Pythag Expansion (October 11, 2003)
This post will appeal only to math-lovers. Please skip otherwise.
RS = X + i
RA = X - i
win%
= RS^2 / (RS^2 + RA^2)
= (X^2 + 2Xi + i^2) / (2 * (X^2 + i^2) )
win% over .500
= .5 x (X^2 + 2Xi + i^2) -.5
-----------------
(X^2 + i^2)
= .5 x ( (X^2 + 2Xi + i^2) - 1 )
-----------------
(X^2 + i^2)
= .5 x 2Xi
-------
(X^2 + i^2)
= Xi / (X^2 + i^2)
Runs-per-win converter
= (run differential) / (wins over .500)
= (RS - RA) / (wins over .500)
= (2i) / (Xi / (X^2 + i^2)
= 2 * (X^2 + i^2) and let i approach zero
---------------
X
= 2 * X
http://groups.google.com/groups?q=%22pythagorean%22+%22glass%22++group:rec.sport.baseball.*+group:rec.sport.baseball.*&hl=en&lr=&ie=UTF-8&oe=UTF-8&group=rec.sport.baseball.*&selm=7r9ilk%241k2q%40enews4.newsguy.com&rnum=1
Posted 10:15 a.m.,
October 11, 2003
(#2) -
Patriot
He was using 1.83. Would your math still hold with a different exponent, or is it unique to 2?
Posted 10:39 a.m.,
October 11, 2003
(#3) -
Patriot
That link doesn't work. Just search rsbb for "pythagorean" and "glass".
Posted 11:43 a.m.,
October 11, 2003
(#4) -
studes
(homepage)
Just had my son, the math genius, look this over. He understands the math, but says it wouldn't work for other exponents. The binomial expansion in the first step would be different for other exponents, and the logic wouldn't hold.
Posted 12:07 p.m.,
October 11, 2003
(#5) -
Patriot
Hmm. I wonder why Glass came up with it using 1.83 then.
In the Pyth/Log5 article in one of the non-national Abstracts, Bill said that 2 was the most natural exponent. I don't know what the heck he meant, but it does seem like there are some nice tricks you can do with 2 that you can't do with others.
Posted 12:24 p.m.,
October 11, 2003
(#6) -
tangotiger
(homepage)
The above is Patriot's link.
memo to Pat: easier to shove that in the "Homepage" box when posting.
memo to All: The PythagenPat equation is the best most reliable equation we currently have.
Posted 7:04 p.m.,
October 11, 2003
(#7) -
David Smyth
That is, I think, equivalent to what I did when I came up with the Base Wins formula in the early 1990s. Base Wins uses 2*X.
Anyway, it is true, is it not, that the variations in the R/W converter are due to the run distribution patterns at various levels of R/G. So, is there a "natural" or inherent R/W converter which would be correct if there were no luck involved in how the runs fit together? Well, if you have 4.5 R/G, then the only distribution where we would know for sure what the R/W is every time is team A=9 R/G and team B=0 R/G. You have 162W, no losses, every time. So the R/W converter is 729RAA/81WAA, which is 9.0.
And this also works at the 1 rpg level, which is the level where all the formulas break down except for PythagoPat. It's just that at 1 rpg, there cannot be any distribution other than the one in which no "chance" is involved.
Posted 11:44 a.m.,
October 12, 2003
(#8) -
Patriot
I think David makes a great point in his post, and this creates a question for player evaluation methods. How should we convert RAR or RAA or whatever into wins? Should we use the approach that David does in BsW, using this method, or should we use a Palmer type converter?
Personally, I'd still prefer the Palmer type(also exemplified by Ben .91, etc), but I'm very open to being convinced otherwise. Maybe it all goes back to the underlying philosiphy behind your system. But to me, if teams with an RPG of 11 convert runs to win at 12.1:1 as Ben suggests rather than 11:1, then we should use 12.1. Each RPG level would presumably have a certain "average" level of luck.
Posted 1:45 p.m.,
October 12, 2003
(#9) -
David Smyth
Patriot, I wasn't trying to advocate using 2*X as a standard R/W converter. But I think it does have a place, depending on the question you are asking.
Posted 2:05 p.m.,
October 12, 2003
(#10) -
Patriot
I understand that. And I know BsW does use it. That's why I was asking the question of which RPW converter we should use.
Posted 10:41 a.m.,
October 13, 2003
(#11) -
David Smyth
"Each rpg level would presumably have a certain avg level of luck."
Sure, that's what you get from Pythagopat (but not from Ben .91).
But if you take the idea of putting a player into an "avg" context as far as possible, that would imply translating their stats to, say, a 4.5 rpg context (with fixed avg component frequencies) with a .5 avg run difference (or whatever the avg diff. is in a 4.5 rpg), and using whatever the fixed R/W is for that (9.2 or whatever). So all the work would be in the "translation". After that, you could use fixed event values and a fixed converter. This approach contains invalid assumptions, of course (that all player performances would translate the same in different contexts), but that assumption is used in lots of other methods (such as Slwts, etc.)
Posted 4:03 p.m.,
October 13, 2003
(#12) -
Patriot
I'm not sure I understand you're first comment. Tell me if I'm getting this wrong. The "luck" as you refer to it is the difference between using RPW=RPG(or 2*X as it has been described in this thread) and the empirically based RPW converters, right? If that's the case, then the Ben .91 does do this too, I think. It's saying that RPW=RPG/.91. So RPW is related linearily to RPG, but there is a difference between RPW and RPG. Then Palmer or our method goes a step further and makes it a non-linear relations.
The second part of your post makes sense to me. It's kind of like Davenports' "translations". He takes everyone and puts them into an ideal context with a .270 Lg BA, a Pyth exponent of 2, etc.
Posted 4:41 p.m.,
October 13, 2003
(#13) -
Patriot
If that's the case, then the Ben .91 does do this too[incorporate luck], I think
Posted 4:42 p.m.,
October 13, 2003
(#14) -
David Smyth
The Ben .91 only properly accounts for the luck in an avg context (or, more properly, in the weighted avg of all known contexts). It is not handling the luck of the run distribution patterns any differently than 2*X is, in principle, because that is not linear.
Posted 4:52 p.m.,
October 13, 2003
(#15) -
tangotiger
(homepage)
Right click on the above link, and "save picture as" to your PC. Then double-click the file that you just downloaded.
(If you exceed my bandwidth limit, try again tomorrow. I'll try to upload it to baseballstuff.com, but I'm having problems from work right now.)
I used the PythagoPat (with the .28 value) to figure out the expected win% for RS+RA from 2 to 16, and from RS-RA from 0 to 5. Then, I simply figured the RPW (runs-per-win) converter as (RS-RA)/(win% - .500)
What are you going to see?
You are going to see the "black line" where the run differential is 0, and the RPW converter starting at "2" when RS+RA = 1. And you will see it go up, in an almost straight line, to almost "15" when RS+RA = 16.
You will see a few more lines at other RS-RA lines that follow the same pattern. (This corresponds to the legend on the right.)
As I mentioned in other threads, the PythagoPat corresponds the closest to the Tango Distribution, and that distribution is the best one to model actual win matchups.
Posted 7:30 p.m.,
October 13, 2003
(#16) -
David Smyth
I did as Tango says and when I click on the icon I get a "no preview available" window. If someone can tell me how to fix this I would appreciate. I am not very computer savvy.
But the idea is that it does always answer every question to use the Tango distribution or Pythgopat. This is not a discussion on which R/W converter is most "accurate"; there are some questions where a less accurate R/W converter might be better suited, due to other features.
Posted 9:19 p.m.,
October 13, 2003
(#17) -
Tangotiger
David,
I wasn't trying to get into the fray here, because I'm not sure what the discussion actually is.
I will start a new thread to only discuss the RPW (and you will be able to see it without any trouble), so as to not hijack your discussion here.
