The reasoning is found in what is called the “nucleolus” of a convex game defined in the original paper by Aumann and Maschler.

This is the best solution in the sense that it solves conflict situations. In a more precise way, the general solution is consisten with the contested garment rule for two persons, i.e, imagine you have a vector solution that has the property that when restricted to only two of the claimants i, j they sum their assigmets x and y respectively to form the state E = x + y. Then, they proceed to deal this state according to their original claims and they get, again, x and y respectively. This is a feature that makes the Talmud rule unique and the best in this sense.

Note. The nucleolus is a solution that exists for all game with characeristic function and in general is not easy to find, but in the case of the game proposed by Aumann the nucleolus is easily computable by the method of vessels.

Basically, what is the algorithm solving for? In the perfect algorithm, the money should be divided in such a way that there is ‘minimization of regret’. i.e. in any other division; the sum of ‘regrets’ of all parties will be more than this solution. Only then can it be socially acceptable. One way of looking at ‘regret’ can be %loss incurred. So, if I was entitled to get 100, and I got 30; my ‘regret’ is 70%; and it is same as the person who got 300 out of a 1000 he was originally supposed to get.

How does this solution ensure that? How is it the best solution for all the parties involved COMBINED? Would be great if someone can throw some light at that? Can it be proven (to the parties) that in no other way will the overall regret decrease?

Aakash