In my time in the mathematical community I've formed the subjective impression that it's noticeably less common for mathematicians of the highest caliber to engage in status games than members of the general population do.

Higher status will not in itself help you solve hard mathematical problems. You need economic security, good conditions in which to think, and so forth, and society can help or hinder you there. But when you face a problem that no-one else has ever solved, it's you versus the universe. There's no big brother around who will tell you the answer if only you can win his favor. So the psychology of how to make progress in such a situation is existential rather than social.