I've been to a lot of chess tournaments, and sex differences in chess ability are not being discussed before the games begin. This is another reason to doubt the "stereotype threat" explanation for sex differences in chess.
I disagree. A stereotype need not be explicitly discussed to affect behaviour. Simply observing a biased sample from the population that is touted as representative alters people's heuristics and behaviours.
There's an endogeneity problem with your idea. Is chess performance the input or output to observing prevailing social norms? Observing just chess outcomes can't tell us the answer, so to conclude a priori that outcomes are highly correlated with ability for both genders is just supposition.
I would caution that we're not truly observing chess
ability but, rather, chess outcomes. Ability might factor into observed outcomes but, as with gifted underachievement, the choice of extent of expression of ability is under the individual's control. There is enough evidence that some high IQ girls mask academic ability (that is, play "dumb") to adhere to gender stereotypes and fit in a coed classroom. With the bias toward female teachers, it wouldn't be hard to argue that most clasrooms are biased in favour of female socialization. If chess clubs are disproportionately male dominated, one would expect high-ability girls to self-select away from performing at full ability in even greater proportions than we observe in traditional classrooms because the social gradient is even steeper.
