Or: Fun With Math

Or: The Unified Theory Of Baseball

Or: Winter Break Is Boring As Hell

With my recent delve into Offensive Winning Percentage, I thought I'd take a look at whether or not the replacement value of .350% was correct. In order to do this, we must reconcile the OWP baseline with the winning percentage baselines set by Tango for SPs and RPs. Allow me to take you through my mathematical process. First, we have the major assumption: A replacement team will win 30% of its games - that is, win% = .300. Now, some more assumptions. Any numerical data is taken from the 2008 season, and is averaged between NL and AL.

Win% = from Pythagenpat - repR = replacement runs scored, repRA = replacement runs allowed.

lgRA = lgR = 4.61

x = this is our pythagopat exponent

So we have everything we need to make the Pythagenpat calculation. Now, recall the formula for Offensive Winning Percentage

so

OK, now we need to figure out how we'll find runs per out. Well, to make it simpler, we only need to find league runs per out because we will use repOWP as our independent variable. So let's find lg(R/outs). In order to do this, we need runs/162 and outs/162. These are both pretty easy, as long as we make the assumption that each game is 27 outs. This isn't necessarily true, due to only having anywhere from 24 to 27 outs as the home team, but I feel that this is offset enough by extra inning games for any error to be significant.

lgR/G = 4.61

lgR/162 G = 4.61*162 = 746.82

Outs/G = 27

lgOuts/162 G = 27*162 = 4374

So then we have lg(R/Outs) = 746.82/4374 = .1707. Now let's plug this into our repOWP equation.

Great. Now we're down to just R/outs as our variable on the right side. Let's eliminate outs by multiplying the top and bottom of the fraction by outs^1.89. Remember, outs = 4374 from above.

Yes, that's 269317.23 in the denominator. A little more rearranging...

There. repR has been isolated. So let's see what we get when we use the .350 that Bill James gives as replacement level as our repOWP.

Alright! We're done with step 1! Now, we can finally use our first assumption - win% = .300. Using Pythagenpat we have the following (note that 538.158 R/162 = 3.32 R/G):

So we now have our runs allowed for a replacement team where OWP = .350. Let's convert this to Defensive Winning Percentage (DWP). (Note that repRA/g = 842.57/162 = 5.20)

lgRA = 4.61, repRA = 5.20

The current accepted values for DWP are .380 for starters and .470 for relievers. In order for this to add up to .442, relievers would have to pitch 68% of innings. Clearly, that means that our repOWP is too low.

I created this spreadsheet where you can just change the OWP baseline in the lower right and it'll spit out RS, RA, and DWP numbers, as well as allowing you to change innings pitched by starters and relievers. I assume that a replacement team would average 5 IP per start, resulting in 810 IP from starting pitchers and 648 from relievers (again, this is assuming 9 IP per game, 9*162 = 1458).

Using the above values for starter and reliever DWP and our inning distribution, we can find what our expected DWP is.

Finally, we can figure out what our OWP should be based on this. This is where the spreadsheet comes in. It's a real pain to figure out repRA using the DWP formula, and since DWP and OWP are dependent on each other, I just input OWP values until I got the correct DWP. It turns out that the correct OWP is EDIT: .370%, which gives 562.9 RS and 881.3 RA, or 3.47 RS/g and 5.44 RA/g. Now, applying this to my data from my last post, I get the following (I also fixed the data using the current simple WAR estimator - I had the average wOBA set too low).

Basically, what we see here is that the WAR values we get here are much closer, overall, to what we get using the simple WAR estimation.  It should be noted that the replacement wOBA that I get using OWP is .304, which strikes me as being a bit high (but maybe not that high), but that's why I'm looking for some discussion on this.

Edit: So I realized that when I was figuring replacement wOBA I forgot to divide by 1.15.  It turns out that replacement wOBA works out to .306, which I think makes a little bit more sense.  This correction is also reflected in the table (total WAR is lower because I had forgotten the 1.15 in the players projections as well).

Edit 2: Yet another stupid mistake:  Due to a computing error, the correct OWP% should be .370%.  This leads to a replacement wOBA of.304.  The necessary changes were made to the article.