As an active participant in Cross Validated/SE, I know the importance of a question having an answer -- even an answer that kicks the can down the road -- so that visitors to the site can see that a question is not as hopeless as a message in a bottle, but rather aspires to be a shout from a tenement window, where you know that someone, somewhere will shout back.
Call it an echo, but I'm shouting back at myself. In topical area, my question regards organizational behavior, but the spirit of my question is game theoretic, and I think the body of literature I am searching for is in game theory in organziations -- every over-achiever's favorite sport! To that end, Zhu and Weynant (2003), looks promising . . . ?
And while it is only tangentially relevant, I think I might prime the pump of other responders by citing the most famous what-do-you-think-he-thinks? problem of all time: The Room with Two Doors.
[OK, that was a spoiler, but really you should seek your chuckles elsewhere. We're doing research here! ;) ]
Zhu, K., & Weyant, J. P. (2003). Strategic decisions of new technology adoption under asymmetric information: a game‐theoretic model. Decision sciences, 34(4), 643-675.