Last Sunday, upon logging into Twitter I was met with the following tweet:

For me, this was like Christmas in April! And the answer to your question is “yes.” I am a disturbed individual.

So, in honor of the update to the run expectancy matrices, today begins a multi-week series about decision making using the RE24 framework. As such, I feel it is important to have a post about RE24 itself! This is that post! (Perhaps, “this post is that?”) We will explore the following: what is RE24, where do the numbers come from, and how to read an RE24 chart. Then I’ll present the path ahead!

To begin, RE24 stands for “Run Expectancy for the 24 Base-Out States.” What is a base-out state (now to be referred to as BOS)? It is one of the specific alignments of base runners and outs in an inning. For example, a runner on 1^{st} base with 2 outs is one of the states; the bases loaded with 1 out is one of the states; bases empty with 0 outs is another. There are twenty-four of these states possible, hence, RE24.

As a baseball fan, a lot of RE24 will be intuitive to you. Consider the following example: a runner is on 1st with 1 out and the batter walks. Now, there are runners on 1^{st} and 2^{nd} with 1 out; this is the new BOS. The new BOS is more likely to score a run (and more likely to score multiple runs) than the old BOS, right? Well, of course! There is a man in “scoring position” now! See? Sabermetrics can be intuitive!

Now, however, we have to quantify that intuition. Well, *we* don’t have to quantify it because someone already did it for us. That was nice of them! I don’t know if Tom Tango and his cohorts were the first people to figure all this out, but they are the source I’ll be using. Tango publishes tables of all 24 base-out states on his blog, which are basically just excerpts from *The Book: Playing the Percentages in Baseball*.

I will present one of the tables, slightly modified for effect, and then we’ll discuss it. The below table shows the average amount of runs that scored from each base-out state until the end of the inning from 2010 to 2015:

As expected, we have an 8-by-3 grid filled with glorious numbers; one for each unique base-out state! Your eye is likely drawn to the red box at bottom-left. This is the “bases loaded with no outs” BOS. It is red because more runs score, on average, in that situation than any other situation.

Before we go further, you may be thinking “where did these numbers come from?” It’s a good question. The answer is simple. People with a lot of time and database know-how took every single plate appearance in the last 6 seasons and then figured out how many runs scored from that plate appearance until the end of the inning. Since each plate appearance has to start in one of the 24 specific BOS, it becomes an easy algebra problem after that to arrive at the final run expectancy number.

[Note: from this point on I will start referring to base-out stats with the following notation: 1-3_0 means “first and third, no outs,” 123_2 means “bases loaded, two outs,” and —_0 means “bases empty, 0 outs.”]

This is perhaps one of the most interesting things about RE24. It is based on reality; what actually happened. A big gripe of the “old school” or “traditional” fan base (from my point of view) is that sabermetrics often deal in theoretical situations, or perhaps focus on things that *might* happen rather than what has already happened. Here, we have a cornerstone of modern sabermetrics based entirely on historic data. There are no estimates, no crazy formulas, and no hard-to-understand acronyms. The simplicity of RE24 is as astounding as its application. These tables are the building blocks for slightly more complicated concepts, but we will save those topics for another day.

So, why do we use this chart and how do we use this chart? We (and by “we” I mean “major league managers”) should use this chart to aid the decision making process! How do we use it? Very easily!

Imagine for a moment you are leading of an inning. Obviously, our BOS sits at —_0. From this point until the end of the inning, an average of 0.481 runs will score. That simply won’t do. We need more runs than 0.481. Let’s have our leadoff hitter smack a double down the line. Now, at -2-_0, you can see from the chart our run expectancy has increased from 0.481 runs to 1.1 runs! Outstanding!

But wait! We haven’t actually scored a run have we? No? No. So what good is this? Here comes the sabermetric thought process. If we attempt to maximize the net change in run expectancy from one plate appearance to the next, over the long run we will also maximize our run scoring. Maximizing our run scoring will maximize our winning. Winning is good.

[Note: The generalized expression for calculating the change in run expectancy (∆RE) is as follows: ∆RE= RE_{f}-RE_{i} + Runs, which is final run expectancy minus initial run expectancy plus how many runs scored on the play.]

Corollaries of the first chart also exist. The below chart shows not run expectancy until the end of the inning, but instead shows the odds that at least 1 run will score from that position until the end of the inning. Behold!

On this chart, you will notice every box is occupied by a number between 0 and 1. The odds are never 0% and never 100% that you will score a run. Perhaps most interesting is the bases loaded with no outs box; there is only an 86.1% chance of scoring a run from this BOS. As it turns out, strikeouts, pop outs, and double plays happen!

Also interesting are the four boxes ranging from –3_0 to 123_0; each has a value between 84.3% and 86.1%. This shows us that once a man is on 3^{rd} with 0 out (or 1, for that matter) adding base runners doesn’t have an appreciable effect on the likelihood of scoring a single run. This is why many times you’ll see managers walk the bases loaded in the 9^{th} inning of a tie game, setting up a force at any base. The total runs don’t matter anymore; only the first one.

Notice that when going from 1–_0 to -2-_1 our run expectancy decreases from 0.416 to 0.397. This is one situation in which a sacrifice bunt is oft employed, and this simple calculation is one of the methods used by certain folks to show why sacrifice bunts are generally poor decisions. [The last part there is a sneak-peek into next week’s article entitled *How Productive are Productive Outs?*]

One final chart and this primer will be at an end. The below chart shows the total percentage of at-bats beginning in each BOS.

As you’d expect, the —_0 base-out state dominates, largely due to the fact that a minimum of 17 of these plate appearances are guaranteed in each game. The most unlikely BOS is –3_0, accounting for a whopping 0.2% of all plate appearances. This particular chart, when combined with the above charts and some math, help determine optimal lineup construction.

As promised, this primer is now at an end. I know it was quite dry, but presenting all this information, along with some pertinent examples of how this can and should be used in a real baseball game by our beloved Redlegs, turned out to be far too long for a single post, hence the decision to make this a series. I promise next week will be more fun to read!

As alluded to above, next week’s post will be entitled *How Productive are Productive Outs?* Tune in next week to find out!

Great stuff Patrick. I’m looking forward to this series!

Oh Patrick…you should write television scripts for a series with cliff hangers every week. This isn’t dry; it’s invigorating. So the next post in the series can be expected on Friday?

Does this at all account for who’s on base? I’m assuming there’s less of a chance a pitcher will score from first with two outs than, say, Billy Hamilton.

This does not. It is entirely derived from every real situation that has occurred. So we can always assume average hitters and average base runners.

I will, however, attempt to devise a system to adjust RE for the batter the plate. I think we’ll always just have to treat Billy as a special case.

I don’t know about the league average hitter part. That really shouldn’t be the case in some of these situations when you think about it.

MLB lineups may not be fully optimized, but pretty much every manager gets their best hitters more PA with runners on compared to their worst hitters. Put it like this: who is more likely to be batting with the bases loaded and no one out? Mesoraco batting fourth or Billy Hamilton batting ninth? If you’re being even somewhat smart about your lineup, the answer should be Meso. So, in the base out states with runners in scoring position, it should be a slightly better than average hitter.

Same goes for.pitchers. The worst pitchers spend more of their time pitching with runners on than the best ones do. This is slightly mitigated by the fact that teams try to pitch their best guys the most, but still this is one of the reasons often cited for why hitters put up better numbers with runners on base. Because if runners are on base, they are more likely to be facing a weaker pitcher.

I think the most accurate thing to say is that these numbers are the average results of these different base out situations. But I could be wrong.

You are correct on all points. Better players have a larger effect on the overall numbers and saying what you said in your sentence would certainly be more accurate. 🙂

Patrick: Since the percentages are derived from past events, not projections, I assume that they are affected by teams employing old-school strategy (I’m mostly thinking of sac bunts). If so, the percentages should change a bit as more teams use newer analytical tools to dictate game strategy.

Yes. As strategies and run-environments change, the charts change. If you go to Tangotiger’s blog, he’s got more charts for different time periods. Up until Sunday, our most recent chart was something like 1995 to 2010, which included the steroid era. So our new 2010 to 2015 chart is a much more accurate reflection of what we see on the field today.

Good stuff Patrick. And some interesting, counterintuitive data there too. I’m fascinated that run expectancy is higher (albeit ever so slightly) for 13_0 and 13_2 than it is for 23_0 and 23_2 respectively. Meaning that if the runner on first could steal second with 100% success a manager should at least in theory tell him NOT to go. Even though it would eliminate the force at second. How can first and third and no outs or two outs be a more productive situation than second and third and no outs or two outs? Weird. Now I want to know WHY that is. Can’t wait for the next installment.

Think you may have that backwards. 😉

1-3_0 is less than -23_0 (1.784 to 1.964).

This is where I think these tables are confusing me though. Going from the 1-3_0 to the -23_0 base-out state increases the average runs scored from 1.784 to 1.964, but it decreases your chance of scoring at least 1 run from 0.86 to 0.852. Those 2 things seem counter-intuitive to me.

See the same explanation regarding going to -2-_0 to -12_0 you posted below.

Patrick – You’re right, sort of. I was looking at the table for the odds of scoring at least one run, not the total run expectancy. Still, odd that the odds of one run go down from 13_0 to 23_0, and from 13_2 to 23_2. So my thinking would apply, at least in theory, if a team needed at least one run to tie a game late, or only one run to win a tied game late. I’m not suggesting the numbers or odds are wrong, just that it’s counterintuitive in a few small ways, which I find fascinating.

Ahh, gotcha! Yeah.. it’s even confusing sometimes just to talk about the subject!

(Speculation incoming, so beware). I speculate that this is because early in games, when it is 1st and 3rd and no outs, teams will trade a double play chance and allow the run to score over trying to cut down the play at home (on an infield ground ball).

The question is, based on these numbers… SHOULD they do that? Can’t wait to find out!

Let me also use that as a transition into why the retrospective data of RE24 is flawed. As team strategy (if my speculation is correct) has altered the data — we are assuming with RE24 data that defenses NEVER would permit a run to score and are ALWAYS trying to prevent runs. Therefore, when they turn a double play in inning 1 in a 1st and 3rd situation, that gets recorded into RE24 the same as when the team draws the infield in in the 9th inning to try to cut that 3rd base runner down. As such any conclusions made based on RE24 data points must be made with the caveat that there are multiple confounders.

Good stuff nonetheless Patrick!

I am by no means in the “Joey needs to walk less!” boat, but interestingly I notice that when going from 2_0 to 12_0 our run expectancy decreases from 0.614 to 0.61. Let’s say Suarez leads off with a double. Joey should then be less inclined to take a walk in this situation, correct?

This is correct.

But, you always have to weight the opposite outcome. For example, if Votto walks, it decreases RE, as you stated, but if he changes his approach to try and get a hit, perhaps it will lower his actual chances of both getting a hit and getting a walk. In that case, it’s just one of those “minimize the effect” situations.

And also, in that situation, I’d expect most pitchers to pitch around Joey, figuring he does less damage walking than he may well do with a good pitch to handle. He’s disinclined to swing at bad pitches, as he should be, because he’s unlikely to have a productive result when he does. Reasonable? Great work, by the way, Patrick.

Very reasonable.

Thanks!

Also, the situation must be taken into consideration. Down one, maybe Votto should bunt here. 3_1 run expectancy increases to .66. However, down two he should probably walk, increasing average runs scored from 1.1 in 2_0 to 1.437 in 12_0

Thanks for this post, Patrick. These charts are a lot of fun. I’ll definitely pull them up during the game tonight.

I wonder what, if any, percentage error would be considered to be in these numbers.

Some it seems counter-intuitive. The case you mention for example:

Suarez leads off with a double. The base-out state of 2_0 indicates that 1.1 runs will score on average by the end of that inning, with a 61.4% chance that at least 1 run will score.

Now Votto takes a walk, we have the 12_0 base-out state and now on average 1.437 runs will score by the end of the inning but there is only a 61% chance of at least 1 run scoring.

How does the average number of runs go up but the percent chance to score at least 1 run go down?

Those situations are not necessarily counter-intuitive. With a runner on 2B & 0 outs the chance of scoring more than 1 run relates to the chance of another runner reaching base without an out being recorded. That can happen via a walk that would probably not have occured if no one had been on base with 1B open or possibly a failed fielder’s choice trying to get the runner at 3B rather than 1B.

The other side of that coin is the vchance to scor 1 run. With 1B open, there is no force out or GIDP potential. As soon as 1B is occupied, then the force out and GIDP potential comes into play.

Yep. GIDP!

I think I’d take the 0.004% less chance of scoring one run for the +0.337 runs. Unless it is the bottom of the ninth in a tie game. That is really the only time that a non-pitcher bunting would be worth it.

The other thing is, the chart you’re looking at shows only the chance that at least 1 run will score, not runs per inning. So really the only scenario that Votto theoretically hurts the team by walking is last at-bat, tie game, because that is the only time the possibility of scoring more than one run is insignificant. Looking at the first chart, Votto walking after a lead off double increases average runs per inning from 1.1 to 1.437. That leaves the question, is the opportunity to raise runs per inning by .337 worth the risk of lowering the chance of scoring at least 1 run by .004? Yes! That is a no-brainer trade off.

Great information. I am so glad you are doing this series. Your earlier posts about RE24 generated some interest in this for me. I was a little hesitant to delve into it this year, because of my unfamiliarity with it. But a series done in doses like this is just the ticket. The charts make it an easy and enjoyable read. Easy to understand too. I will look forward to your next installment. Nice work.

Thanks, WV.

If this goes well I’ll look into some extra posts in the future about other topics like this.

It’s perfect timing for an article/series like this since there isn’t a lot of meaningful 2016 data yet!

Remember I am below a novice in this area, but do Stolen Bases vs. Caught Stealings enter into this and your later discussions on RE24? Just curious.

Yep, SB/CS will be part of a future post!

Cool. I’ll save my Q’s for then.

I just wonder though, if there is enough room on Price’s and Riggleman’s legal pad and clipboard for information like this during a game?

Good stuff! Keep it coming!

I shall attempt to do so!

Loved this post. Do you think that Pre-2010 RE-BOS #’s would be similar? I imagine they’d all be higher during the steroid era, but these are figures that should be valid over a longer period of time, right?

As has been noted by a few people, the RE24 chart being shown is generic, and won’t necessarily be used unadjusted for various strategies.

For example, here’s a chart broken down by various inning and score classes:

http://tangotiger.net/re24_innScore.html

And here’s one when facing Kershaw:

http://tangotiger.net/re24kersc001.html

Obviously, as you slice and dice the data, your sample size starts to play havoc, which is why you can’t rely on ACTUAL data. Once you buy into the idea using actual data, you can rely on data based on probability instead.

Great note, Tom! Thanks for stopping by.

I’ve publicly stated I’m not a hard-core stats fan before, but once again I have great appreciation for those who are and can explain it to old dogs like me.

+100, Patrick. Which of course comes with the requisite +200 for Little Jeter.

This is a cool start, and a nice easing into the topic. As mentioned above, this is purely an average pitcher vs an average hitter in the average ballpark with average weather during the average day-night and with an average number of hot dogs being scarfed in the bleachers. The trick is applying it to a given situation, and that’s where a top-10% manager running on instinct probably loses to an average manager whose front office has data on a few more variables.