Barbanel & Brams on Slicing the Pie
J. B. Barbanel and S.J. Brams have posted Cutting a pie is not a piece of cake. Here is the abstract:
- Gale (1993) posed the question of whether there is necessarily an undominated, envy-free allocation of a pie when it is cut into wedge-shaped pieces or sectors. For two players, we give constructive procedures for obtaining such an allocation, whether the pie is cut into equal-size sectors by a single diameter cut or into two sectors of unequal size. Such an allocation, however, may not be equitable—that is, give the two players\ exactly the same value from their pieces.
For three players, we give a procedure for obtaining an envy-free allocation, but it may be dominated either by another envy-free allocation or an envy-causing allocation. A counterexample shows that there is not always an undominated envy-free allocation for four or more players if the players’ preferences are not absolutely continuous with respect to each other. If we do make this assumption, then the existence question remains open for four or more players. For three players, the question of whether there exists an undominated envy-free allocation is open whether or not the players’ preferences are absolutely continuous with respect to each other.