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Introduction

Doesn’t it always seem to happen to your team? Down by two in the 5th, your team gets a lead-off double followed by a walk. First and second, no one out – this is the start of a big inning. Next thing you know, after a lazy fly-ball to shallow left and a strikeout, your only hope is a solid base hit. One weak grounder to short later, you curse your team for another wasted opportunity. The Run Expectancy Matrix helps us determine how much of a wasted opportunity the above scenario truly was.

There are 24 unique states describing the position of runners and the number of outs in an inning. There are 8 unique runner positions (bases empty, men on 1st and 2nd, bases loaded, etc.) and 3 out possibilities.

For each state, we are interested in determining the expected number of runs that are scored in the rest of the inning. If one goes to the Statistics page of Baseball Prospectus and clicks on “Run Expectancy Matrix” a table like the following will pop up:

```Runners    Exp_Outs_0    Exp_Outs_1    Exp_Outs_2
000          0.526         0.281         0.108
003          1.520         0.951         0.362
020          1.165         0.708         0.334
023          2.017         1.425         0.600
100          0.908         0.536         0.228
103          1.772         1.566         0.496
120          1.558         0.944         0.461
123          2.349         1.596         0.803
```

Back to our scenario, this table tells us that, with runners on 1st and 2nd (Runners = 120) and no outs, the expected number of runs is 1.558. That’s fine and good, but where did this 1.558 come from? More importantly, what were the assumptions?

To create the Run Expectancy Matrix, the sabermetric gnomes (you thought real people did this?) look at the play-by-play data for some time period, typically a given year or set of consecutive years. At the beginning of each play, they look at the state and then determine how many runs were scored in this half-inning from the beginning of the play to the end of the inning. Then, they calculate the average of the number of runs that follow for all plays with the same beginning state.

To illustrate the point, let’s walk through an example inning. On April 5, 2008 in the bottom of the fifth inning of the RangersAngels game, there were six distinct plays:

1. Matthews, Jr. singled to left
2. Guerrero singled to left. Matthews, Jr. to 2nd.
3. Anderson pops out to second.
4. Hunter doubles to left. Matthews, Jr. scores. Guerrero to 3rd.
5. Kotchman hit by a pitch.
6. Kendrick grounds into a 6-4-3 double play.

The table below shows how the above six plays would be translated into the state and the additional runs to be scored that inning.

```Play  Runners   Outs   Additional Runs Scored
1      000       0        1
2      100       0        1
3      120       0        1
4      120       1        1
5      023       1        0
6      123       1        0
```

At the beginning of the third play, we have the state of 1st and 2nd, no outs. After this state, one run scored during the rest of the inning (on the fourth play). Throughout 2008, there were 2,520 times a play started with 1st and 2nd and no outs. These plays plus all following plays in the inning led to 3,925 runs scored. If we divide 3,925 by 2,520, we get 1.558 expected runs.

There is some filtering of which plays are included and excluded in the calculations. Typically, a Run Expectancy Matrix focuses on innings when the batting team’s objective is to maximize runs. When the batting team’s objective is to play for a single run (tied-game in the ninth) or the inning is partial (a walk-off hit in the bottom of the ninth), these innings are typically excluded, because they underestimate the true run potential of a given state. In the numbers presented in this article, I have used each play in 2008 that occurred between the first and eighth inning as a simple filtering.

Uses of the Run Expectancy Matrix

There are two significant uses of the Run Expectancy Matrix:

• Improved performance measurement through advanced statistics
• Strategy analysis.

In regards to improved performance measurement, imagine the following: a reliever enters the game in the middle innings with his team up by one, with the bases loaded and only one out. He gives up a sacrifice fly that ties the game and then strikes out the next batter to end the inning. With traditional statistics, this reliever would be charged with a blown save and allowing one inherited runner to score. Did he provide value to his team? Actually, he did. Based on the run expectancy matrix, in this situation, we would expect the opponent to score 1.596 runs. In this example, the reliever improved his team’s game position by 0.596 runs, while the traditional statistic of blown saves suggests a poor performance. This type of situational analysis in comparing the actual runs scored to the expected runs scored is the basis of statistics INR, WX, and WXRL.

The other important use is to analyze the value of certain strategies (stolen bases, sacrifice hits) based on a likely game situation. By comparing the beginning state to a few possible end states, we can determine if a given strategy is likely to create more runs. Joe Sheehan wrote an excellent analysis of the stolen base for the Baseball Prospectus Basics series back in 2004 using the Run Expectancy Matrix for 2003.

Limitation #1: Expected Runs vs. Run Frequency

Relying only on the Run Expectancy Matrix in evaluating strategy, however, can lead to flawed conclusions. We would dismiss the sacrifice bunt entirely, because in all situations a successful sacrifice bunt decreases the expected number of runs. If a team is playing for one run, however, a Run Frequency Matrix (something I first saw on Tom Tango’s website) shows that it can be a sound strategy. The table below shows the probability of scoring at least one run given the situation:

```Runners   Exp_Outs_0   Exp_Outs_1   Exp_Outs_2
000          28.2%        16.5%        7.1%
003          86.3%        65.5%       25.3%
020          62.8%        41.0%       21.9%
023          83.9%        69.0%       26.6%
100          42.4%        27.1%       12.7%
103          83.3%        62.5%       26.7%
120          64.5%        42.1%       22.6%
123          86.6%        67.0%       32.4%
```

With a runner on 2nd and no outs, sacrificing him over to 3rd slightly improves the odds of his scoring from 62.8% to 65.5%, although the overall expected runs go down from 1.165 to 0.951. We are increasing the odds of scoring at least one run by 2.7%, but at the cost of lowering the probability of scoring more than one run. Interestingly enough, the very common situation of sacrificing with a runner on 1st and no outs is typically not beneficial, as the probability of bringing that runner home goes down from 42.4% to 41.0%. This counter-intuitive result highlights one other key issue of using the Run Expectancy Matrix (or Run Frequency Matrix) blindly. It is important to understand who is bunting, who is on deck, and who is the runner.

Limitation #2: Understanding Lineup Position

The Run Expectancy Matrix essentially assumes that an “average” hitter is coming up to the plate in every given state. However, it’s obvious that the average number of runs scored will be different in a 1st and 3rd, no-out state with Mauer, Morneau, and Kubel due up versus Buscher, Gomez, and Punto.

I looked at the play-by-play data for 2008 and calculated the Run Expectancy Matrix for each lineup position. In the table below, I pulled some examples that will help us determine the required breakeven success rate of a stolen base when a man is on first with no one out. In the first column, we see the run expectancies for the beginning state. The average is 0.908 runs, but this varies from 1.052 (for the #3 hitter) to 0.783 runs (for the #9 hitter).

```Lineup     Expected Runs   Expected Runs   Expected Runs
Position   (100 - 0 out)   (020-0 out)     (000-1 out)
Overall       0.908           1.165          0.281
1             0.923           1.192          0.323
2             1.018           1.254          0.325
3             1.052           1.232          0.328
4             0.938           1.201          0.311
5             0.827           1.162          0.270
6             0.813           1.090          0.260
7             0.819           1.022          0.234
8             0.873           1.098          0.217
9             0.783           1.108          0.225
```

If the steal is successful, we have a man on second with no outs (the 2nd column). If our base stealer is caught, we have the bases empty and one out (the 3rd column). The overall numbers (the first row) suggest that the benefit of a successful steal is an additional 0.257 runs (1.165 – 0.908). However, if caught stealing, it costs the team -.627 runs (0.908 – 0.281). Therefore, the breakeven percent (after some simple math) comes out to be a 70.9% success rate. If we do the same math for each lineup position, we see that the breakeven percent varies based on who is up to bat, from a low of 62.4% when the #5 hitter is up to 80.1% when the #3 hitter is up.

```Lineup        Benefit of       Cost of
Position    Successful SB    Caught Stealing   Breakeven Percent
Overall        0.257           0.627               70.9%
1              0.269           0.600               69.0%
2              0.236           0.693               74.6%
3              0.180           0.724               80.1%
4              0.263           0.627               70.4%
5              0.335           0.557               62.4%
6              0.277           0.553               66.6%
7              0.203           0.585               74.2%
8              0.225           0.656               74.5%
9              0.325           0.558               63.2%
```

We are, of course, assuming that our current lineup positions match the average historical lineup positions of all teams. A better mousetrap would be for a manager to have these breakeven percentages based on his current lineup and the likely production from each spot. The best mousetrap would also have these percentages adjusted based on the current game state (are we maximizing runs or going for simply a single run), and the likely success rate based on our man at first and the current opposing battery-mates. To create something like this, we would need to use more sophisticated tools from optimization theory like dynamic programming, but that is a different article altogether. Ah, to dream the impossible dream!

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kgoldstein
5/24
Real quick -- the sabremetric gnomes comment was my first laugh at loud moment. After that, I have more positive things to say, as I REALLY like the fact that you took something we all know about and added a bit of your own research to make it even more accurate by measuring the quality of hitter. Nice overall job.
wcarroll
5/24
The title? Umm, no. Too much. The rest is really solid work, but it's a bit more 201 than 101. Actually, its more 301 since there's a lot of original work here. That's not a bad thing at all and I feel like I learned a lot here. I just don't feel like he aimed at the right goal. Call it iffy song choice. In the end, I'm left with two opinions here. It was good for what it was, but I don't think what it was is what we were looking for. I'd like to see him focus a bit more if he makes it to next week.
ckahrl
5/24
The title was funny, but not in what I'd call a publishable way. That said, this was a tough task that Tim took on: talking about run expectancy matrices is basically a matter of demonstrating that you, as a writer, can make a collection of very straightforward tables sing, whether that's stylistically or with dancing bears or sabermetric gnomes. It's important to note the level of difficulty involved, which I admire, but it's also worth noting that I found this a bit dry, and for what is supposed to be an introductory piece, the question is whether it really captures a new reader's imagination. I could be fairly accused of having a sabermetrically jaded palate, but it's because I've seen enough exercises like this that I'm glad Tim showed an ambition in taking it on.
Oleoay
5/24
Overall I loved the writing and thought it was a good article. I found it easy to understand and methodically laid out. I loved the example of the reliever who allowed a sacrifice fly and it showed a practical application of this concept. My main nitpick would be that I, if this is an introduction article, to spend less time on the limitations and more time on its usefulness. Talk about stolen bases, the risk/reward for a runner on first trying for third base on a single, etc. The other nitpick was the title didn't work for me, maybe because my mind kept mixing it up with "karaoke". Still, it's not a great idea for a Basics article to have a title that would intimidate or confuse a reader...
Oleoay
5/24
This article was one of the five that I gave a thumbs up to on my initial read through.
gersh22
5/25
I like the topic. Adding, or at least linking to, a fangraph would be a helpful visual.
chunkstyle
5/25
I liked the title and have generally wished for some more humor in general. I enjoy analysis, but it's nice to have some quirky/geeky quips along the way. Come to think of it, the same goes for regular BP writers too...

There's nothing really inappropriate about the title - it's just bathroom humor. Beats the heck out of watching a bunch of football analysts in ties laughing like hyenas at one another's non-jokes.
Kongos
5/25
The first one so far that I've given an immediate thumbs up. At first I thought it was going to be really lightweight, but the original research in the latter part of the article swung me around. It's the best-written so far, too.
cdkasdin
5/26
My sentiments exactly.
acmcdowell
5/25
It may seem like a minor quibble, but the line describing the runner states after each play is practically illegible. If you're coming in without a good understanding of how run expectancy matrices work and see that line of numbers I doubt you make it through the rest of the article. It cropped up in another article or two, but formatting tables is key.
tkniker
5/25
Yes, when I first read the article when it was posted, I'm not sure if this is one of my tables (I sent it in the same format as the other tables), that missed the formatting guru.

It should be (Hope it comes out in the comment section):

Play Runners Outs Runs Scored
1 000 0 1
2 100 0 1
3 120 0 1
4 120 1 1
5 023 1 0
6 123 1 0
tkniker
5/25
Drat. Comments section removes any additional white space, so it's tough to recreate what it should be (and what it was in the original Word document).

Basically the table shows what the simple db table would be.

First column: Play Number (corresponding to the numbered list above)
Second column: Runners at beginning of the play (000 = bases empty, 120 = Runners on 1st and 2nd)
Third Column: Outs - Number of outs at beginning of play

Fourth Column: Additional Runs scored. Number of additional runs scored in the inning starting with the current play to the final play inclusive.
metty5
5/25
Tim - read some of the chats on The Book. Good Luck- the lack of tables was the first thing I noticed with this piece. Maybe its just me, but I think you need more of a goal and explanation of what you are trying find. I understand this was more of a teaching piece, but I think you can teach about the 24 out states and still draw out some different conclusions.

Overall, as a reader of the Book Blog, and sabremetric fan. I loved the piece.
dpease
5/25
This has been fixed in the article text. Sorry about that.
dtrainmets
5/25
I really enjoyed Tim's into article, and I think this submission is again among the best of the bunch. He's got a very inviting and humorous style and I found this to be an incredibly interesting and thought provoking article. Great job overall.
jtrichey
5/25
Yeah, this line:

Play Runners Outs Additional Runs Scored 1 000 0 1 2 100 0 1 3 120 0 1 4 120 1 1 5 023 1 0 6 123 1 0

I'm sure did not come out how the author intended. At least I hope not. Still, thumbs up from me. Run expectancy has always fascinated me anyway, and the added bonus of what changes by lineup position makes this a big time winner. Basics yes, and I learned some things. great job!
tkniker
5/25
I'm guessing this is a bit of my academic/research side (and something that probably was the biggest decision for me on whether or not it should be included) but I didn't want to do just a "basic" piece that didn't add a little something to the literature at the end.

Maybe this wasn't exactly in the spirit of The Basics theme that they were hoping for, but I didn't think it was so far of a departure. I felt that most of the article truly lined up with the goal.
Oleoay
5/25
I realized the table wasn't formatted properly and just skipped it. That's a good reason to support tables with analysis and interpretation (which you did). Even though I couldn't read the table, you explained well what you were talking about.
SkyKing162
5/25
I enjoyed the first few paragraphs, as it presented simple introduction to the topic accessible to anyone. There was a high degree of difficulty with tackling a lot of related topics later, but I don't think that was a necessary choice. Interesting stuff, but not part of a Basics article.
tkniker
5/25
I have found the "it's not really the basics" comments interesting, given that Run Expectancy Matrix was used as a building block for one of the old Basics articles, (the Joe Sheehan article that I site)
SkyKing162
5/25
I see your point, but that SB article wasn't an introduction to run expectancy. It leaned on it and explained eough about it to use it, but was in depth on the SB strategy. THIS article could have been the resource that people who wanted more information on RE after reading Joe's piece could have clicked through to.

Again, these mostly weren't in the style I was expecting. I enjoyed most of the advanced information. And just because I wasn't expecting it doesn't mean I get to have the final word.
hotstatrat
5/25
terrific article â€“ well organized, clearly written, pros & cons a great touch

This stands out as clearly the best entry on this round - and I happened to like Tim's qualifying entry on organizational depth the best, too.

Tim is clearly the Adam of this competition. For those of you who are either musically deadened or never watch American Idol, Adam was by far the best singer this season, although, Tim's superiority might not be quite as obvious. I could see Ken Fuchs and Brian Cartwright doing some collosal work and some of the other writers strong enough to keep Tim from getting overconfident. Heck, most American Idol viewers prefered that nice handsome boy Chris.
hotstatrat
5/25
woops: "colossal" and "preferred".
pferrington
5/25
wow - maybe I'm just tired but I didn't like it. Well I'm sure all the comments are correct but I tried to pretend I didn't know what a run expectancy matrix was.

Instead of using a game story to good effect it is immediately abandoned for dry research. the average fan isn't coming in with a sabermetrics or scientific background and will, I think, do what I did: read a paragraph or two... start skimming and never find a way back into the article.

Sure it might be great original research but that wasn't the assignment now was it?
gwguest
5/25
I wouldn't really consider this a 101 / Basics article. There aren't many average baseball fans (the ones I know anyway) who even know that this concept exists.

As an article, I love it. As to the assignment, I'm not so sure.
gwguest
5/25
Re-reading the assignment again... I would consider this a "BP Basics" but not a baseball basics. Fine by me.
Aaronjs27
5/25
I agree with you completely. I have a small group of friends who are very into this stuff and do understand the concept of run matrices, run expectancy, etc. But many people think I'm a jerk when I tell them that having Jose Reyes or Jacoby Ellsbury run so frequently in front of star hitters like Wright, Beltran, Pedroia, and Youk is potentially costing them runs, big innings, and sometimes games. As somebody who's a large believer in sabermetrics, I thought the article was fantastic, but that's probably because I completely understood it. And I give Tim much credit for being able to cover such a great deal of material in a short article. If he was given an entire chapter in Baseball Between the Numbers, I think he probably could have also brought in those unfamiliar to the matrices more gently, and I'd like to see what else he's capable of if he's given a less technical assignment, which is why I voted for him to move on.
harderj
5/25
Another piece to have introduced as a rationale for paying attention to run expectancy (though maybe more advanced than 101) would be how astute managers can use their "closers" earlier than normal in high leverage situations (i.e., in the 7th inning if a high leverage situation, rather than saving for a 9th that may not come).
daiheide
5/25
I think this is a very good piece. Contestants shouldn't be judged more poorly because they decided to "introduce" one of the more difficult concepts around. Those need introductions, too, and their introductions are accordingly going to be a little more difficult than introductions to some of the simpler statistics.
jsnell
5/25
Nailed it. Starts at a real 101 level and then ramps up as you go, but it's a fairly gentle curve. I do think there might be two articles here -- one on the more basic side with a little more description (my favorite way to explain run expectancy is "baseball as a board game") and then a second part that delves into more detail with original research.
llewdor
5/25
Solid, solid piece. I enjoyed both the tone and the content.
mbodell
5/25
I thought this was absolutely excellent. Everything after the title was pretty much perfect. I really think the limitations are a very important thing to cover when doing a basic stat as it helps tell when the stat is useful and when it isn't and it also helps answer any "Yeah, but..." questions someone has on the stat.
BurrRutledge
5/26
First article I've read that is an unequivocal thumb up. Covers the BP Basic on run expectancy like a blanket, and then tucks it in with a nice piece of new research on lineup position. Exemplary.
DrDave
5/26
Before I quibble, I'll first say that I thought this was well done, and at the right level of sophistication. The title was borderline; the gnome joke was fine.

Pure nerd quibble: if you're going to talk about using Run Expectancy or Run Frequency to evaluate whether or not to bunt, you need to mention the possibility that the bunt attempt does not result in a successful sacrifice. Strikeout, pop up, lead runner forced on FC, both runners safe on error... all of these things happen more often that most people think.
tkniker
5/26
Yep, good point. I thought about that but didn't put it in, because I didn't want to add yet another layer of complexity that would ding me more by some for not being basic enough.

jdavlin
5/26
I've read 7 of these so far, and this is the first one that I've immediately loved. I really don't get the criticism that this is a "201" or "301" level piece. To paraphrase a previous commenter, this is "BP Basics" not baseball for dummies. The idea is to present a complex concept in a manner that is understandable to a reasonably knowledgeable baseball fan, preferably in a style that is entertaining. For me, Tim succeeded on all counts.

I did not find the writing style "dry" in the least, as others have commented. I guess it's all in then eye of the beholder, but for me, Tim writes with an ease and confidence, both here and in his initial entry, that is impressive.

Oh, and the title...umm...YES.
dpowell
5/26
I think this was clearly the best. But it's traditional for commenters to point out the one thing they found annoying so I'll mention this: "more sophisticated tools from optimization theory like dynamic programming." This seemed a bit much. Would you really ever need anything more than a straightforward simulation?

But, really, great job!
tkniker
5/26
Cool! A discussion of optimization theory vs. simulation. Now we're cooking with gas.

My point on this was not so much simulation or dynamic programming, but as one wants to get more sophisticated, one could not simply do an averageing of play-by-play data because one would run into data sparseness issues, i.e., the whole sample size issue. Therefore, if one desired a tool/mehodology that took into account: the current lineup, the opposing battery-mates, the situation, etc. one needs to go down the path of DP or simulation.

Personally, simulation is fine and dandy, but it does determine what kind of time constraints. It's easy enough to run a simulation of 1,000,000 runs if you have several minutes to wait (depending on the granularity of the simulation), however, one could use a DP approach if by plugging in a new pitcher or pinch-hitter, one needs to have a recommendation in just a few seconds.
greensox
5/26
If for no other reason, the title should get him into round 2.
The article was interesting too.
mketchen
5/26
This was an awesome piece of writing. This is pretty heavy stuff and I have spent a fair amount of time reading up on it. With that said, this might be the clearest illustration of this information being broken down I have seen.
iolair00
5/26
Overall, I like this article. I thought that the first half was well written. I also feel that the topic is a good one for an intro article.

I felt that the writing was less clear (and less entertaining) in the second half of the article. I also felt that the numbers suffered from a common problem in sabermetrics - too many digits. I'd be astounded to find out that the run expectancy numbers had more than two significant digits. The table looks to be sourced directly from BP, but that sort of nit still needs to be picked.
abskippers
5/26
I can pretty much tell which articles I'm going to like just by looking at the titles. The ones titled like high school science papers turn out to be just about as interesting. Yours might be a little much for a family website but you clearly win the week's prize for creative title. And I would say that was predictive of what I thought was also a solid job of taking the basics the assignment called for and trying to incorporate interesting applications for the stat that the more sophisticated BP audience would still enjoy. Good work.