually better to use a closer for a cheap save than to try to stop a deficit from getting bigger.

Throwing Good Pitchers After Bad
There's something rather heroic about these sorts of situations. A team that has a come-from-behind walk-off win in the bottom of the ninth will be the lead on Baseball Tonight, assuming that no one took steroids that day. And after the big home run is shown five times, someone will point to the yeoman's work by the guy who held the line in the top of the ninth to keep the game within one run. And for what it's worth, pitching a scoreless inning is a good thing, and I encourage all relievers to do so as often as they can.

The problem is that we might have a skewed view of how often these efforts are rewarded rather than simply in vain. A team that enters the ninth inning down a run wins the game less than 20 percent of the time. It's seductive to believe that "This team is never really out of a ballgame," but in fact the rewards that a team is chasing in that situation are small enough that it's not worth using your best reliever to chase them. A manager wouldn't use a good reliever when down by 15 (unless he needed to get some work in), because in that case, the game is essentially over and tiring out an effective arm would be throwing good money after bad. And the ugly truth is that "down by one" is actually a lot like "down by 15" in that it's not a good place to spend a scarce resource—even worse than the much-derided three-run save. If it works, it might make for a great storyline. It just doesn't work often enough for it to be worth it.

The full reason is one that you might not expect. The difference in win probability between being tied and being down a run in the last inning of play is actually greater than being down one run and being down 87. That means that the marginal decrease in win expectancy for each individual run actually gets smaller with each run allowed. So while our closer is better than our third-best reliever at not giving up extra runs, that difference doesn't matter as much. In the three-run scenario, it's true that most relievers would convert the save, and that closers are somewhat more efficient at doing it. A lesser reliever might be more likely to give up a run or two, but since the lead is three, it doesn't matter. What drives this finding is that lesser relievers are more likely to give up three or more runs than closers, and those particular runs are valuable in that they tie the game or give up the lead. We’re not really valuing the closer's ability to put up a goose-egg (which even bad relievers do the majority of the time). We're valuing the fact that he's not as likely to implode, and as infrequent as that event is, it's a killer when it does happen.

But if there's a hidden message in here, it's this: teams continue to undervalue tied games in late innings. They do understand that small leads are precious, but they haven't yet caught on about ties. I have previously made the case that a closer should be in the game for the road team during a tie game in the ninth inning or later (because if you give up a run, you lose). However, home teams should likely do the same. A home team that allows a run in the top of the ninth to give the visitors a one-run lead does have some life left, but the swing in win expectancy is still pretty big. That's the kind of situation where you want the best available pitcher on the mound.