Mike, I feel like we are just going in circles now and perhaps part of the reason is that we just haven’t been clear enough about just what our positions are. These are mine:
1. Participation rate alone is not the sole cause for women’s under performance in pool (or in chess or just about anything else) even though many people continue to try to argue precisely that.
2. I do not think the belief that participation rates explain the majority of the performance gap is a reasonable one to hold considering the current evidence, but at the same time I don’t think it would be reasonable quite yet to dismiss the possibility for that completely out of hand either.
3. Women and men are inherently biologically different and unequal in respective strengths and skills and this is almost always going to at least partially explain performance disparities, and as a result of these biological differences performance gaps in most things will never disappear completely.
I'll take (1) and (2) together. But first I'll make a general comment.
While my experience here is you generally are a smart and thoughtful person, I believe on this issue you are not rational. The backdrop is many people don't even want to TALK about biological differences between men and women that might impact policies and opportunities and the like (true), that some find the discussion repulsive (true), that some have a veneer of rational discussion but really are apologists for the first two groups of people (true), and that many people have a strong desire for reality to be a certain way on this issue (true).
This is where you enter. You fancy yourself a person who is not going to be swayed by the way you WANT the world to be. You are the one who is going to accept the slings and arrows and stand in the public square and say "sorry..it is what it is; the emperor has no clothes..."
But there is another group of people, and it includes many in the scientific community. It is people who really DO view this issue rationally, who are thoughtful and open to new ideas, who are capable of separating the way they might want the world to be from the way the world is. I think when and where this group aligns in conclusion with the delusional crowd above, it--to you--becomes the delusional crowd. And you are blind and deaf to its arguments.
(1) and (2): Participation Rates: This, once again, is not a variable. It doesn't explain things and it is not the cause of things. It in our discussions is essentially a catch-all for "all else being equal."
Again, when we point out that Alberta CAN produces far more good Hockey players than does Alabama, USA, that is just an observation--one we can check and perhaps all agree is true.
And when we say the Ukraine produces far more good chess players than does Japan, that once again is an observation--a measurement--that we can agree is true.
And when we say the male sex produces a lot more good pool players than does the female sex, that is just an observation that we can agree is true.
In none of these cases does the observation suggest or imply that the former is superior in some base fundamental way than the latter. If you don't want to make a claim about genetic propensities toward chess in the Ukraine vs Japan or genetic propensities toward pool in men vs women, then there is nothing to do.
However, if you DO want to make such a claim, then the burden is on YOU to make the case these and other observations/measurements are actually measuring what you claim they are. The default position in all of the above is we are observing nothing relevant about fundamental genetic differences. Once again, if you want to make a case otherwise, it is your burden to satisfy the "all else being equal" requirement. It is your burden to take into account participation rates, general popularity of the activity amongst the group, differences in recognizing and nurturing talent, barriers to entry, reinforcement, and all sorts of things.
I believe you approach this backwards. I think you say clearly men are better at pool, canuks are better at hockey, and Ukranians are better at chess, as though it is a given these simple observations are measuring something fundamental about innate propensities and you challenge others to refute that. That's unreasonable. The burden goes the other way. I know you don't like the hockey and chess analogy and you didn't make those claims about hockey and chess. But I bring them up because the same issue is at play, and I think there is less noise and it is easier for people to see.
If you made a claim that Ukranians are genetically more suited for chess than Japanese and cited the number of grand masters as support, I an--others--would simply throw the ball back into your court and say you haven't effective made the case that the number of grand masters is measuring what you claim it is.
(3) Yes men and woman are biologically different in many different ways. We are of course all biologically different in many different ways some of which are readily apparent to us and most of which are not. Tall people are better at picking apples. Men are taller on average. So, all else being equal, men on average are expected to be better at picking apples. But if all across the world 5'9" men are hired over 5'10" women to pick apples because we all know men are better at picking apples, then something not good is happening. That's a practical problem that exists even when there IS a fundamental genetic difference.
Given the myriad other things that affect observed performance differences, it will be a long time before we get even close to assessing whether women or men on average are better in some subtle way at mathematics or art or physics or philosophy or business or corporate efficiency or military strategy and any of a host of other human endeavors. In fact we will probably never get there given most things of value depend on many different individual traits and excellence can be achieved employing core competencies with different weights.