While we are talking about the deterministic rules working for setting up controlled decision environments, everything might seem very clear and understandable so that we all might be persuaded about the conditions of the environment and motivations of the rules which are to solve the problems across the environment.
Real life is much more vaguer than the well defined solution environments and because of this, most of the solutions devised can't satisfy the stakeholders of the problem domain. For instance, if I were the party A in the Selfridge-Conway procedure, I could have still been envying to a piece other than mine by arguing that "Why am I chosen for cutting the cake first? If cannot cut the cake in 3 equal pieces, it will be a disadvantage for me! I'd like to be the first picker!" And nobody can criticise me becuse of my opinion about the procedure.
One can say that, "Okay, you have to agree on the pillars of the procedure firstly!".
Yes, it is a good starting point. And a powerful statement that ties me up about obeying the rules. However, even if I agreed the rules of the procedure, after the procedure ends, I might still be unhappy with my piece and crying about it. It is possible!
Then, would it be my psychological problem or would it be the truth that such problems are unsolvable if you are dealing with human beings and their feeling of justice and satisfaction? Is there a way for highly intellectual people to share things without using brute force? In personal interactions, human beings had come a significant way when it comes to solve the problems in a civilized way. On the other hand, in international relations brute force is being used vastly between parties.
I believe that we are standing just besides a threshold!
After a revolutionary step, which has been led by "pure" science, over this threshold, the big question marks in our minds will disappear very fast. And we all are going to take our most relieved breath! ...before finding us new and bigger question marks :))
Think of this!