The ultimate conclusion of which is likely that both parties will decay to picking the secret value/first guess randomly (although I'm not sure if the optimal distribution is perfectly flat?), which is also something that we can model.
The game theory here is similar to another quiz "Guess 1/3 of the average".
You are really trying to guess how deep the other has thought about the problem, so you can tell which strategy they settled on, and then you adapt your strategy based on that. Of course, it's a loop.