A Simpler Way to WAR (long post)

Since discovering Wins Above Replacement, I've had basically two ambitions:  one, to create an uber-stat which I could use to combine peak and career weight when having a Hall of Fame discussion.  I believe I've already done that with WARSCOR, particularly in its new revision (though it's admittedly much more convoluted than the Hall of Stats or JAWS methodologies).  The second goal I've had is to make calculating WAR a simpler process.  In other words, to be able to calculate it myself quickly and efficiently.  Well, I'm here to say that, while I haven't quite "done it," I've gotten much, much closer.  I now have a very good (and reasonable, I think) way to calculate WAR for offense and for pitching.  Defense, not so much, but that's okay (and we'll explore what that might look like, if there ever were such a thing, at the end of this post).  This is a start.  It doesn't really do that, but it does give you a way to compare offensive players to pitchers by giving them a "won-lost" record which matches up with a pitcher won-lost record - in other words, there are 162 decisions for the pitchers, but also 162 "decisions" for the hitters.

You may recall my last post that I used ERA+ to give relievers a "record."  Well, we're not going to worry about Fibonacci wins like I did in that post (though you'd certainly be welcome to play with them, if you feel like it).  But the rest of the methodology stays the same, pretty much.  Except that now, we're going to be comparing to replacement level.  So, what is a reasonable replacement level?  How about, just picking something out of the air, .310?  If you'd like to use something different, you're welcome to it.  Just follow the same steps, only with a different product side of the equation.  We have to find out that, if a team's run prevention and run scoring are equally bad (compared to average), what would they be for a .310 winning percentage?  The equation looks like this:

           (100-x)^2                     -     31
(100-x)^2 + (100+x)^2           -    100

100(100-x)^2 = 31(100-x)^2 + 31(100+x)^2
69(100-x)^2 = 31(100+x)^2
69(10000-200x+x^2) = 31(10000+200x+x^2)
690000-13800x+69x^2 = 310000+6200x+31x^2
380000-20000x+38x^2 = 0

Solve that equation, and you get (roughly) 20.  Actually, using 20, you get a winning percentage of .307, but that's good enough for me.

So, basically, since we have measures that can tell us how to compare players to average (where 100 is average) that are consistent (if not perfectly linearly related) to run scoring, we can actually tease out individual records from this exercise.  We want to know what a player would be like if he were on an average team.  We could actually do it with a replacement-level team instead, but average will work fine well.

So now, for pitchers, we figure out the record.  It's easy:  divide the number of innings pitched by 9.  This will be the number of decisions.  Then take 10000/(ERA+) [or, alternately, just use ERA-, which needn't be adjusted].  Take this number and insert it for "x" in this formula:

    100^2     
100^2 + x^2

Now, multiply by the number of decisions.  This gives you a number of wins.  You can get "losses" by subtracting wins from decisions, if you felt so inclined.

So, let's look at two pitchers:  Justin Verlander in 2011 (251 IP, 172 ERA+) and Justin Verlander in 2012 (238.1 IP, 160 ERA+).

2011:
10000/172 = 58
100^2/(100^2+58^2)=.748
251/9=27.9
.748*27.9=20.9
2011 Verlander, by this method, "went" 20.9-7.0

2012:
10000/160 = 63
100^2/(100^2+63^2)=.716
238/9=26.4
.716*26.4=18.9
2012 Verlander, by this method, "went" 18.9-7.5

So, let's do hitters.  They're pretty much the same, except that we use OPS+ or wRC+, and we're adjusting the numerator and denominator.  They'll do different things:  wRC+ will include SB and some other offensive events (including sacrifices, double plays, etc.); OPS+ will only consider hitting properly.  But they're basically the same.  Still, I'll show them separately, because they're just different enough to cause a kerfuffle.  For OPS+:

First, we take batting outs (AB-H) and divide by 25.5 to get the number of "decisions."  Then, we just plug like we did last time, with OPS+ standing in for x, but this time, it looks like this:

       x^2      
x^2 + 100^2

Then, we multiply by "decisions."  Here are two hitters, Miguel Cabrera in 2011 (572 AB, 197 H, 179 OPS+), and Miguel Cabrera in 2012 (622 AB, 205 H, 165 OPS+):

2011:
572-197=375
375/25.5=14.7
179^2/(179^2+100^2)=.762
.762*14.7=11.2
2011 Cabrera, by this method, "went" 11.2-3.5

2012:
622-205=417
417/25.5=16.4
165^2/(165^2+100^2)=.731
.731*16.3=12.0
2012 Cabrera, by this method, "went" 12.0-4.4

-------------------
Aside:
I'm gonna take this opportunity to say a word about replacement level.  A replacement level player on an average team will be considerably better than replacement level.  That just makes sense, doesn't it?  Since we're only comparing offense or defense, and making the other average, we will get the overall to be higher than .307, which is what we used as replacement level.  By this method, a replacement level offensive player would be stuck into this formula:

80*80/(80*80+100*100)=.390

A pitcher actually has a different replacement level, for this exercise, since:
100*100/(100*100+120*120)=.410
Personally, I don't see this as any reason to really care, because no one pitches enough innings for this to even make up a full win.  If you feel differently, please feel free to do the math to normalize this discrepancy.  Otherwise, keep in mind that this is just a fun, silly exercise by a person who only took one math class in college.

As you can see, the result is not .307, but .390.  Of course, this means that, comparing, say, 2012 Miguel Cabrera to replacement level, we'd do (replacement level) * (number of "decisions"), and then subtract that number from Cabrera's own wins.  In other words:

.390*16.3=6.4
11.9-6.4=5.5 "Wins Above Replacement"

Of course, there's another alternative.  We could have put Cabrera on a team that gave up runs at a replacement-level rate, and then simply subtracted wins at a rate of .307, our initial rate.  Like this:

165*165/(165*165+120*120)=.654
.654*16.3=10.7
.310*16.3=5.0
10.7-5.0=5.7 "Wins Above Replacement"

I've been working off the first method, but if you were to work by the second method, I wouldn't begrudge you.  It's probably actually a little better.  A little cleaner for the comparison to replacement, anyway.  But it's up to you.
End of Aside
-------------------

Finally, we'll look at what it looks like if you use wRC+, instead of the baseball-reference stats.  We'll use two players:  Ryan Braun in 2011 and Ryan Braun in 2012.  In this method, we look at all the outs the offensive player made, instead of just batting outs.  So the formula looks like this.

First, we figure total outs, by taking batting outs (AB-H), like before, and adding GDP, SH, SF, and CS.  Then we divide that by 27 to get "decisions."  The rest of the formula is identical to the OPS+ version.  So here's Brauny.

2011:
563-187+9+3+0+6=394
394/27=14.6
173^2/(173^2+100^2)=.750
.750*14.6=11.0
2011 Braun, by this method, "went" 11.0-3.6

2012:
598-191+12+5+0+7=431
431/27=16.0
162^2/(162^2+100^2)=.724
.724*16.0=11.6
2012 Braun, by this method, "went" 11.6-4.4

So, there you go.  You can see that we can, pretty easily, produce a "pitcher-like" record for an offensive player.  Obviously, it's not on the same scale quite, since even top players end up with under 20 "decisions," making comparisons difficult.  But it's still fun, I think, to look at.

So now, we get to imagination land.  How would I change this, if I could, to make it more like actual WAR?  Well, first of all, I would want a defensive system.  What we'd need to develop, of course, is a system by which we measured, basically, the number of "plays" that a player made (or perhaps better, runs saved on plays made, or whatever), relative to the expected number for his position, just as we would have for OPS+ or ERA-.  Once we have that number relative to 100, just as we do for other parts of the game, we can determine the number of decisions and then the number of wins.

Of course, it would be silly to have a number of defensive wins and losses that equalled 162, as well as a number of offensive and pitching wins and losses.  So what do we do about it?  Well, for my money, we would divide the offensive number by two, the defensive number by six, and the pitching number by three.  That would give us a much better basis for comparison.  We could actually see this already.  Take Cabrera in 2011 and Verlander that same year.  If we take Cabrera's record in half, we get 5.6-1.8; Verlander's as a third and we get 7.0-2.3.  Those are a lot more comparable, and would be even moreso if we were able to add in Cabrera's defensive wins and losses.  We'd see something much different from what we're used to seeing.

There is, of course, one problem that I must address.  If one were to implement the system I just suggested with, say a shortstop who was slightly below average fielding and hitting (let's say a 99 in each), he would grade out as a below-average player.  However, everyone knows that a shortstop who is basically average defensively and basically average offensively is a HUGE asset.  This is why, perhaps, it would be good in creating a defensive system to compare the runs saved, not to position, but to all positions on the field.  That would make shortstops automatically very valuable, while it would make first basemen very low in value.  But that's just an idea.  As far as I know, there's no such stat out there, so maybe that's another project for me.  But I doubt it.

So, that's my big brainstorm.  If you made it this far, wow.  Just wow.  Because I'm really, really impressed.  It was a ridiculously long post.  But I hope you enjoyed it.  Suggestions?

Thanks to baseball-reference and fangraphs for the stats in today's post!