Nothing in your post or the proceeding discussion is of a "rather mathematical nature", let alone a precise specification of a mathematical problem.

Given a problem A, find an analogous problem B with the same payoff matrix for which it can be proven that any possible agent will make analogous decisions, or prove that such a problem B cannot exist.

You do realize that game theory is a branch of mathematics, as is decision theory? That we are trying to prove something here, not by empirical evidence, but by logic and reason alone? What do you think this is, social economics?