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Wednesday, July 28, 2010
SABR 111 - Base values
Part 2.
Yesterday, we were able to figure out, with the barest of efforts, the number of bases each event moves the runners and the batter himself:
5.40 HR
4.40 3B
3.22 2B
1.81 1B
1.42 BB
And we reasoned that since four bases equals a run that we could just divide those numbers by four to get the run impact:
1.35 HR
1.10 3B
0.81 2B
0.45 1B
0.36 BB
Analysts like myself have spent tens or hundreds of hours pouring through hundreds of thousands, if not millions, of events to come up with numbers that are within .05 runs of each of those numbers. What a waste, right? Yeah, probably.
Anyway, while it’s easy enough to accept that four bases equals one run, is it a given that each base is worth the same 0.25 runs?
Let’s try something. Remember when I said that if you have a high run environment, say 0.9 runners on base instead of 0.6, that we’d increase all the base-advancement values by 50%? So, this previous chart:
1.40 HR
1.40 3B
1.22 2B
0.81 1B
0.42 BB
Becomes this:
2.10 HR
2.10 3B
1.83 2B
1.21 1B
0.63 BB
So, a HR advances baserunners on base a total of 2.1 bases, and walks add 0.63 bases to the baserunner. Adding in the bases for the batter himself, and we get:
6.10 HR
5.10 3B
3.83 2B
2.21 1B
1.63 BB
Divide by 4, and we get:
1.53 HR
1.28 3B
0.96 2B
0.55 1B
0.41 BB
Let’s go crazy, and say that we’re going to have ALOT of runners on base. Not the standard 0.6, but 1.8 runners on base for each at bat. We had 0.3 runners on 1B originally? Now we have 0.9. We go from 0.2 to 0.6 runners on 2B, and end up with 0.3 runners on 3B. That’s 1.8 runners on base. Again starting with our standard base-advancement values, we’ll triple them to give us this:
4.20 HR
4.20 3B
3.66 2B
2.43 1B
1.26 BB
Once again adding our bases advanced by the batter, and we have:
8.20 HR
7.20 3B
5.66 2B
3.43 1B
2.26 BB
Divide by 4 and we have:
2.05 HR
1.80 3B
1.42 2B
0.86 1B
0.57 BB
Does it make sense though that a HR adds +2.05 runs, while a walk will add only 0.57 runs?
Imagine an even crazier run environment where just reaching first base is an almost guarantee to scoring a run. (Think of your crazy softball league where the team OBP is .750 or something.) Having a single advance a runner from 1B to 2B is (almost) no less advantageous than moving him to 3B. In short, while four bases are still worth one run, each base is not worth the same 0.25 runs. That is, the important base is the first base, while the other bases are small stepping stones. They may as well not even be there, and it would just be a two-base game.
This is where all those hundreds of hours processing millions of events pays off (to the extent that a payoff is even possible).
Since runners on 3B eventually score from 3B happens 60% of the time, the value of taking the extra base to home plate is worth 0.40 runs (as opposed to the original 0.25 runs). Scoring from 2B eventually happens 43% of the time, so the run value of 2B to home plate is 0.57 runs. Since we already figured that scoring from 3B to home plate is 0.40 runs, then going from 2B to 3B adds 0.17 runs. Similarly, the 26% times scoring from 1B leads us to a value of 0.17 runs of going from 1B to 2B. And, home plate to 1B is 0.26 runs, to match the 26% chance of scoring from 1B. Here are the run values of each base:
0.26 home to 1B
0.17 1B to 2B
0.17 2B to 3B
0.40 3B to home
---- -----------
1.00 home to home
With me so far? Well, don’t proceed until you are.
Let’s go back to what I said yesterday:
On average, there are 0.3 runners on 1B, 0.2 runners on 2B and 0.1 runners on 3B. Therefore, the total number of bases advanced by the triples and homeruns is 0.3 times 3, plus 0.2 times 2, plus 0.1 times 1, or a total of 1.4 bases. This is the base-advancement value of a triple and home run.
Rather than counting the number of bases, I’m going to count the number of runs each of those bases represent. The 0.3 runners on 1B score, and each of those runs is worth 0.74 runs (0.17 + 0.17 + 0.40, or 1 minus 0.26), for a total of 0.3 x 0.74 equals 0.222 runs. The 0.2 runners on 2B that score are worth 0.57 runs each, or a total of 0.114 runs. And the 0.1 scoring runners on 3B are worth 0.40 runs each or a total of 0.040 runs.
Add them up, and the base-advancing run value of a HR and triple is .222 + .114 + .040 = .376 runs. Since the batter who hits the HR always scores, his run value of 1.000 runs, making the combined total value of the HR at 1.376 runs.
The guy who hits the triple scores 60% of the time, giving his run value 0.60 runs, plus the .376 runs for the runners he moves over, for a total of 0.976 runs.
Following the same process for doubles, singles, and walks, we get these run values:
1.38 HR
0.98 3B
0.73 2B
0.45 1B
0.34 BB
These numbers are much closer to the standard linear weights values we have all come to know and love.
(They are a little low, only because I used a lower run environment than we are used to. There are not 30% runners on 1B, but 33%. Remember I said I wanted to use nice rounded numbers. Using 33% as the frequency for runners on 1B, and this is what I would have gotten:
1.40 HR
1.00 3B
0.75 2B
0.46 1B
0.34 BB
Happy? Yes, me too.)
Now, what run values would we get if we had 1.80 runners on base, instead of 0.60? We’ll have to take a leap of faith here, as we need to know the chance of being able to score from each base. Just for the sake of illustration, let’s say you score 70% of the time from 1B, 80% from 2B, and 90% from 3B. Following the same process as outlined, we get these run values:
1.42 HR
1.32 3B
1.17 2B
0.94 1B
0.83 BB
As you can see, the walk earns a big jump, while a HR barely moves. That’s because there’s little value in base-advancement, as just getting on base is almost enough to score. It’s that first base that contains almost all the value.
If we made the chance of scoring as 85% for first base, 90% for second base, and 95% for third base, and bump up the number of runners on base to 1.9, we get these run values:
1.22 HR
1.17 3B
1.09 2B
0.98 1B
0.92 BB
As you can see, all the run values start to be pulled toward 1.00. That is, just getting on base is enough to know how many runs you are going to score, as there’s almost no differene between how you got on base.
That’s how you value each base in baseball.