Interesting idea, and intuitively it makes sense. I can think of examples of both types of player in my own life.
I enjoy that your solutions boil down to “Sociologists, mathematicians’ are toddlers, distract them with a toy” and “Mathematicians, try using your social skills”.
In my experience though, rather than trying to manipulate people, it usually works better to be explicit about intentions. As a really blatant example, our local pickleball club hosts separate competitive and casual nights. These both are attended by many of the same people, and they adjust their play style accordingly.
The job of making explicit the play style intention is usually the facilitator of the game. Because they will hold the power to sanction players that aren’t playing in the right spirit.
Dave: What’s your favourite colour?
Alice: Green
Dave: That’s not very precise.
Alice: Well that wasn’t a very precise question. If you wanted preciseness you should have asked for hue, value and chroma, or the numerical value of my favourite colour.