First, I think a team that scores 800 and gives up zero: 8002 / (0 + 8002) = 1.00, which is intuitive.

But, on the whole I think Aaron is right. Why isn’t the Formula [RS / (RS + RA)] or [RS3 / (RS3 + RA^3)]. I suspect the answer is because it happens to fit better. Because this doesn’t contain any connection between the winning/losing process and the formula, there is no reason to read into it, unless you are trying to figure out why it works.

This is akin to noting that Mars tends to orbit the earth, except that it changes direction for a short period of time every now and then. You notice that it happens to be in July (hypothetically), and decide that the war god of Mars hates American Independence, and therefore turns around out of anger every July. It makes sense, but there’s not reason to believe it to be true.

Instead, you should use the deviations (like the twins or july) to examine the things that cause teams to deviate. If you were designing a team based on runs scored and runs allowed per dollar of payroll, you would want to see which teams win more relative to that predicted by the formula. You would notice that teams with winning records do better, but that probably doesn’t help you. Then you would notice that strong bullpens increase the chance of winning relative to the simple pythagorean formula. This may indicate that each bullpen run prevented is worth more dollars compared to starting rotation, and would cause you to refine you’re formula based on starting and relieving runs.

Or, that’s what you should do.

Lastly, I don’t think I believe that records in one run games matter. I think they are the exciting (devastating?) ones to watch, but if you score 5 in the ninth, coming from 6 runs down, I don’t think it means that the loss was any more meaningful than losing by 6. I just don’t believe that teams/players are that different in how they perform in different innings, or based on how close the score is. If anything, one run games come down to how good your best reliever is. Our record in one run games is good, because Joe Nathan is good, and he tends to pitch in one run games. If you believe that one run games are important, than you should also subscribe to the theory that closers many times more important than the other relievers. Many people do, but I’m not convinced.

It is not correct that the Pythagorean Formula does not have a theoretical explanation. It is found right here:
sjmiller/math/papers/PythagWonLoss_Paper.pdf” target=”_blank”>http://www.math.brown.edu/sjmiller/math/papers/PythagWonLoss_Paper.pdf

By assuming that runs scored and runs allowed are independent for each team, and that runs follow a so-called Weibull distribution, it is fairly easy to derive the formula. As the above-mentioned paper shows, these assumptions are pretty reasonable according to the data.

Actually, the best-fit exponent in the formula is somewhere around 1.79 to 1.82 instead of two.

It is true, however, that originally Bill James devised the formula empirically.

But this is a great thread. First, thanks Phoenix for providing the link. I did not know that it was based on a specific distribution, which I do find interesting. However, the problems that still exist are that a) there is no reason to believe that weibull is appropriate, unless someone has shown that it provides a good fit (maybe it does), and b) the squared term is arbitrary. It turns out that a better fit is ^1.82.

Newton is a perfect example, however, about the value and danger in creating these models. It is important to understand the difference between empirical evidence and understanding the process (cause/effect). If you like those stories that Newton got hit on the head with an apple, that is a great example of emprical evidence that gravity exists. A better example is Newtons basic laws of motion (f=m*a). This was very interesting, and created a new way to understand the process of motion. However, his ideas about how motion works are based on recognizing a pattern. When Einstein (et al) started researching higher velocity physics, they discovered that Newton’s formula no longer applied. It takes more force to effect the same accelaration, given a higher present velocity. This led to the discovery of relativity (still baffles me).

Bill James has a very good reason to propose the pythagorean formula. He noticed that it seems to be a good fit. He probably backed into the weibull distribution, or tested a few distributions before deciding that Weibull had a better fit. Why couldn’t it be any other common distribution, or another distribution that is too mathematically complicated for people to actually use?

The point is that it is very cool that this fits well. Further research will likely show a number of modifications that would improve the model (who wants to be Einstein to Bill James’ Newton?)

Aaron’s suggested dissertation is a great idea. There was an Actuary (don’t remember his name) who quit his job with an insurance company to start doing fantasy baseball projections. All the stats geeks out there can take a minute to imagine that as a full time profession. The dissertation gets tricky though, as there are really so many underlying issues you have to test. First is modeling the correlation of consecutive pitches, at-bats, innings etc. Once you do that, i suggest researching the relative importance of different factors like closers, starters/relief, number of pitches thrown/taken, slugging percentage, home runs, using generalized linear modeling. Anybody in a grad-school program that would fund them for this research? If so, I’ll apply there to help out.

It makes intuitive sense that good relief pitching would help you beat your Pythagorean projections, because some runs clearly have more influence on the outcome of a game than others. Giving up 5 runs when you’re already down five doesn’t hurt as much as giving up one when you’re tied in the ninth.

Maybe one way to tweak the Pythagorean formula would be to work in Win Probability Added numbers. So instead of just comparing runs scored to runs allowed, you could weight runs allowed according to situational WPA. That is, you could multiply runs allowed times some coefficient representing how high-leverage the situation was. So runs scored in a blow out would be multiplied by some number less than one, and runs allowed close and late would be multiplied by some number higher than one. You’d add together the resulting numbers, instead of just runs scored, before popping them into the formula.

Or you could just tweak the total runs scored by some factor representing bullpen quality before applying the formula. If you could figure out the influence of the bullpen somehow -that is, if you could document that teams with 5% above average bullpens, or closers, or whatever, typically outperform their pythagorean by 2%, say - then maybe you could tweak the pythagorean projection with some standardized bullpen correction.

I don’t know any of the math for this but it seems like bullpen quality and the deviation from the pythagorean projection are quantifiable and so could be correlated, at least hypothetically.

sorry for alternating between “runs scored” and “runs allowed”—I should have just said “runs allowed” since I was talking about pitching stats