Can we solve fairness?

EDIT: thanks to Wei Dai for the next step! Now I know that any "purely geometric" construction that looks only at the Pareto set will fail to incentivize players to adopt it. The reason: we can, without changing the Pareto set, give any player an additional non-Pareto-optimal strategy that always assigns them higher utility than my proposed solution, thus making them want to defect. Pretty conclusive! So much for this line of inquiry, I guess.

Well, of course you can't restrict your attention to the Pareto Set. Every presentation of the bargaining problem characterizes the problem using both the Pareto boundary and a "zero-point", "threat point", or "non-agreement point". The additional strategies that Wei suggests also change the zero-point. That is, they change the problem.

As to whether we can solve fairness, it is already solved - at least in the 2 party perfect information case. And it has been solved since 1953.