THE BOOK--Playing The Percentages In Baseball

Now, let me try it by forcing the same 5.20 runs per game.

Our two hitters have an EQUAL impact to team runs per 27 outs, with the following:

Linear Weights per out:
.082 RICE
.100 PIERRE

And Linear Weights per PA:
.060 RICE
.052 PIERRE

***

Also note that the PIERRE team gets 1.44 more PA per 27 team outs (and 0.16 more PA for himself).

If we take .052 times .16 we get… .008.  That’s the difference to make up between him and RICE.

So, the reconciliation starts with Linear Weights per PA, and then you have a further adjustment for the extra PA that he gets over RICE.

And that should conclude that!

***

I’ll give you the short-hand formula as soon as I can…

Could you possibly use a more extreme example? It might be easier to follow along. For instance, start by giving Pierre and Rice +50 LWTS.

Basically what you’re doing is trading in singles for home runs to make the hitters “match up,” right? What happens when you trade doubles instead of home runs to force the fit?

Let’s take it step-by-step.

Create a player name RICE, such that when added to a team of 8 BASE players, you get 0.25 more runs scored than a team of 9 BASE players.  RICE is worth +.25 runs per 27 team outs.  Make RICE top-heavy in HR and low on walks.

Do the same with PIERRE, but make him top-heavy in walks and low in HR.

We’ve established that both players are worth the same to his team (according to BaseRuns).

What does Linear Weights say?  It says their LWTS per out is:
0.084 RICE
0.103 PIERRE

And their LWTS per PA is:
0.061 RICE
0.053 PIERRE

So, both are “wrong”, in that we expected to find a metric that finds them both equals.

A team of 9 BASE players would give each player 4.50 PA.  And so, RICE’s +.061 runs per PA would normally give +.2745 more runs per 27 team outs.  And PIERRE’s +.053 runs per PA would give +.2385 more runs per 27 team outs.  That’s if we give them each 4.50 PA.

My BASE player gets 0.55 runs per 4.50 PA, so we would have expected 0.55*9 plus .2745 with RICE on the team, or 5.2245.  We actually have only 5.20 runs, and so, we need to remove .0245 runs here from RICE’s tally per 27 team outs.  So, his +.2745 needs to be depressed to +.250 runs.

Similarly, PIERRE’s contributions needs to be bumped to +.250 from +.2385, or an extra +.0115 runs per 27 team outs.

***

Peter should be happy now.  He was right in terms of the extra PA contributions not being handled quite right, but couldn’t put his finger on it.

Colin: actually, I’m trading walks and singles for HR.  Let’s finish off this particular discussion before I try other combinations.

It’s just that I don’t think that the only thing you’re capturing here is the difference in outs per PA - we know that linear weights holds the run environment constant, which isn’t what happens when you change the underlying events.

I would, I think, prefer to reconcile a player’s affect on his teammate’s PAs seperately from reconciling the linear weights values. I don’t know, I’ll have to think about it.

Colin’s first point in #6 is exactly what I was going to say.  This approach does not fully isolate the effect of the extra PA, it seems to me.

Tango, can you give the LW/PA and Team PA/G for Rice+ and Pierre+?  Assuming a base OBA of .333, I get Rice+ at 37.41 team PA, an average team at 37.78, Pierre+ at 38.53.  I am curious how (LW/PA)*(PAR) would match up with the target values, where PAR is the PA-ratio (team PA/37.78 in this example).

When you, DSmyth, and myself agreed upon LW/PA at FanHome, we also agreed that it would have to be adjusted for the PA of the league (ie in order to compare players from different seasons, you would have to scale their LW/PA by the Lg PA/G).  Perhaps this same principle applies to adding a player to a reference team, although I still have the same concern that Colin articulated.

I’m using 27 batting outs per game, so your base is 40.5 PA.

The difference in PA for the two teams is 1.44, using OBP of .2727 and .4842 (with a rest-of-team OBP of .333333).

***

I think you guys may be right, and maybe I should look at things in terms of adding just 1 HR, etc, instead of the big jump I’m using.

Tango -After much thought I have a finished logical proof that linear weights above average perfectly includes the extra PA’s created by non out batting events and consequently that the method of converting linear weights above average to runs created by adding the average runs per PA to every PA, both outs and non outs of a batter, is the correct method of converting a batter’s linear weights above average to absolute runs created without any further adjustment being necessary and for all run environments.

Tango - So I think you should be the one who is happy now!

Well, I hope you can provide the proof, so that you can save me an hour of time testing whether I was wrong to begin with.

I am working on writing it up.  I hope to have it done by tomorrow.

Cool, looking forward to it.

“Only Nixon could go to China”.

You are assuming that BaseRuns is near-perfect at estimating runs scored for a lineup of players?  I would certainly question that assumption.

A Markov or sim would be perfect, by their very nature.  After I was going to do my initial study with BaseRuns, I was going to run it through my Markov calculator to see the impact.

I’ll await Peter’s research before proceeding.

I am going to guess that a sim would show that lwts per PA is almost perfect at predicting how many extra runs a player adds to a lineup of average players.  I showed that to some extent on that other thread, I think.  Didn’t I?

After spending too much time on this, my original file stands:

http://www.tangotiger.net/reconcile.html

Tango - Sorry for the delay in responding.

My mistake was starting with the theoretical principle that non out batting events create both runs and extra batting opportunities for other players and out batting events create no runs and use up batting opportunities by using up the outs in an inning.  Both statements are true, of course, but they only serve to create misunderstanding when we try to apply them to a linear weights metric.  That is because linear weights is a closed system.  Once we establish the particular dataset that is to be evaluated by linear weights, whether it is data from a single team in a single year or multiple years for an entire league, the amount of expected PA’s for an inning are fixed for every inning of every game of every team just as the run expectation values are fixed and the individual linear weights are fixed.  The entire concept of a linear weights metric (or any metric) is to have fixed average values for different events that can be applied to any individual in that dataset to allow direct comparison to other individuals in the dataset.  For the major leagues from 2005 through 2007 the average number of PA’s in an inning can be established empirically as 4.33.  But the same number can also be computed from using the ratio of the number of out batting events to the total number of batting events (.6595) and the ratio of out batting events (total outs –outs on bases) to total outs (.9526).  The relationship is PA/INN = (1/.6595)*(3*.9526).
The 4.33 number is the fixed expectation of the number of batting events that it will take to complete an inning at all times before any outs are made.  It is true at the beginning of an inning, and it is also true if the first 10 batters in the inning get hits.  When the eleventh batter comes to the plate it is still expected to take an additional 4.33 batting events to complete the inning.  Only when a player makes an out does the number of expected batting events decrease.  The computed expected rate of decrease is a constant 1.44 PA’s per out, but the actual empirically calculated decrease varies because out ratio per batting event varies for outs 1, 2, and three.

If we multiply (PA’s/Inn) by (Runs/PA ) we get (Runs/Inn).  In this case that is 4.33 *.12299 = .532.  That number is very close to the number .537 that is the RE value for the 0 on 0 out state in the RE table.  The difference being due to the double counting of innings where there is a home run with 0 outs.  This is important because the RE value for the 0 on 0 out state is the starting value that is subtracted from any event occurring during that state to calculated that event’s linear weight.  If it is correctly reflecting both the number of PA’s per inning and the average runs per inning than the linear weights derived from the RE table the runs created above average where the average is including all the PA’s that are expected to remain in the inning.  If that is so than all we need to do to get the correct value for absolute runs created for an event is add in the average value per PA; no other adjustment is necessary.
And that is exactly what we find.  The empirical method for computing an RE table by calculating the number of times a base out state occurs and dividing that number into the total number of additional runs that score following that event exactly compensates for the number of PA’s left in the inning and the average runs per PA.  The only adjustment that is needed is that the .12299 value of Runs/PA includes total runs since we want the metric to measure the player’s offensive contribution of his batting skills only we adjust the average value that is added to each PA to .12 which is the computed value of batting runs per PA leaving out the runs created by passed balls, stolen bases, and other none batting events.
One way to check whether this methodology is correct is to see how it affects the absolute run value of an out.  I started this discussion by saying that an out that doesn’t advance a runner creates no positive value but as a negative value equal to the lost opportunity due to using up 1/3 of the outs in an inning.  This lost opportunity cost should then be 1/3 of the average run value of an inning which was calculated above as .532 runs. One third of that is a cost of .1774 runs.  The above average linear weight value of a strikeout with no stolen base or caught stealing is -.2912 runs.  Adding the average value of a PA to that gives an absolute run value for an out of -.1712.  Pretty close with the difference being caused by K’s with passed balls and wild pitches a BaseOut distribution for K’s that is slightly different than for the average PA.

I really don’t understand this whole R/PA thing when it comes to reconciling Linear Weights.

Peter did point out a flaw in Run Expectancy charts. Perhaps people will now begin to believe me when I say the sum of Linear Weight runs do not sum to exactly zero:

“If we multiply (PA’s/Inn) by (Runs/PA ) we get (Runs/Inn).  In this case that is 4.33 *.12299 = .532.  That number is very close to the number .537 that is the RE value for the 0 on 0 out state in the RE table.  The difference being due to the double counting of innings where there is a home run with 0 outs.  This is important because the RE value for the 0 on 0 out state is the starting value that is subtracted from any event occurring during that state to calculated that event’s linear weight.”

Terps - That is not a flaw in the run expectancy charts nor is it a reason for the linear weights values not to sum to zero.  The run expectancy values are what they are.  The only flaw lies in the reasoning of the people who assume that the RE value for the 0 on 0 out state should be exactly equivalent to runs per inning.

To the extent that I understand Peter’s post, I think he is right. The key is the “closed system” part. And all of the extra “refinements” that people worry about, such as the extra PA created, etc., exist because they are (improperly) going outside of the closed system

I’ve come to the opinion recently that the typical theoretical team procedure is not valid (at least, for a neutral evaluation of a player), and I think what Peter is talking about is exactly the reason why.

I will list what I understand so far about this topic. Please correct me if I am wrong.

1. When Linear Weights, stated in terms of the average run value of each play, are applied to a player with an above average OBP, the additional runs created (marginal runs) from the extra PA’s he generates are already accounted for.

2. When Absolute Linear Weights, that are calculated by adding the average/empirical Runs per Out to each play where an out is recorded, are applied to a player with an above average OBP, the additional runs created (absolute runs) from the extra PA’s he generates are not accounted for.

For point #2, couldn’t we simply figure the run-contribution of the extra PA’s and then add that number to the player’s initial RC estimate.

Terps - In linear weights above average all runs above average or below average including extra PA’s generated by a non out PA and extra PA’s lost by an out PA are included in the linear weight above average.  Adding the average run value of a batting event (.12 runs) perfectly to the runs avove average linear weight perfectly converts to absolute runs created without ANY further adjustment.

I think we can all agree that linear weights above average properly compensates a player for the extra plate appearances he generates for his teammates. Many common theoretical team models based on dynamic run estimators seemingly do not, although they compensate a player for the added value his events have in changing the run environment for the other players, which LWTS (in any form) does not. This does not have to be true, Marginal Lineup Value (for all its faults) achieves this, and it should be relatively simple to substitute BsR for Basic RC.

The trouble I have with attempting to preserve this relationship when moving to “absolute” LWTS or “runs created” linear weights is twofold.

First, I’m not sure it’s entirely appropriate to do so. If we were to break down the difference between the method, I’d say that:

* Linear weights above average (when applied to a hitter) presume that, without the actions of the player being evaluated, an average number of runs would score from this point forward.
* Runs created (when applied to a hitter) presumes that, without the actions of the player being evaluated, no runs would score from this point forward.

When you reconcile them at the Runs/PA level (instead of Runs/Out) you end up with a system that isn’t quite either.

A possibly related point - linear weights reconciled this way are simply not as accurate at estimating team run scoring totals. I reran my test set of LWTS, reconciled both ways, against all innings 1994-2007, after correcting the foul error issue terpsfan referred to. The average error for LWTS_RC was .394 runs per inning, compared to .451 for LWTS_PA. That may seem like an entirely pedantic concern, and I wouldn’t necessarily disagree. But I think it’s necessary if you do it this way to be very clear that this is only for evaluating individual hitters, not teams/games/pitchers/etc.

Is it appropriate for any Runs Created method (RC LW, BsR, RC) to consider the effect of PA generation? By PA generation don’t we mean the value of keeping the inning alive by not making an out, or reducing the scoring potential of the inning by making an out(s). My logic on this is probably faulty, but aren’t RC methods not supposed to take into account the “inning killer” portion of run creation.

My last post makes me look like an idiot. I didn’t mean to compare the “inning killer” portion of run creation to the idea of PA generation in a literal sense. It was just an analogy.

I actually think that an absoulte RE table (an RE table starting from zero) would be the best way to calculate RC LW. After we applied the RC LW to a player, we could then make a seperate adjustment accounting for the run-value of his PA generation that would be based on the hitters OBP compared to the league average OBP. Patriot and Sibelius (where in the heck did this guy go) have a formula for this.

I may be double-posting here.  I think my last post got eaten, but I’m not positive.

I largely agree with Colin/24.  Also, Terps, I don’t think you were off base in #25.  PA generation = out avoidance = inning killer.  Different ways of saying the same thing.  I have always preferred to explain it as PA generation, because it emphasizes the positive result of the batter’s actions rather than his avoidance of a negative.  But that’s semantics, and the approaches are equivalent.

It can be shown that traditional absolute RC/Out, or absolute RC/PA with the PA generation adjustment that terps referenced, results in equivalent Runs Above Average to traditional, Palmerian Linear Weights.

It can also be shown that absolute RC/PA with the PA adjustment is equivalent to the Tango-style reconciled LW.

So I only see two distinguishing factors between the approaches:

1) baseline--it is not as easy to apply a non-avg baseline to “R+/PA"--you have to define what the replacement player’s out rate is in addition to his run production rate.  This can be worked around easier in the Tango approach since you don’t have to worry about the out rate directly.  So this is really a non-issue as best as I can tell.

2) presentation/personal preference--some people might like to have an all-encompassing RC figure that can be used on a PA basis.  Others might like to have an absolute RC figure that is 1) calculated equivalently to team absolute RC and 2) works with outs as the denominator.

In the end, I think this is a semantics/presentation issue.

I agree with Patriot.

If we agree that Linear Weights per PA is what we ultimately need, then this is how we should express it: in terms of something per PA.  This saves us from worrying about per out and making further adjustments to account for that.

wOBA works nice in that respect.

I recall seeing David Smyth apply MLV to Baseruns. Browsing through Tango’s archive of Fanhome and Scout.com postings, I can’t find the post though. I recall seeing it in the Scout.com postings, which would be more recent than Fanhome. I don’t know, I guess I’ll have to open up all 400+ postings until I find it.

This freeware program searches for text inside any kind of file. I highly recommend it.

http://www.fxsearch.com/ldw_eng/

I was able to find all files in my baseball folder that contained MLV in them.

#30, wow, I have always wished for a program like that!  Thanks!

Nice, good job.

EditPlus is good too…

Lookdisk is a nice little program. A bit user un-friendly, but it gets the job done.

I don’t know why the Windows search function can’t look inside files. We shouldn’t even need a program like this.

I don’t know why the Windows search function can’t look inside files. We shouldn’t even need a program like this.

Actually, it does, doesn’t it?  Windows XP.  I just went to “search” and it has a box for “a word or phrase in the file...”

I just played around with the search function in XP. It does search inside text files and Microsoft Office files. It doesn’t search inside html, zip, and cab files.

html files are just text files with a bunch of “tags,” right?

I guess that would be a way you could describe an html file. I should have gotten a degree in the computer sciences. It would be nice to know programming, sql, html, etc....

This is a response to Colin’s comments he made about reconciling LW and RC in this thread:

http://www.insidethebook.com/ee/index.php/site/comments/best_performances_of_2008/

Here is what Colin said:

“For everyone’s benefit (or at least my tenuous sanity) here’s the last thread we had on converting between absolute runs and runs above average.

For the sake of brevity I’ll try and say something that (I hope) won’t get me killed, that should summarize the issue:

Whatever you reconcile your linear weights by is what you need to measure playing time with. If you are measuring playing time by PAs then your LWTS need to be reconciled at the PA level (which should give an out value of around -.14 for the modern offensive era).

It looks like jinaz reconciled by out instead. This is probably not a huge deal, but it is an area for potential improvement.”

My comments:

The problem I have with reconciling with PA’s, is you inflate the results of all offensive events. For instance, if LW says a home-run is worth 1.40 runs, if we add the average PA run value to this, a home run is now worth 1.52 runs. This doesn’t seem logical. It makes sense to reconcile by R/O. Then, a HR is still worth 1.40 runs. The only values that change are categories where an out can be recorded (1B, 2B, 3B, AB-H, SO, CS). And since outs are rarely recorded on singles, doubles, and triples, their RC run-value is only slightly higher than their LW run-value (within .005 runs). I wish we could come to some consensus on this issue.

I guess the problem is how are we going to express the raw Runs Created number. If you reconcile by R/O, then you have to express the RC number in terms of Outs. If you reconcile by R/PA, then you can express the RC number in terms of Plate Appearances.

Couldn’t we use Plate Appearances as the denominator if we added Patriot’s “PA Generation Adjustment” to the Runs Created figure that was derived from Linear Weights that were reconciled to runs scored using R/O?

Suppose you have an average player with 80 RC, with 400 outs in 600 PA.

You have the player you are interested in with 100 (explicit) RC, 300 outs in 500 PA.  His LWTS will be +40.

His RC is 100 per 300 outs, whereby the average is 80 RC per 400 outs, or an average of 60 RC per 300 outs.  And 100 minus 60 is +40.

Given 500 PA, our player is +40 runs above average.

The average player has 80 RC in 600 PA, or 67 RC in 500 PA.

67+40 = 107.

So, our player is 107 RC (100 explicit and 7 implicit for extra PA or fewer outs used), and is +40 compared to an average hitter.

Any way you do this, you CANNOT have someone with the same number of PA change the ordering of their performance, as you switch between LWTS and RC.  If you come up with a translation where you have one guy as +40 and another as +38, while the first guy is 100 RC and the second guy is 102, all the while they both come to bat the same number of times, then you are doing the conversion wrong.

***

And the HR being +1.40 runs means 1.40 runs ABOVE AVERAGE.  The average PA would generate 0.12 runs, so the HR generate 1.52 runs.  The HR is not worth 1.40 absolute runs.

If we reconcile Linear Weights to Runs Created using R/PA, how do we reconcile non-PA events? Should we use Runs/Event?

I thought it would be a good idea to see what a Baseruns equation would look like that used our new RC method for the inputs. Actually the method isn’t new. Gary Skoog used it 20+ years ago in his Value-Added article.

Instead of using R/PA, I used R/Event. You have to use R/Event if you are including non-PA categories. This might make your eyes bleed:

http://spreadsheets.google.com/pub?key=pzy9IhjJPqavcyrLYpqLsSw

There are still a lot of loose ends with using R/PA to reconcile Linear Weights. Let me see if I can identify them.

1. How do we treat non-PA events? If we use R/PA, then we can’t reconcile non-PA events. We need to use Runs/Event, so we can properly reconcile SB and CS. You can’t apply R/PA to only PA events and then use the LW run-value for SB and CS.

2. Where does Baseruns fit in? Can we use these new Runs Created values in Baseruns equations? Will Baseruns break down if we use these new values? Can we still use Theoretical team methods to apply Baseruns to individual batters?

3. Prepare to deal with negative Runs Created. Since the value of the out categories are going to be more negative, we will have more players with negative runs created. I don’t have a problem with this. I’m just pointing it out.

1. You use the LWTS values, exactly as shown in The Book, and as I show on my site.  You reconcile from LWTS to RC by adding .12 runs per PA.  That means the non-batting events stay frozen.

The “opportunity” given a player is the number of PA.  It is not PA+SB+CS+WP+PB+BK+PK.  That’s why we add .12 for each PA, because that’s the average number of runs per PA the average player creates.

2. I’m not exactly sure of your question.  But, the marginal values that come out of BaseRuns must reconcile to whatever you get in #1.

3. Negative runs created are a reality.  There’s nothing disbelieving in accepting that a player can be so bad as to undo the good of other players.  A guy who is 1 for 600 has created negative runs because he has unleveraged almost all the opps presented to him.  People who don’t accept negative runs created have simply not been properly educated on the matter by those who accept negative runs created.  It’s a tiny obstacle that can be used as an opportunity for education.

I understand that the “opportunity given a player is the number of PA,” but the values of SB, CS, WP ... are MARGINAL values, just like the run-values of PA events. In #39 you wrote:

“the HR being +1.40 runs means 1.40 runs ABOVE AVERAGE.  The average PA would generate 0.12 runs, so the HR generate 1.52 runs.  The HR is not worth 1.40 absolute runs.”

A SB is .18 runs ABOVE AVERAGE. A CS is -.44 runs ABOVE AVERAGE. Why is it OK to add an average run value to the HR, but not the SB or CS? Please give a different answer than: “A SB or CS is not a PA.”

I guess you can tell that I don’t like this method. Just like I dislike the idea of moving out of my parents house. Convince me to “move out.” Show me the new way is better than the old way.

Suppose you have someone with 100 PA, 33 BB and 66 SB. 

Another guy has 100 PA, 33 triples.

Let’s presume that this is a leadoff hitter, and his SB occur on pickoffs.

In both cases, the guy will end up at 3B 33 times, out of 100 PA.

The run value of a walk with bases empty 0 outs is +.40 runs.  The run value of the SB are +.25 runs.

The run value of a triple is +.90 runs.

So, the first guy add +.40 runs for his 33 walks and +.25 runs for his 66 walks for a total of +29.7 runs.

The second guy adds +.90 runs for his 33 triples, for a total of +29.7 runs.

Their 67 outs are both worth the same, which we’ll call -.30 runs each, or -20.1 runs.

Both players are +9.6 runs in 100 PA.  One guy does it in 100 PA, and the other guy does it in 100PA+66SB.

Now, you want to scale it to absolute runs created.  The only constants you have to work with are the 100 PA or 67 outs.  You can’t use PA+SB, since now, you will put everything out of whack.

***

You can also consider a single that’s stretched into a double, or a single with a SB.  It’s the exact same thing.  You can’t call one one opp, and call the other two opps.

***

You should move out of your house once you are too dependent on your parents.  You need to grow as a person to reach your potential, and that’s sometimes easier done without someone catching you every time you fall.  They should be there to provide support only when the circumstances are more dire than more routine.  I’m presuming that you are out of high school.

OK, your explanation for using Plate Appearances makes sense. It still seems like we are mixing apples and oranges since we are mixing runs-above-average values with runs-created values.

The only reason I brought up moving out of parents house was I had spent the previous 2 hours fighting with them. They had a right to be upset considering I was fired yesterday (3rd time this year).

Tango, there is nothing wrong with living with your parents as a young adult. I don’t consider myself “dependent” upon them, although technically I am, since I live in their house.

I don’t particularly like runs created, so trying to reconcile to them is always a bit of a pain. 

***

I never said there was anything wrong with living in your parents house as a young adult.  When did I say otherwise?  I said: “once you are too dependent”.  That is, when you are mooching off them more than you are returning back into the household (in financial or emotional support).  Lebron James, as an extreme example, would be fine if he lived with his parents.

R/PA runs-created have similar run-values to Linear Weights generated from TECH-1 Runs Created. Patriot generated applied the +1 method to TECH-1 RC for the 1990 NL. Look how similar the values are:

R/PA, RC TECH-1
0.573 0.564 1B
0.864 0.855 2B
1.159 1.146 3B
1.492 1.437 HR
0.414 0.348 NIBB
0.280 0.273 IBB
0.431 0.348 HBP
-.027 0.032 SH
0.088 0.032 SF
-.122 -.112 AB-H
-.152 -.112 SO
-.344 -.382 GIDP
0.173 0.143 SB
-.433 -.382 CS

I guess we can’t bash Runs Created anymore, at least the old TECH Versions.

For the record, I didn’t get fired. I was laid off for the 3rd time this year. Although I consider getting laid-off and getting fired to be the same thing. You are being fired in a cordial manner.

Nobody ever argued that Runs Created will not from time to time produce acceptable outputs. Bill James may be stubborn but he’s not stupid. In fact, from time to time RC will outperform BsR, depending on what tests you run, which versions you use and what dataset you’re testing on.

The problem is that the Runs Created model is inherently flawed, and so even when it’s more accurate it is less correct. BaseRuns is a more useful - and correct - model of run scoring and thus serves our purposes better.

terps, the problem is that James does RC per out rather than RC per PA.  So, you are not comparing apples-to-apples. To that end, his weights are wrong.

Tango,

Shouldn’t James express Runs Created as R/PA. The opportunity component (C Factor) of Runs Created is plate appearances, not outs.

I’m having a hard time understanding how this relates to the linear weights generated by Runs Created. What do you mean by “To that end, his weights are wrong.”

James expresses it as RC per out, not per PA.

Which as a model of team run scoring is absolutely correct. And there’s no such thing as a model of individual run scoring.