I Cut, You Choose

Envy-free cake-cutting

Divide 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 cuts the cake into two pieces, and the other person chooses his piece first.

Divide-and-choose is mentioned in the Bible. In Genesis, 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.

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: The cutter can cut the cake into two pieces that she considers equal. Then, regardless of what the chooser does, she is left with a piece that is as valuable as the other piece.

The chooser can select the piece which she considers more valuable. Then, even if the cutter divided the cake to pieces that are very unequal (in the chooser’s eyes), the chooser still has no reason to complain because she chose the piece that is more valuable in her own eyes. To an external viewer, the division might seem unfair, but to the two involved partners, the division is fair – no partner envies the other.

The protocol works both for dividing a desirable resource (as in fair cake-cutting) and for dividing an undesirable resource (as in chore division). Divide and choose assumes the parties have equal entitlements and wish to decide the division themselves or use mediation rather than arbitration. The goods are assumed to be divisible in any way, but each party may value the bits differently.

The cutter has an incentive to divide as fairly as possible: for if they do not, they will likely receive an undesirable portion. This rule is a concrete application of the ‘veil of ignorance’ concept (a thought experiment by American political and moral philosopher John Rawls), that asks individuals to consider what social choices they would want made if they didn’t know in advance where in society they would fit.

The divide and choose method does not guarantee each person gets exactly half the cake by their own valuations, and so is not an exact division. There is no finite procedure for exact division but it can be done using two moving knives; see Austin moving-knife procedure.



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