Divide and choose

Divide and choose (also Cut and choose or I cut, you choose) is a procedure for envy-free cake-cutting between two partners. It involves a heterogeneous good or resource ("the cake") and two partners which have different preferences over parts of the cake. The protocol proceeds as follows: one person ("the cutter") cuts the cake into two pieces; the other person ("the chooser") chooses one of the pieces; the cutter receives the remaining piece.

Divide-and-choose is mentioned in the Bible, in the Book of Genesis (chapter 13). When Abraham and Lot come to the land of Canaan, Abraham suggests that they divide it among them. Then Abraham, coming from the south, divides the land to a "left" (western) part and a "right" (eastern) part, and lets Lot choose. Lot chooses the eastern part which contains Sodom and Gomorrah, and Abraham is left with the western part which contains Beer Sheva, Hebron, Beit El and Shechem.

Divide-and-choose is envy-free in the following sense: each of the two partners can act in a way that guarantees that, according to her own subjective taste, her allocated share is at least as valuable as the other share, regardless of what the other partner does. Here is how each partner can act:

To an external viewer, the division might seem unfair, but to the two involved partners, the division is fair - no partner envies the other.

If the value functions of the partners are additive functions, then divide-and-choose is also proportional in the following sense: each partner can act in a way that guarantees that her allocated share has a value of at least 1/2 of the total cake value. This is because, with additive valuations, every envy-free division is also proportional.

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