When should a team give up?

It's not been a good day. It's 15-1 in the bottom of the eighth inning. Unfortunately for your team, you are the "1" part of that equation. Worse, part of the reason that your team gave up 15 runs is that your starter had nothing tonight and gave up 8 of those runs while only getting 7 outs. That's not a good ratio. Three relievers have already pitched in what amounts to garbage time. This one isn'€™t going in the win column. Maybe it's time to tell the fifth outfielder to go out to the bullpen and warm up. He hasn't pitched since Little League, but he'€™s about to face Major League hitters. Or at least the AAA guys that the other manager has already subbed in. Why waste another reliever on a loss?

It's easy to think a team in this position should simply give up. MLB doesn't have a mercy rule, but sometimes things get a little silly. Position players pitching is the ice-cream truck of baseball. You might not need ice cream, but you're going to stick your head out the window and look. Because it's an ice-cream truck! There's a certain joy in the fact that Adam Dunn once pitched, despite having very few qualifications for being a Major League pitcher, except that he was a major leaguer. I think some of that joy comes with the fantasy that maybe I too can pitch some day in the majors.

It's not just me, though. Everyone loves it when this happens. So why don'™t we see more of these guys?

One thing is pretty clear. When a position player comes in to pitch, a team has basically given up. Sometimes, there are games where a position player gets called into duty because it'™s the 20th inning and there's no one left in the bullpen, but usually it's a manager saying, "I'm not going to throw good resources after bad. Let'€™s go through the bare minimum of the motions of playing a game and get to the post-game spread." Why bring another reliever out to throw an inning and have him be a little more tired for the next game?

It'™s one thing to say "Never give up!" But it'™s another to be reasonable about the matter and realize that there'™s a line in there. Somewhere. Perhaps we can figure out where that line is. But before we do that...

Warning! Gory Mathematical Details Ahead!

There's a statistic called win probability. We can summarize the situation in a baseball game by knowing a couple of pieces of information: The inning (and whether it's the top or bottom), the number of outs and where the runners are, and who'™s winning (and by how much). For example, if two teams are tied at the beginning of the third inning (no runners, no outs, top of the third, tie game), we can look back over baseball's recent history and see how many times that situation happened (17,260 times between 1993 and 2014) and how often each team won after that (54 percent of those games won by the home team, 46 percent by the visitors). We can say that the home team has a win probability of 54 percent.

Going by what'™s happened in the past isn't a perfect system for figuring out win probability. Run scoring has dropped nearly 20 percent since 2000, so a three-run deficit is a much bigger issue now than it used to be. Plus, not all teams are created equal in how they score runs, so a three-run deficit might be an unclimbable wall for some teams and a mere trifle for others, but the general idea is fairly sound. For a moment, let's assume that the historical average is good enough.

Let's say that a manager is in a situation where his team has a 1-percent chance of winning the game. All hope is not lost, but 99 out of 100 times, his team will lose. Yes, every once in a while an amazing comeback happens, but it's rather rare. Our longshot team might make it interesting or close, but close only counts in horseshoes and atomic weapons. Should the manager give up? By giving up, we mean that he would put in his reserves from the bench into the field and perhaps let the utility infielder toe the rubber. Let's just assume that by doing this, he completely surrenders of any chance he had left.

Subbing in position players likely doesn't have much effect for the next day. We know from research that players do play a little bit better when they'€™ve had a full day off, but that the effect is rather small, and here we'€™re only talking about a couple of innings. One could make the case that if the rest of the game is truly meaningless, it's better to get the good players out of the game, lest they risk injury, even though that risk is pretty small as well. All told, if a manager gives up on the game, he might as well empty the bench, but that's not going to have as much effect on his chances going forward. Every-day players are used to playing every day.

The question of who will pitch during garbage time is a much bigger one. Before a game, most managers sort their relievers into piles. There are the guys who are unavailable for one reason or another. There are the guys who are available, although sometimes at differing levels (available only in an emergency, available only for an inning, available for whatever). And then there are the "need to pitch" guys. That'™s why sometimes you see the closer come into a game when his team is up by nine. He needs the work. A blowout is a good time to get a "œneed to pitch" guy into a game. But what happens when you run out of those guys and now you're faced with using guys who are regular relievers? Should a manager send out a real pitcher to pitch the ninth inning or send out the back up catcher?

Here's where we start getting into the math. If a team has a 1-percent chance of winning and chooses to chase after that, it had better not incur a bigger penalty the next day. We know that relievers who pitch on back-to-back days lose a little bit of their velocity on the second day, but gain a little bit of movement. The question is how likely that is to result in a bad outcome. There's a general rule that ten runs equals one win, so a 1-percent chance at a win equals a tenth of a run. If you could make the case that pitching in garbage time might make a pitcher a tenth of a run worse over the course of his next outing, then it's foolish to send him out there.

What does it take to make a reliever lose one-tenth of a run worth of value? We estimate that the difference between your average out in baseball and your average single is something around .75 runs. Let'™s just focus on that. Since we don't know what will happen tomorrow, we'€™re going to focus on probabilities. A reliever might only face four or five batters in an outing. Let's assume that tomorrow he will be needed to face five batters. If, because of his garbage time outing, our reliever's overall chances of having an out turn into a single for one of those five batters increases by 20 percent overall (20 percent * .75 runs), then the penalty for his outing the day before is .15 runs, and therefore a bad investment.

But let's look at what that really means. We can get to an overall 20 percent chance of the reliever giving up an extra hit if we assume that each batter has an increased chance of 4 percent. That doesn't sound like much, until you realize that it's a bump in expected OBP against of 40 points. We're talking about a case where a reliever is 40 points worth of on-base percentage worse than he normally is simply by virtue of the fact that he'™s now pitching back-to-back days. A guy who has that much trouble pitching back to back doesn't make it to the majors to begin with. Even if we buy that such a pitcher might exist, his manager still has a hole card. The next day, knowing that Smith is the kind of guy who gives back 40 points of OBP on the second day, Smith's manager will simply make sure to sort him into the pile marked "don't use him in a critical situation, so if he really is that bad, he'll be bad in a situation where it doesn'™t matter.

But hang on, this is all based on a 1-percent chance of winning. What are some situations where a team has a one percent chance of winning? Using data from 1993-2014, I came up with a few examples:

- The bottom of the eighth inning, with one out and runners at second and third, and the home team up by a mere three runs.

- The bottom of the inning, with one out, a runner at third, and the home team up by four.

These don't seem like situations where a manager would ever give up (even though his odds are actually pretty remote of his team winning) and there are situations which are much worse. For example, in cases where they'€™re going into the top of the ninth inning and the home team is down by seven runs or more -- something that has happened in 2,971 games during the 22 years I looked at -- the home team has won exactly one of those, for an expected win percentage of 0.03 percent.  Now, all of a sudden, using the math above, if we sent out a reliever to pitch the ninth inning and the next night, he would be needed and he'd be a point or two of OBP worse in terms of our expectations, then sending him out there to pitch garbage time is a bad idea. If he doesn'€™t need to pitch, why chance it?

A lot of times, position players come on to pitch when the difference between the two teams is in the double digit range. Again, let's look at the beginning of the top of the ninth inning (no outs, no runners) with the home team trailing by X runs and look at their win expectancy.

Around a four-run deficit, we get back to the 1-percent mark (surprised?), and at a five-run deficit, we still round up to 1 percent. But past that, we get into territory where we wouldn'€™t have to expect much of a penalty for pitching back-to-back for keeping the real reliever in the bullpen and sending out the fake one instead. Sure, a a manager could just say "œWell, I'™ll just shuffle the pen so we don't need him tomorrow," but in these sorts of blowout games, the losing team often has burned through a few relievers already just to get to the eighth or ninth inning. The manager can't put all of them on the "won't use them" pile tomorrow. At some point, the manager might look at his roster and realize that there'€™s a very real chance that Smith will also be needed tomorrow night. He might want to start thinking about the kid who pitched a little bit in college who'™s sitting on the end of the bench, instead of Smith.

It's true that tomorrow's game might be another blowout in one direction or the other and it won€'™t matter how well or poorly the reliever would fare, but in 2014, almost two-thirds of games (63 percent) ended with the teams within three runs. Most games actually have some drama in them, even if it's mild drama, and because starting pitchers average only about six innings per start, some of that drama will involve the bullpen. Now, all of a sudden, we see that once we get to about a seven-run deficit, especially if a team has no off-day tomorrow and an even somewhat tired bullpen, and no one left who '"œneeds to pitch," the position player pitching starts to make sense. The effect isn't giant. Using this strategy won'™t make a team 10 wins better, but it does makes sense mathematically.

More Ice Cream Trucks, Please!

In 2014, the visitors found themselves down by 7 or more runs in the bottom of the eighth inning (they probably won't see the bottom of the ninth) and needing to field a pitcher 103 times. The home team was in a similar situation in the top of the ninth 96 times. It means that the average team found themselves in a situation where a position player pitching might make sense 6 or 7 times last year. In 2014, we did see an uptick in the number of position players who did their best pitcher impersonations, but if we assume that even a quarter of these situations fit the mold of "œno off-day, tattered bullpen," we should have seen roughly 50 position players pitching last year. Managers don't seem to be giving up quickly enough.

So, those of you who just can't get enough of utility infielders hanging out at the one spot on the infield that they never seem to get around to, rejoice. If the statistical revolution ever takes hold in baseball, someone will realize that there's a tiny (and I mean very tiny) advantage to be gained from putting a position player on the mound much more often than it is currently done. But it's an advantage. And maybe someone will actually get around to doing it more often.