# The True Cost of Sub-Optimal Lineups

*Warning: This post will have plenty of numbers and math and “advanced” things. I have to make up for this debacle.*

Before the season started, I wrote a piece about lineups. If you want another affirmation for lineups not mattering *that* much, you can click on the link and read what I had to say.

When talking about lineups not mattering that much, no one ever puts a number on it. Probably because it’s a lot of estimation and speculation. If that’s the case, you may not be convinced of the veracity of the analysis. You’re in luck today, however. I happen to love estimation and speculation! I also happen to love trying to convince people of the veracity of my analysis. I might have my work cut out for me. Onward!

In order to try and determine the true cost of a sub-optimal lineup, I’m going to examine a specific case; Brandon Phillips batting 3rd behind Joey Votto. This will represent somewhat of an outlier for an example given Votto’s prolific career at the plate. Their nearly 100-point spread in wOBA is about as big as you’ll ever see in the middle of lineup.

For this analysis, we’ll use the RE24 framework. I also wrote a piece about RE24 earlier in the year. If you need a refresher, please check it out. I’ll write the rest of this piece assuming you’ve read the previous piece. It’s a pretty good piece, if I’m going to toot my own horn for a minute. One Mr. Tom Tango, the new Senior Database Architect for MLB Advanced Media, agreed:

Ok, enough about that! Let’s get into it!

In order to try and measure the true cost of a sub-optimal lineup, we’re going to look at every (or some approximation of “every”) situation that can occur following a Votto at-bat in two different situations. The first situation will be a lineup of Phillips, Bruce, Duvall, and Suarez following Votto. The second situation will be a lineup of Bruce, Duvall, Suarez, and then Phillips following Votto. For this analysis, it doesn’t really matter whether Votto is batting 2nd or 3rd.

The reason why we can look at this problem in this specific way is because of some work done by the fantastic Jonah Pemstein and Sean Dolinar. They developed what they’re calling the “Batter-Specific Run-Expectancy Tool.” Essentially, it modifies the “league average” tables we’ve grown accustomed to looking at using data and math so we can get a more accurate view of RE based on the hitting skill (measured in wOBA) of the batter at the plate.

Here are the players in question, as well as some wOBA information and the “wOBA bucket” I’ll be using on the Tool to determine their personal RE24 table:

On this table you can see the normal stratification you’d expect from these players. I’m slightly weighting the bucket I’ll use based on recent performance. So what does one of these player-specific run expectancy tables look like? I’ll show the extremes to give you an idea:

Votto, as you’d expect, gives a higher RE in every situation. That is the power of not making as many outs at the plate.

How should we use these tables to answer our question, though? Getting to a specific, quantifiable cost is a bit tricky and takes some brute force.

First, we need to determine how often each batter in the analysis will be put in each situation described by each base-out state. To start, here is a simple example. If Votto were to lead off the inning, we know he’ll get out about 60% of the time (this assumes an OBP of .400). So, we know that in this situation Phillips’ at-bat will occur with 1 out and the bases empty 60% of the time. Makes sense? Excellent! Here are the values I’ll be using. You can probably justifiably quibble with some of the numbers, but I assure you, they don’t change the magnitude of the final answer in any non-trivial way:

Armed with these numbers, we can now begin a brute force attack on the question!

As a reminder, this analysis assumes a sequence beginning with Joey Votto batting, followed by some order of the other players. Votto is our constant. First thing we need to do is determine all the things that could happen after Votto bats with 0 outs, 1 out, and 2 outs.

Doing this is a series of probability calculations, which I did using a tabular method so I could keep everything straight. Here’s an example of one of those calculations straight from my spreadsheet:

The first chart shows the odds that Phillips comes to the plate in each given situation. The box in the top left filled with 4.2% represents what? Well, how many ways can Phillips come to the plate with the bases empty and 0 outs assuming Votto just batted? A home run, of course! You’ll notice the 4.2% in the box matches Votto’s HR rate from the previous table.

If we look at the same box in the 2nd chart, you’ll see it filled with 0.416. That is the player-specific run expectancy for a player with a .305 wOBA in that situation. Since BP is in that situation 4.2% of the time, we multiply 4.2% by 0.416 and we get 0.017, which you can see in the corresponding box in the 3rd chart. This process is repeated for every box and the results are then summed. The bold 0.413 you see on the side is the total of all RE for each situation Phillips can be in following a Votto at-bat which started with 0 outs. The way to interpret this number is “Phillips creates an average of 0.413 runs when he bats following Votto when Votto batted with 0 outs.”

Ok, so that was a lot of work for a single, tiny situation wasn’t it? Good thing for you I am a disturbed individual. The next chart you’ll see is a glimpse into the larger process. I can’t really explain it without showing it, so bear with me. We need to be able to account for each permutation of things that can happen in an inning and assign a somewhat accurate likelihood of each event occurring. Here’s how I did it:

On this larger slice of the same spreadsheet, you can see how I’ve laid out the rest of the batters. Bruce following BP has a lot more percentages filled in. This is because there are a lot more things that can happen after 2 at-bats (Votto then Phillips) than with just a single at-bat. I specifically highlighted the top-left Jay Bruce cell for a reason. It says 1.00%. If you look at the Excel formula bar, you see (0.042*0.025) + (0.358*0.025). This formula accounts for two things. First, the odds that Votto and BP will homer back-to-back. That would bring Bruce up in a bases empty, no out situation. Also, if Votto reaches in any non-homer manner ((0.400-0.042)=.358) and then BP homers (0.025), that would also get Bruce to this situation.

Without belaboring the method anymore, I recorded as many of these permutations as I could and then summed them up to see how close I got. In each case, I hit more than 98% of all possible things that can happen in an inning. The ones I missed are likely very fringe cases, like a player advancing from 1st to 3rd on wild pitch that occurred on a steal attempt. Since things like that don’t happen often, missing them shouldn’t change our results.

I used the above method to work up BP batting 3rd, as well as BP batting 6th, thus moving all the other players up one slot. Remember, we’re still only working with the situation where Votto begins with sequence with 0 outs. Here are the results:

In the “BP 3rd” column, you see BP’s cumulative RE from the spreadsheets above of 0.413, followed by Bruce’s 0.459, Duvall’s 0.340, and Suarez’s 0.086. Seems like a large drop-off a the end, and it is. The odds that the Suarez will even come to the plate in this situation is only 21%. Another way to say that is there exists a 79% chance that Votto, Phillips, Bruce, Duvall will create 3 outs before Suarez can bat. The rest of the lineup can be ignored, since the odds of each extra player coming up to bat in this sequence drops very quickly.

Ok. We are going to use some estimation and math to finish out quickly here before I lose the 5% of folks who are still with me.

Since Votto will come up in varying situations, we need a way to account for that. The way I am going to estimate the rest of the situations is as follows. First, we’ll estimate the likelihood of Votto coming up in each of the 0, 1, and 2 out situations. Over a large sample, we’d expect a somewhat close relationship, with 0 out being the smallest given the nature of not batting 1st in the batting order. I chose 30% for 0 outs, and 35% for 1 out and 2 outs. Then, we need to estimate the decrease in overall run expectancy in all situations given that sometimes the sequence starts with 1 out and sometimes it starts with 2 outs. To do that, I used the league average RE tables and reduced the “BP 3rd” and “BP 6th” numbers by the same proportion from 0 outs to 1 out as the normal league average RE decreases when going from 0 out to 1 out.

We then sum all those figures together, which gives us a “per sequence” run expectancy. We can then assume we’ll see this sequence 4 times per game, so we multiply by 4, giving us a “per game” run expectancy. Then, we simply multiply by 162 for a “per season” run expectancy. That’s what we were looking for, right? I think so… Here it is in tabular form, in all its glory!

The final answer is 14.75 runs per season. That is the true cost of batting BP 3rd instead of somewhere like 6th. Given FanGraph’s current measure of 9.64 runs per win, Bryan Price’s decision to construct the lineup in this manner is costing the Reds a whopping 1.53 wins per season.

See? Lineups don’t really matter *that* much. We all knew that before this long and arduous Friday morning article. However, the point I’m trying to make is that sometime in the future, 1.5 wins will probably matter. If a simple decision like this that often gets shrugged off can result in a swing of more than a win, so can other decisions that occur during a season. A good manager (and front office) need to identify things like this, low hanging fruit, we’ll say, and try to eke out as much advantage as they can. As sports fans, we know one thing is true… good teams don’t leave wins on the table.

*Note: wOBA data and tables generated from the wOBA Tool are courtesy of FanGraphs. The rest of this mess is courtesy of my noggin’.*

Lose 1.5 wins with the lineup. Lose 1.5 wins a year with the baserunning. Lose 1.5 wins a year by habitually playing shorthanded. Lose 1.5 wins a year by adding 1-2 guys annually to the roster that both the analytics guys and the old-time scout wannabes agree couldn’t outplay a corpse (Pacheco, Schumaker, etc. etc). In the end even if they actually put together some real talent then they end up rolling the dice in the wildcard game like the Choo Reds or the Pirates and crap out?

Bottom line it just gets old if you’re a fan of this team and its all correctable?

Agreed. Things that are easily correctable should be corrected.

I have no problem rebuilding and losing. I understand the process is something we have to go through as fans of a small- to mid-market team.

However, when we say day-in-and-day-out the same mistakes being made with seemingly not a care in the world, it certainly gets old, as you say.

Indy,

In any given situation of any given game, there is no right or wrong answer.

Votto walks with Cozart on 2B with no outs and Phillips then hits a single up the middle scoring Cozart and sendiong Votto to 3B. The result indicates that the decision to hit Phillips 3rd behind Votto was a good decision. Votto walks with Cozart on 2B with no outs and Phillips then GIDP moving Cozart to 3B with 2 outs. The result indicates that Phillips hitting 3rd behind Votto was a bad decision.

With a runner in 1B, 1 out and the Reds leading by 2 runs, JC Ramirez enters the game and stikes out the next 2 hitters. The result indicates that the decision to bring in Ramirez was a good decision. With a runner in 1B, 1 out and the Reds leading by 2 runs, JC Ramirez enters the game and gives up a 3-run HR to the first hitter he faces. The result indicates that the decision to bring in Ramirez was a bad decision.

As you point out, over the course of a season, repeating such decisions will have a cumulative effect, good or bad, based on the statistically anticipated outcome. You failed to include the repeated use of the sac bunt and giving up valuable outs in your argument. These decisions cost the team wins and the cumulative impact of such decisions by the manager and/or front office can have a significant impact on the success of the team.

The problem I have with analyses like these is that they assume players’ stats will remain constant after a major lineup change occurs. If BP were dropped to 6th or 7th, I believe he would sulk, but also be determined to prove the manager wrong. His performance would likely be affected one way or another. Maybe he starts swinging for the fences even more than he already does, or maybe he becomes more selective and starts trying not to make any outs. Conversely, a player suddenly moved to the top of the order will want to prove the manager right. Maybe he starts swinging for the fences, or maybe he becomes more confident and more selective. We just can’t know what will happen until it’s been tried. Stats are great tools — I’m a big fan — but we can’t lose sight of the fact that these are human beings and not video game characters with pre-programmed hitting abilities.

BP has never changed his approach before, so I don’t know why he would now. I wrote about that very thing earlier in the year. You can click on my name to get to all my posts and scroll down to find it if you want.

Also, in the absence of a crystal ball, I’ll take data over speculation any day of the week.

Also, analysis doesn’t say “I guarantee with 100% certainty that this change will cost the Reds 1.5 wins.” Analysis says “given the most likely outcomes, barring any unforeseen circumstances, this is likely to cost the Reds around 1.5 wins.” There’s always caveats with analysis.

Everyone is speculating that BP isn’t being moved because he’ll sulk. If so, it isn’t at all crazy to think this will affect his performance one way or another. Early in his career, being shifted from 1st to 2nd to 5th would likely not have changed his approach. But we don’t know how he would respond to a demotion by this manager at this stage of his career. The stats all say Duvall can’t possibly be as good a bitter as he seems to be. But maybe his confidence changed after realizing he was the everyday left fielder, and maybe he actually improved. Humans can be funny that way.

Hitter, that is. I doubt Duvall is bitter.

I’m not one of the people who thinks BP would sulk. I think he’d be the same, below average hitter regardless of where he hits. And the data backs that assertion up. Like I said above, speculation is ridiculous. We have data, so there’s no reason not to use it as an aide.

Regarding Duvall… it’s not that the stat says he CANT be a great hitter… it’s that no other hitter in history had as bad of a walk-to-strikeout ratio as Duvall has, that also went on to have a productive career.

Duvall could certainly be the first. That is a real possibility. He could also change his approach to walk more and strike out less. That is also a real possibility. But until he actually makes a change, there’s no reason to think he’ll be the one exception in the history of baseball. It’s far more likely that he won’t be that one exception.

I think the improvement Duvall made this past off-season in compacting his swing has been responsible for a lot of his dramatic improvement this season. If he can focus on better plate discipline this next off-season, Duvall could make the argument regarding his once-in-baseball history results irrelavent. Minor league power hitters routinely see there ticket to the show as simply hitting more HR.

The Old Cossack’s perspective is that Duvall saw his opportunity after 2015 and made a serious effort to improve as a major league baseball player, both offensively and defensively. Jay Bruce made the same step-wise improvement to his hitting by gradually making off-season adjustments to hit to all fields with power rather than just trying to jerk the ball into the Ohio River. He was making good press along that path until his knee injury derailed his progress. I like Duvall’s focus, dedication and professional approach to his craft. I’m not ruling out his ability to continue his improvement at the plate.

There have been lots of studies done on this very thing and I am unaware of any that showed players performing significantly differently when batting in different lineup slots (assuming enough PAs to give a good sample). The exception, I believe, is batting in front of the pitcher which artificially increases a player’s walk rate for obvious reasons.

I might be tempted to agree if we were talking about two players with similar stats. Of course, if we were talking about two players with similar stats, Patrick probably wouldn’t have written this.

The spread in wOBA between Duvall and BP is greater than the spread between BP and Ivan DeJesus/Tyler Holt type hitters. No one would suggest BP be dropped that low in the order.

But the other issue is what kind of hitter BP is. He’s a ground ball hitter (46.7%) who hits the ball hard less often than Tucker Barnhart. That’s not really a guy you want up with a runner on 1st. In fact, BP has had a runner on 1st with less than 2 outs in 58 plate appearances and hit into 10 double plays. An astonishing 17% of the time.

Duvall, on the other hand, is a fly ball hitter (46.2% FB), who hits the ball hard (38.7%) more often than every Red not named Votto or Burce. He has hit into a DP in 3 of his 40 chances, 7.5% of the time. He is a guy you want to bat with runners on base as much as possible. Odds are he will strike out (27%), or fly out, or get an extra base hit (15% of all plate appearances). So even his outs are likely not going to hurt too bad, or possibly advance a runner.

Nice work Patrick. That must have taken hours to do. Thanks. I started to read this before having coffee this morning. Mistake.

You say in the end that the difference is about 1.5 runs. But does having BP bat following Votto, then have a negative run effect on the other spots in the lineup for Bruce, Duvall and Suarez by having them bat in a less optimum spot? Is there a cumulative effect where when all added up, it is costing the Reds much more than 1.5? Thanks, good work.

Yes, the cumulative effect you mention is real. The analysis takes that into account.

If you look at the table called “Situational RE” you see the numbers corresponding to each batting slot, 3 to 6, for both situations of BP batting 3rd and BP batting 6th.

For an example, let’s look at the 4th slot. When BP bats 3rd, that is Jay Bruce. When BP bats 6th, that is Adam Duvall. Because of Bruce having a higher wOBA than Duvall, moving Duvall from 5th to 4th, and Bruce from 4th to 3rd, actually decreased the productivity of the 4 spot.

If you look at all the other spots, you see the same thing. Every spot except 3rd has decreased in run expectancy. That’s why we sum it all up!

Basically, the benefit of moving Bruce to 3rd to bat behind Votto would outweigh the rest of the negative effects of giving guys like Suarez more at-bats.

Thanks. I wasn’t sure. My brain was still soggy this morning after all the rain we got yesterday. I very much enjoyed the RE24 article and should have re-read it first as you suggested. After your RE24 article I wanted to make that the advanced stat I wanted to learn more about this year. Much to my chagrin, I haven’t had the time to start on that yet. I’ll wOBA to that. Thanks for getting me straight.

Patrick’s analysis had two distinct options to compare and contrast and both started with Votto. Option #1 had Phillips, Bruce, Duvall and Suarez following Votto, in that order. Option #2 had Bruce, Duvall, Suarez and Phillips following Votto, in that order. It wasn’t just Phillips batting behind Votto or not batting behind Votto, so the results were as you questioned, cumulative.

Thoroughly enjoyed and appreciated the work and presentation Patrick, although I also enjoyed and appreciated your post regarding little Jeter’s first MLB game.

Sorry for the duplicate response. I was interupted after beginning my response.

No worries. You’re helping my march towards 1,000,000 comments! On this pace, I’ll be about 357 years old when that happens.

Patrick,

This is very interesting work. I consistently enjoy reading your posts. In grand baseball terms, I don’t know enough to say whether they’re methodologically groundbreaking, but they certainly feel that way to me.

I do have a question about this post: did you think about considering the following inning?

Let me explain (without numbers!). It’s not a surprise that an individual inning with a Votto-Bruce-Duvall-Suarez-Phillips sequence would result in more runs than an inning with a Votto-Phillips-Bruce-Duvall-Suarez sequence. As your RE24 table shows, Phillips is the least productive hitter of the 5. Not all innings–even run-scoring innings–see 5 hitters. And teams score more runs when their most productive hitters are batting.

However, assuming that the Reds are retired in order that inning, the expected production of a Suarez-Phillips-?? lineup in the following inning is lower than a Duvall-Suarez-?? lineup. The same holds if the Reds are retired in four hitters; Suarez-?? would outpace Phillips-??.

The above scenario occurs in a non-trivial amount of innings led off by Votto, but your analysis also builds in a consideration of when Votto comes to the plate with 1 out and 2 outs. In these circumstances (especially 1 out), it’s even more likely that Phillips will bat in the current inning but not the following one, further increasing expected production in the subsequent inning.

I’m confident that having Phillips bat behind Votto is particularly detrimental, as he is receiving higher-leverage ABs, and typically underperforming the other Reds hitters. But, when you build in the higher expected value of a subsequent inning where Phillips does not hit but the Reds’ better hitters do, I’m not sure that the run difference nets out to 14.75 runs over the course of the season.

And, of course, some of the time, there is no subsequent inning, and the expected productivity boost from this inning is not offset at all by a decline in the following inning. I believe this is the level at which lineup evaluation is typically performed, though–much less interesting than your approach.

Just my 2 cents, and I wouldn’t be surprised if I missed an adjustment in your analysis to take this very scenario into account. Thanks so much for the consistently great work!

Eli

Thanks for the thoughtful and well-written post.

You are correct with the general tenor of your post. The things that happen outside an inning in which Votto bats are certainly affected by the reshuffling of the order.

In my estimation, however, the effect is minimal. Here’s why I think this…

The absolute value difference of consecutive batter wOBA is essentially what is driving the need to have an optimized lineup. More plainly, stacking your best hitters in a clump together has a multiplicative effect on run production (the whole is greater than the sum of its parts, or something like that).

The rest of the Reds order not mentioned here (Cozart, Hamilton, Barnhart, Peraza, Whoever else) all are solidly below average hitters at this point. Cozart is an exception this season, but we might consider all these other guys to be true-talent .285-.315 wOBA hitters. Their spot in the lineup generally doesn’t matter since they provide similar (certainly not identical) value regardless of how they are shuffled around.

This is also somewhat true for Suarez and BP, who hit near the end of the sequence in the scenario where BP bats 6th. So really, from 5th (Suarez) all the way through the top of the order (Cozart, based on career number) and not counting the pitcher, you have hitters between .285 and .315 wOBA.

Hope this somewhat alleviates your concern. If I had to ballpark the error introduced by what you bring up, I’d say probably no more than 15%, which would put the true RE difference between 12 ish and 17 ish. But that’s off-the-cuff with no calculation.

Thanks again for reading and posting.

Too Long, Didn’t Read Version – The fact that Votto’s wOBA is so much higher than anyone else’s renders what the rest of the lineup does around Votto largely irrelevant. What occurs directly after Votto (stacking your other best hitters) is what really matters.

Yes, this is why batting orders don’t make much difference. In other words, Brandon Phillips (or bad hitter X) has to bat sometime. Patrick has already provided one response, the value of stacking good hitters being a multiple of the sum of the parts. A second response is that putting relatively worse hitters lower in the lineup results in them batting fewer times. Over the course of 162 games, batting 7th instead of 3rd results in ~72 fewer plate appearances.

Great stuff Patrick. Clearly there are two easy conclusions here. First, Phillios should not bat immediately after Votto. Sixth or seventh in the order seems about right. Second, the hitter that should bat after Votto is… Joey Votto. A couple of years ago Shin Soo Choo played the role of Joey 1 while Votto himself played Votto 2 and that team had a pretty good offense. So: time to move Phillips down. Time to promote Jesse Winker and install him at lead off. And time to package Zack Cozart and some pitching for Joey Votto, or at least the closest thing among the young prospects in MiLB. Imagine a staff of Bailey, Reed, DeSclafani, Finnegan and Stephenson with a solid bullpen and a lineup that starts Votto, Votto, Votto, Bruce, Duvall. Start printing the playoff tickets. Okay, I got a little giddy there but one can dream, no?

“And we head to the bottom of the 4th! Due up… Votto, Votto, and Votto.”

Maybe dreams can come true?

It’s a real life game with real life humans. Some of the math is interesting, some amusing and some down right boring and useless. I’ll enjoy a game and you can analyze the percentage of waste when pushing the toothpaste from the top instead of the bottom,

If you don’t enjoy the numbers then why not simply skip an article such as this one and enjoy some of the articles that have a more of a “isn’t baseball a beautiful game” and “baseball as art” feel? A lot of people don’t care for the analytics and they just read the myriad of pieces on RLN that don’t dive into the numbers as this one does. Many other readers and commenters really enjoy this kind of stuff and focus on these articles. Many other like myself enjoy both kinds of articles and find ourselves spending WAY too much time on this great Reds site!

Before these useless stats, how did a team decide whom should bat where in the lineup? Why was Mantle and Ruth middle of the order hitters and not 8th or 9th?

Yeah. Well said by LWBlogger2.

I always find it amusing that people like you assume people like me don’t understand that baseball is played by humans.

I’ve seen that comment more than a few times.

Here’s what I know. As a human, BP is not as good a hitter as many other humans who should bat in front of him.. That would be an even more boring article, wouldn’t you say?

Big night tonight. Cody Reed making his home debut. Some guy named Pete is getting honored tonight too.

Patrick, your article apparently didn’t make it to the Reds dugout.

Tonight:

Votto>Phillips>Bruce>Duvall>Suarez are in the #2-6 spots.

Not a giant Pete Rose fan but he should be in the Reds HoF and I’m happy for him and for the many, many fans that support him.

“Patrick, your article apparently didn’t make it to the Reds dugout.”

Sorry to refute your theory WV, but I clearly saw the article printed out and laying on Price’s desk in his office as he was filling out his lineup card. I also noticed the report of historical batter performance against Colin Rea laying right beside this article and it apparently trumped this article because Phillips is hitting .667 against Colin Rea.

Beaten by the old 2-3 against Colin Rea analysis. We just can’t compete with such mountainous evidence!

If the Giants are interested in Bruce, who are some prospects that stick out in SFG farm?

According to FG, their 3rd ranked prospect, and 18-yr-old Lucius Fox, is a middle infielder with blazing speed and a weak bat. That seems to be who the Reds might target… Haha! Really, though, the Giants don’t have that many marquee type guys, so I’d bet any potential Bruce deal with SFG would involve two lower end guys. The Reds are seeming to value quantity lately.