There’s another compromise position. Namely, two can form a coalition against the third and treat the problem as dividing a pie between two individuals with different claims. For example, Xannon and Yancy have a combined claim of 2⁄3 to Zaire’s 1⁄2. Proportional division according to those terms would give Zaire 3⁄7 to the duo’s 4⁄7, which they can then split in half to get the distribution {2/7, 2⁄7, 3⁄7}. As it turns out, you get this same division no matter how the coalitions form. This sort of principle dates back to the Talmud.
Of course, this only works if Xannon agrees that Zaire can justly claim half the pie. If not, Xannon and Yancy could compel Zaire to accept one-third of the pie.
Rational actors would each claim the entire pie for themselves, and then any fair division scheme would result in each getting a third.
There’s another compromise position. Namely, two can form a coalition against the third and treat the problem as dividing a pie between two individuals with different claims. For example, Xannon and Yancy have a combined claim of 2⁄3 to Zaire’s 1⁄2. Proportional division according to those terms would give Zaire 3⁄7 to the duo’s 4⁄7, which they can then split in half to get the distribution {2/7, 2⁄7, 3⁄7}. As it turns out, you get this same division no matter how the coalitions form. This sort of principle dates back to the Talmud.
Of course, this only works if Xannon agrees that Zaire can justly claim half the pie. If not, Xannon and Yancy could compel Zaire to accept one-third of the pie.
Rational actors would each claim the entire pie for themselves, and then any fair division scheme would result in each getting a third.