In the introduction, I explained a set of tests I did to find out the true value of an APBA baseball card.  This section will deal with the valuation of those numbers that the hitter controls to some extent: hits, walks and hit by pitch.  The error and rare play numbers will be in a future post.

Before we get started, let’s establish what the Major Leagues did on a divided by 36 plate appearances for the 2012 season:

Test	R/36PA	1B/36PA	2B/36PA	3B/36PA	HR/36PA	BB/36PA	SO/36PA	HP/36PA	GDP/36PA
2012 MLB	4.11	5.46	1.61	.18	.96	2.88	7.12	.29	.71

I played 5 seasons with the 5.75 computer game using Buck Jr. as the manager. The numbers were fairly close:

Test	R/36PA	1B/36PA	2B/36PA	3B/36PA	HR/36PA	BB/36PA	SO/36PA	HP/36PA	GDP/36PA
Baseline	4.04	5.40	1.53	.18	.94	2.80	7.84	.23	.59
Difference	-.07	-.06	-.08	+.00	-.02	-.08	+.72	-.06	-.12

Runs are down a little bit, mostly due to a combination of my over-zealousness of making sure I didn’t have to manually fiddle with lineups by giving some scrubs more playing time, and the micromanager using the better pitchers a little more often than he should.  For comparison purposes, these are fine numbers and can be used as a baseline when I start adding numbers to the cards and running the simulations.

First up we will test out everyone’s favorite number, the 1.  At minimum, this will be worth one run.  Based on the base-running situations from actual 2012 data, it’s not that hard to jump to exactly how many runs it should be worth:

• Bases Empty: 56% of the time, so .56 runs
• Runner on 1st: 18%, Runner on 2nd: 9%, Runner on 3rd: 3%, so .60 runs [(.18 + .09 + .03) * 2]
• Runners on 1st and 2nd: 7%, Runners on 1st and 3rd: 3%, Runners on 2nd and 3rd: 2%, so .36 runs [(.07 + .03 + .02) * 3]
• Bases Full: 2%, so .08 runs (.02 * 4)

The sum of the four items above is 1.50.  And compare that to the simulation:

Test	R/36PA	1B/36PA	2B/36PA	3B/36PA	HR/36PA	BB/36PA	SO/36PA	HP/36PA	GDP/36PA
Extra 1	5.54	5.42	1.55	.18	1.93	2.80	7.81	.22	.56
Baseline Diff	+1.50	+.02	+.02	+.00	+.99	+.00	-.03	-.01	-.03

That’s pretty much spot on.  Notice a slight uptick in the other hits as well. More runs means more downgraded (and fewer upgraded) pitchers. This was causing an issue with valuing hits a little too high, so a scaling factor was introduced, which basically knocked the runs down by the ratio of new hits over baseline hits for numbers 1 through 7. The scaling factor for the 1 was 1.03, so the expected run value (1.50) was divided by the scaling factor (1.03) to give us 1.46. We can now start calculating the card worth from my two subjects presented in the introduction: Jay Bruce and Torii Hunter:

• Jay Bruce has a first column 1: 1 * 1.46 = 1.46 runs per 36 PA (so far)
• Torii Hunter has 11 second column 1’s behind 3 0’s: (11 [1s] / 36) * 3 [0s] * 1.46 [R/36PA diff] = 1.34 runs per 36 PA (so far)

Moving on to the power numbers, we start to hit some uncharted territory.  With the exception of the 3, the power numbers on average are more powerful when there are runners on base, meaning that these values can be season-dependent.  In a pitching year, they’re going to be not as powerful.  However, since we’re looking at 2012, we’ll just have to use what we have.  Here are the differences for 2 through 6 as compared with the baseline:

Test	R/36PA	1B/36PA	2B/36PA	3B/36PA	HR/36PA	BB/36PA	SO/36PA	HP/36PA	GDP/36PA
Baseline	4.04	5.40	1.53	.18	.94	2.80	7.84	.23	.59
Extra 2	5.12	5.43	1.64	1.06	.98	2.81	7.74	.23	.58
Extra 3	5.02	5.57	1.75	.72	1.04	2.81	7.77	.23	.57
Extra 4	5.00	5.59	2.03	.46	1.02	2.80	7.73	.23	.59
Extra 5	5.12	5.51	2.07	.27	1.22	2.82	7.77	.23	.59
Extra 6	4.94	5.42	2.44	.19	1.02	2.82	7.74	.24	.58
Diff 2	+1.08	+.03	+.09	+.88	+.04	+.01	-.10	+.00	-.01
Diff 3	+.98	+.17	+.22	+.54	+.10	+.01	-.07	+.00	-.02
Diff 4	+.96	+.19	+.50	+.28	+.08	+.00	-.11	+.00	+.00
Diff 5	+1.08	+.11	+.54	+.09	+.28	+.02	-.07	+.00	-.01
Diff 6	+.90	+.02	+.89	+.01	+.08	+.02	-.10	+.01	-.01

Right off the bat, we see that a 2 and a 5 seem to be equivalent.  The difference between a 3 and 4 does not seem to be that significant despite the fact that you get more triples and homers and less doubles and singles.  3s, 4s and 5s all have times where they can be converted into singles, and it’s obvious that happens a little more often than we realize.

Of note is that the numbers for 2 and 6 represent an extra first column 2 or 6, so there will be a bit of a bump in the homers for both of those in expense of the triples and doubles.  A test I would need to do later on is rig one test so it’s only first column 2 or 6, then second column 2 or 6, and see the difference.

In addition, we had a scaling factor to contend with from 1.03-1.07 depending on the number. The new diff values ended up being 1.02 for a 2, .94 for a 3, .91 for a 4, 1.06 for a 5, and .88 for a 6. Let’s go to the scoreboard:

• Jay Bruce has two first column 5s and a first column 6: 1.06 * 2 [for the 2 5s] + .88 [for the 6] = 3.  Add that to the previous 1.46 and we get 4.46.

• Torii Hunter has a second column 3, a 5 and 17 6s: {0.94 [for the 3] + 1.06 [for the 5] + (17 * .88 [for the 6s])} / 36 * 3[0s] = 1.41.  Add that to the previous 1.34 and we get 2.75.

Right now Jay has a bit of a lead.  Let’s see if Torii catches up with singles:

Test	R/36PA	1B/36PA	2B/36PA	3B/36PA	HR/36PA	BB/36PA	SO/36PA	HP/36PA	GDP/36PA
Baseline	4.04	5.40	1.53	.18	.94	2.80	7.84	.23	.59
Extra 7	4.60	6.44	1.52	.18	.94	2.78	7.75	.23	.61
Extra 8	4.53	6.17	1.52	.19	.94	2.80	7.75	.23	.61
Extra 9	4.37	6.03	1.52	.18	.94	2.80	7.87	.23	.62
Extra 10	4.41	6.10	1.51	.19	.93	2.80	7.76	.23	.61
Extra 11	4.60	6.41	1.52	.19	.94	2.78	7.78	.23	.63
Diff 7	+.56	+1.04	-.01	+.00	+.00	-.02	-.09	+.00	+.05
Diff 8	+.49	+.77	-.01	+.01	+.00	+.00	-.09	+.00	+.02
Diff 9	+.34	+.63	-.01	+.00	+.00	+.00	+.03	+.00	+.03
Diff 10	+.37	+.70	-.02	+.01	-.01	+.00	-.06	+.00	+.02
Diff 11	+.56	+1.01	-.01	+.01	+.00	-.02	-.06	+.00	+.04

Biggest surprise here is the 8 actually being better than the 10, which is the opposite of the Basic game.  As the current grading/carding system is more reliant on having pitchers with single-digit grades, the 10 is converted into an out more than an 8.  For example, with the bases empty, only 22% of the innings pitched in 2012 are a grade of 13 or more which will stop the 8, but the 10 can be stopped by grade 7 (6%), 12 (9%), 15-17 (9%) and 20+ (1%, total of 25%).  Makes me wonder if the Master charts (and the computer game) need to fiddle a bit with the 10 and the 8 to get things more in line.  You also see the effect of having the 18+ pitchers converting 9’s into 13’s with the higher strikeout numbers.

I also added a scaling factor to the 7 and the 11 which knocked the diff value for those to .55 and .54 respectively. Checking in with our rightfield candidates, Torii catches up a bit (for the purposes of this exercise, I’m treating the second column 8’s as 7’s):

• Jay Bruce has 2 8’s, 2 9’s and a 10 all in the first column: (.49 * 2) + (.33 * 2) + .37 = 1.91.  Add that to his previous 4.46 and we get 6.37.
• Torii Hunter has 2 7’s, 3 8’s, 2 9’s and a 10 in the first column, and 3 7’s and 3 8’s in the second column: (.55 * 2) + (.49 * 3) + (.34 * 2) + .37 + (.55 * 6 / 36 * 3) = 3.88.  Add that to his previous 2.75 and we get 6.63.

Finally, let’s check the non-hit-based hitting, walks and hit-by-pitch:

Test	R/36PA	1B/36PA	2B/36PA	3B/36PA	HR/36PA	BB/36PA	SO/36PA	HP/36PA	GDP/36PA
Baseline	4.04	5.40	1.53	.18	.94	2.80	7.84	.23	.59
Extra 14	4.48	5.44	1.53	.19	.94	3.72	7.77	.24	.65
Extra 42	4.60	5.57	1.56	.19	.98	2.86	8.01	1.22	.66
Diff 14	+.44	+.04	+.00	+.01	+.00	+.92	-.07	+.01	+.06
Diff 42	+.56	+.17	+.03	+.01	+.04	+.06	+.17	+.99	+.07

I’m not surprised with the slight upticks with the hits on the 14, since you have another go around at the card if you hit a Z.  But what in the heck is going on with the 42?  There must be some weird double purpose going on that is unique to the computer game.  Everything across the board is higher.  For a lark, I tried giving a few cards only 42s to see if they would get something other than HBP.  Well, yes, they walked against HB0 pitchers.  Other than that, it was HBP’s.  And lots of ejections.  The Tigers had to “forfeit” a game when they ran out of players after their 17th ejection.  On the other hand, Miguel Cabrera had 234 RBI since Austin Jackson was constantly getting hit by the pitch.

Back to the task at hand, let’s check our players:

• Jay Bruce has 3 14s: (.44 * 3) = 1.32.  Add that to his previous 6.37 and we get 7.69.
• Torii Hunter has 2 14s: (.44 * 2)  = .88.  Add that to his previous 6.63 and we get 7.51.

At this point we have most of our positive numbers.  And we even know now that a 14 is better than a 9.  And for some strange reason a 42 is equal to a 7.  But how much will the errors, rare plays and strikeouts make a difference?  We’ll find out in the next installment…

NOTE: Scaling factor and a correction to Jay Bruce’s numbers were added on 7/22/2013.