Your solution is highly unstable under tiny changes to the bargaining set. Consider a two-player setting, with the Pareto frontier of convex but almost-flat form. The bargaining power of the players won’t depend on the tiny changes in the form of the frontier, but the point that meets the plane may vary from the maximum utility to minimum utility for each of the players depending on such changes.
As an example of a solution stable in this sense, you can, for example, rescale the utilities so that the Pareto frontier is inscribed in the unit cube, and select a play that lies on the cube’s main diagonal.
I abandoned this approach for now, instead I’m working an a more clear model of agents as processes, with the goal of finding a way of assigning preferences to the processes, so that the solution for cooperation should appear as just preference corresponding to the process composed of the participating agents, depending on their preferences.
Yes, stability is the big issue. My solution is almost certainly wrong, although instructive :-) The unit cube solution also seems unstable, but in a different sense: a tiny change in payoffs can grow the Pareto frontier a lot. At this point I’m wondering whether there is any continuous solution.
Your solution is highly unstable under tiny changes to the bargaining set. Consider a two-player setting, with the Pareto frontier of convex but almost-flat form. The bargaining power of the players won’t depend on the tiny changes in the form of the frontier, but the point that meets the plane may vary from the maximum utility to minimum utility for each of the players depending on such changes.
As an example of a solution stable in this sense, you can, for example, rescale the utilities so that the Pareto frontier is inscribed in the unit cube, and select a play that lies on the cube’s main diagonal.
I abandoned this approach for now, instead I’m working an a more clear model of agents as processes, with the goal of finding a way of assigning preferences to the processes, so that the solution for cooperation should appear as just preference corresponding to the process composed of the participating agents, depending on their preferences.
Yes, stability is the big issue. My solution is almost certainly wrong, although instructive :-) The unit cube solution also seems unstable, but in a different sense: a tiny change in payoffs can grow the Pareto frontier a lot. At this point I’m wondering whether there is any continuous solution.