Tuesday, December 09, 2008

Demystifying Fibonacci Win Points

In his seminal 1994 book The Politics of Glory, Bill James introduced a simple method for evaluating pitcher win-loss records by combining the two components into one number. He called this method Fibonacci Win Points, and found that it was a fairly good predictor of which starting pitchers would be selected for the Hall of Fame.

You still see Win Points brought up from time to time for that kind of lightweight analysis, but I get the impression that most of the users don’t really know what the results are telling them (outside of the general idea that high numbers are good, it’s a combination of wins and losses, etc.). Since no one is taking the results too seriously, it’s not a big deal. But I always like to look underneath the math hood and see how things work.

So warning: this is a math post, and doesn’t really have much to do with baseball. The formula for Win Points (which I will abbreviate “FIB”) is wins times winning percentage, plus wins, minus losses. Writing it as a formula:

FIB = W/(W + L)*W + W - L

We can also write it as W^2/(W + L) + W - L, and make it abundantly clear that this is a unitless measure. The results bear a resemblance to win figures, but they are no longer a meaningful baseball unit.

To understand how FIB works a little better, let’s express all of the terms as per-decision rates. W% is still winning percentage, but wins are now equal to W% and losses are equal to 1-W%. The formula for Win Point rate (FIBr) becomes:

FIBr = W%*W% + W% + (1-W%) = (W%)^2 + 2(W%) - 1

To convert back to Win Points, we simply multiply FIBr by the number of decisions the pitcher was credited.

FIBr is a useful way to explore the relationship between Win Points and W%. We can see that a .500 pitcher will have a FIBr of (.5)^2 + 2(.5) - 1 = .25. So, a 10-10 pitcher will get .25 Win Points/decision, or 5 points. What does a pitcher have to do to earn .5 win points/decision?

We can find that by setting FIBr equal to .5, and solving for W%. It is a simple quadratic equation, which becomes very easy to see when we use “x” in place of W%:

FIBr = x^2 + 2x – 1

Setting FIBr equal to .5 and solving for x gives sqrt(10)/2 - 1, or .581. A pitcher with a W% of .581 will get a win point for every two decisions, whereas one with a .500 W% gets a win point every four decisions. It’s a very steep function (more on this later).

What is the point at which a pitcher gets zero win points? Solve the equation for FIBr = 0, and you get sqrt(2) - 1 = .414. While the value is superficially similar to a replacement level baseline, win points do not serve as a WAR method. .414 though is obviously the baseline, below which negative values are returned.

Finally, what is the point at which the pitcher gets as many win points per decision as he does wins? Solve FIBr for x, and you get (sqrt(5) - 1)/2 = .618. This consequence of James’ formula is why he called them “Fibonacci” win points, as (sqrt(5) - 1)/2 is the reciprocal of Fibonacci’s number, the golden ratio.

Allow me to go completely down the math digression path for a moment, with a disclaimer that I am not by any means a math professor and that James covered this ground in The Politics of Glory. The Fibonacci sequence starts with 0 and 1; at each step, you add the last two numbers together to yield the next term. So 0 + 1 = 1, 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13, 8 + 13 = 21, 13 + 21 = 34, 21 + 34 = 55, and so on. As this sequence approaches infinity, the ratio between the previous term and the next term approaches (sqrt(5) - 1)/2. You can see this even in the early iterations; 8/13 = .61538, while 34/55 = .61818181… .

This will happen regardless of which two numbers you use to start out with. If we start out with 1 and 155, 1 + 155 = 156, 155 + 156 = 311, 156 + 311 = 467, 311 + 467 = 778, 467 + 778 = 1245, 778 + 1245 = 2023…1245/2023 = .6154, and you can see where this is going.

Writing it out to ten decimal places, (sqrt(5) - 1)/2 = .6180339887. The reciprocal of .6180339887 is 1.6180339888…the only reason why it is not exactly equal to 1 + itself is because I rounded.

The square of the number is .381966011, which has a reciprocal of 2.618033989. So the reciprocal of its square is 2 + itself. I wish I could tell you that the reciprocal of its cube is 3 + itself, but alas… However, the square plus Fibonacci’s number is one; thus it is both the square and the complement. The Wikipedia link above gives some more details on the Fibonacci sequence and the golden ratio and their appearances in art and nature as well as a much more detailed discussion of their mathematical properties.

Getting back to Win Points, we can differentiate FIBr with respect to W% to see just how steep the function is:

dFIBr = 2(W%) + 2

If we look at this specifically at the mean W% (.5 of course), we get a slope of 3. We can write the tangent line at that point in point-slope form as:

y - y1 = m(x - x1) where y = FIBr, y1 = FIBr(.5) = .25, m = dFIBr (.5 at this point), x = W%, and x1 = .5: FIBr - .25 = 3(W% - .5), which can be simplified to:

FIBr ~= 3(W%) - 1.25

Which can also be written as:

FIBr ~= 3(W% - 5/12) ~= 3(W% - .417)

Again, this is just a linearization of the FIBr function for a .500 W%; it is precise at that point, but doesn’t hold over the entire range of W%. However, it does give us some insight into how Win Points work. The baseline is a W% of .417 (although .414 is the actual zero point, a .500 pitcher doesn’t get any win points until his W% reaches .417--and if that doesn't make sense to you, just ignore it, as I'm not quite sure how to express it coherently), and each point of W% above .417 is multiplied by three.

Comparing this to a standard WAR formula, the WAR baseline will be in the same ballpark as .414 (I use .390), but each point of W% above the baseline is equally valuable. A .500 pitcher and a .495 pitcher will have the same WAR gap, given equal decisions, as a .600 and a .595 pitcher. That will not be true for Win Points, which rewards higher W%s more. This may be why James found them useful for predicting the Hall of Fame’s treatment of pitchers, as Win Points put a premium on excellence, beyond its real world win value.

Just to give you an idea of how well the approximation works on the career level, I figured actual Fibonacci Win Points and the linear knockoff for all post-1900 pitchers with 150 or more wins as of 2007. There are 198 pitchers, for whom the average absolute difference between the two is 2.3 win points. The differences are proportional to win points; the biggest differences are for those with the most win points (the largest single difference is 15 between Christy Mathewson’s 433 actual and 418 approximate). I think that this supports my claim that the linear approximation is a decent tool by which to understand how Win Points work.

It should go without saying that the approximation works best for those pitchers with W%s near .500, as that is the point at which I found the tangent line. If you were to find the tangent line at .600, you would get more accurate results for pitchers with W%s near .600, naturally.

Now I will give a freak show stat of my own, which I would never really encourage anyone to use. I include it here as a way to use the career value metric I prefer (given the constraint of working with the actual W-L record) with a career wins pseudo-scale. It is just a quick z-score conversion of pitcher WAR to a pseudo-Wins unit. For the 150 win pitchers, the mean number of wins was 210 with a standard deviation of 55. I figured WAR crudely as (W% - .39)*(W + L); crude because it is using actual wins and losses as the inputs, not because the concept is flawed. This metric has a mean of 65 and a standard deviation of 25 for this group. Setting the z-scores equal and rearranging to isolate wins gives this conversion:

pseudo-Wins = 2.2*WAR + 67

Obviously this is only intended for use with career WAR…saying that CC was worth 7 WAR last year and that is equivalent to 84 wins would make no sense. Of course it would make no sense for a pitcher with 7 career WAR either; this was based on pitchers with 150 wins and shouldn’t be used outside the range of long, reasonably productive careers. Well, it really shouldn’t be used at all, but I needed some original content, even of questionable quality, to make me feel better about a post rehashing James’ method.

The lowest-ranking 300 win pitcher under this formula (and thus, by definition, WAR as defined above as well) is Nolan Ryan (251). There is only one pitcher with 300 pseudo-Wins without 300 actual wins, and that may change this season--Randy Johnson at 319. The other non-300 game winners with 290 or more pseudo wins are Jim Palmer (296) and Whitey Ford (293). In case it is not clear, I have been using actual wins and losses, not “neutral wins and losses” as in the previous post and other posts through the years.

In summation, Win Points may well have value as a gauge of Hall of Fame chances or as a reasonable way to combine wins and losses into one number. However, they are unitless, and they place a very high premium on excellent performance, much more than the actual, tangible win impact of said performance. Despite their baseline of .414, which is a reasonable “replacement” level, the aforementioned premium should dispel any notion that they are a WAR knockoff. And of course, they just so happen to tie in with a fascinating mathematical sequence and ratio, which gives an otherwise forgettable “freak show stat” (I apply that label in the kindest sense of the term, and it is one which Bill James often applied to his own inventions) a little more pizzazz.

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