Last month I made an argument that the NHL salary cap did not have much of an effect on competitive balance in the NHL, but rather the increase in population of NHL caliber players is likely having a much greater impact on increasing competitive balance in the NHL, as measured by the Noll-Scully measure of competitive balance.

I recently found a rebuttal of my blog. Most of my comments that I was planning on making are actually made in the comments of the blog referenced by "doc", which is actually Don Coffin - an economist whose most interesting research (to me anyway) is in baseball economics. So read through those if you want to see why a salary cap should not theoretically affect competitive balance.

There is one slight error stated in the blog above, and that is on the lowest number that can occur with the Noll-Scully measure of competitive balance. The author of the blog states that, "... the lowest you can expect Noll-Scully to be is 1.000, and that's when every team is of exactly equal talent", which is not true. If every team is exactly equal, then every team will have a winning percentage of 0.500, and the standard deviation of all teams having a 0.500 winning percentage is zero. Thus, the Noll-Scully of a league with every team exactly equal in talent (and thus winning percentage) will also be zero, not one. Just thought I would clear up that small error.

## Monday, October 26, 2009

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## 4 comments:

Oh my. It is not correct to say that if all teams in the NHL had equal talent, they would all have .500 records (and SD of zero). That would only be true in a season with an infinite number of games. In the real world, with 82 games, there will be binomial variation. And that would mean a SD of about .055. And in that case the Noll-Scully would be 1.00, exactly as Birnbaum says on his blog. (In any single season, the Noll-Scully could be slightly less than 1.00 due to random chance, but it could never be close to zero. And over time, the average N-S in this league would be exactly 1.0 -- by definition.)

Indeed, that the whole point of Noll-Scully: to compare the actual SD with the SD you would see under perfect balance. How can you use the metric, and I assume teach it to students, when you obviously don't understand what it is or why it's used?

If a league has "exactly equal" talent, and all teams have a winning percentage of 0.500, then the Standard Deviation is zero. That is my point, and thus the Noll-Scully would be zero. While there would be some variation with less than an infinite number of games, the exactly equal (at least to me) implies that all teams would have a winning percentage of 0.500; hence my comment.

The statement you quote, and claim is "not true," is this: "the lowest you can expect Noll-Scully to be is 1.000, and that's when every team is of exactly equal talent." But that statement is exactly correct: 1.00 is the lowest we can expect to observe, and that would suggest a league of perfectly equal teams.

You say that "exactly equal" implies that all teams would have a winning percentage of 0.500. But it doesn't imply that at all. Suppose I have 30 quarters, and flip them each 82 times (replicating an NHL season of equal teams). I assume we can agree that each coin has an "exactly equal talent" for coming up heads (50%). Is it your expectation that the heads percentage will be .50 for each of the 32 quarters? I would hope not. Some coins will be 40%, some 57%, etc., and the SD of head% will usually be around .055 (2/3 of the coins will be between .44 and .56). And the Noll-Scully ratio will be 1.00.

In fact, the "ideal SD" in Noll-Scully is the SD you would see in a league when every team has equal skill. That's the whole point of it! So of course the lower limit is 1.00, not zero.

Wow.....

Stacey-

What do you think of this reasoning?

http://www.fangraphs.com/blogs/index.php/the-super-yankees-theory/

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