Divide and choose (also Cut and choose or I cut, you choose) is a procedure for fair division of a continuous resource, such as a cake, between two parties. It involves a heterogeneous good or resource ("the cake") and two partners who 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") selects one of the pieces; the cutter receives the remaining piece.[1]
The procedure has been used since ancient times to divide land, cake and other resources between two parties. Currently, there is an entire field of research, called fair cake-cutting, devoted to various extensions and generalizations of cut-and-choose.[2][3]
History
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" (northern) part and a "right" (southern) 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, Bethel, and Shechem.
The United Nations Convention on the Law of the Sea applies a procedure similar to divide-and-choose for allocating areas in the ocean among countries. A developed state applying for a permit to mine minerals from the ocean must prepare two areas of approximately similar value, let the UN authority choose one of them for reservation to developing states, and get the other area for mining:[4][5]
"Each application... shall cover a total area... sufficiently large and of sufficient estimated commercial value to allow two mining operations... of equal estimated commercial value... Within 45 days of receiving such data, the Authority shall designate which part is to be reserved solely for the conduct of activities by the Authority through the Enterprise or in association with developing States... The area designated shall become a reserved area as soon as the plan of work for the non-reserved area is approved and the contract is signed."[6]
Analysis
A cake cut into two pieces
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 their own subjective taste, their 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:[2][3]
The cutter can cut the cake to two pieces that they consider equal. Then, regardless of what the chooser does, they are left with a piece that is as valuable as the other piece.
The chooser can select the piece they consider 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 they chose the piece that is more valuable in their 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 partner's share.
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 their 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.
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 that 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: if they do not, they will likely receive an undesirable portion. This rule is a concrete application of the veil of ignorance concept.
The divide and choose method does not guarantee that 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.
Divide-and-choose might yield inefficient allocations. One commonly used example is a cake that is half vanilla and half chocolate. Suppose Bob likes only chocolate, and Carol only vanilla. If Bob is the cutter and he is unaware of Carol's preference, his safe strategy is to divide the cake so that each half contains an equal amount of chocolate. But then, regardless of Carol's choice, Bob gets only half the chocolate, and the allocation is clearly not Pareto efficient. It is entirely possible that Bob, in his ignorance, would put all the vanilla (and some amount of chocolate) in one larger portion, so Carol gets everything she wants while he would receive less than what he could have gotten by negotiating.
If Bob knew Carol's preference and liked her, he could cut the cake into an all-chocolate piece and an all-vanilla piece, Carol would choose the vanilla piece, and Bob would get all the chocolate. On the other hand, if he doesn't like Carol, he can cut the cake into slightly less than half vanilla in one portion and the rest of the vanilla and all the chocolate in the other. Carol might also be motivated to take the portion with the chocolate to spite Bob. There is a procedure to solve even this, but it is very unstable in the face of a small error in judgement.[10] More practical solutions that can't guarantee optimality but are much better than divide and choose have been devised, in particular the adjusted winner procedure (AW)[11] and the surplus procedure (SP).[12] See also Efficient cake-cutting.
See also
Market maker– Stock market trading entity, players in financial markets who offer to either buy or sell at a given price (plus a spread)
↑ Steven J. Brams and Alan D. Taylor (1999). The Win/win Solution: Guaranteeing Fair Shares to Everybody Norton Paperback. ISBN0-393-04729-6
↑ Better Ways to Cut a Cake by Steven J. Brams, Michael A. Jones, and Christian Klamler in the Notices of the American Mathematical Society December 2006.
Related Research Articles
Fair division is the problem in game theory of dividing a set of resources among several people who have an entitlement to them so that each person receives their due share. That problem arises in various real-world settings such as division of inheritance, partnership dissolutions, divorce settlements, electronic frequency allocation, airport traffic management, and exploitation of Earth observation satellites. It is an active research area in mathematics, economics, dispute resolution, etc. The central tenet of fair division is that such a division should be performed by the players themselves, maybe using a mediator but certainly not an arbiter as only the players really know how they value the goods.
An envy-free cake-cutting is a kind of fair cake-cutting. It is a division of a heterogeneous resource ("cake") that satisfies the envy-free criterion, namely, that every partner feels that their allocated share is at least as good as any other share, according to their own subjective valuation.
A proportional division is a kind of fair division in which a resource is divided among n partners with subjective valuations, giving each partner at least 1/n of the resource by his/her own subjective valuation.
Chore division is a fair division problem in which the divided resource is undesirable, so that each participant wants to get as little as possible. It is the mirror-image of the fair cake-cutting problem, in which the divided resource is desirable so that each participant wants to get as much as possible. Both problems have heterogeneous resources, meaning that the resources are nonuniform. In cake division, cakes can have edge, corner, and middle pieces along with different amounts of frosting. Whereas in chore division, there are different chore types and different amounts of time needed to finish each chore. Similarly, both problems assume that the resources are divisible. Chores can be infinitely divisible, because the finite set of chores can be partitioned by chore or by time. For example, a load of laundry could be partitioned by the number of articles of clothing and/or by the amount of time spent loading the machine. The problems differ, however, in the desirability of the resources. The chore division problem was introduced by Martin Gardner in 1978.
Exact division, also called consensus division, is a partition of a continuous resource ("cake") into some k pieces, such that each of n people with different tastes agree on the value of each of the pieces. For example, consider a cake which is half chocolate and half vanilla. Alice values only the chocolate and George values only the vanilla. The cake is divided into three pieces: one piece contains 20% of the chocolate and 20% of the vanilla, the second contains 50% of the chocolate and 50% of the vanilla, and the third contains the rest of the cake. This is an exact division (with k = 3 and n = 2), as both Alice and George value the three pieces as 20%, 50% and 30% respectively. Several common variants and special cases are known by different terms:
The Selfridge–Conway procedure is a discrete procedure that produces an envy-free cake-cutting for three partners. It is named after John Selfridge and John Horton Conway. Selfridge discovered it in 1960, and told it to Richard Guy, who told many people, but Selfridge did not publish it. John Conway later discovered it independently, and also never published it. This procedure was the first envy-free discrete procedure devised for three partners, and it paved the way for more advanced procedures for n partners.
The Brams–Taylor procedure (BTP) is a procedure for envy-free cake-cutting. It explicated the first finite procedure to produce an envy-free division of a cake among any positive integer number of players.
Fair cake-cutting is a kind of fair division problem. The problem involves a heterogeneous resource, such as a cake with different toppings, that is assumed to be divisible – it is possible to cut arbitrarily small pieces of it without destroying their value. The resource has to be divided among several partners who have different preferences over different parts of the cake, i.e., some people prefer the chocolate toppings, some prefer the cherries, some just want as large a piece as possible. The division should be unanimously fair – each person should receive a piece believed to be a fair share.
The last diminisher procedure is a procedure for fair cake-cutting. It involves a certain heterogenous and divisible resource, such as a birthday cake, and n partners with different preferences over different parts of the cake. It allows the n people to achieve a proportional division, i.e., divide the cake among them such that each person receives a piece with a value of at least 1/n of the total value according to his own subjective valuation. For example, if Alice values the entire cake as $100 and there are 5 partners then Alice can receive a piece that she values as at least $20, regardless of what the other partners think or do.
Efficient cake-cutting is a problem in economics and computer science. It involves a heterogeneous resource, such as a cake with different toppings or a land with different coverings, that is assumed to be divisible - it is possible to cut arbitrarily small pieces of it without destroying their value. The resource has to be divided among several partners who have different preferences over different parts of the cake, i.e., some people prefer the chocolate toppings, some prefer the cherries, some just want as large a piece as possible, etc. The allocation should be economically efficient. Several notions of efficiency have been studied:
The fair pie-cutting problem is a variation of the fair cake-cutting problem, in which the resource to be divided is circular.
The Brams–Taylor–Zwicker procedure is a protocol for envy-free cake-cutting among 4 partners.
Equitable (EQ) cake-cutting is a kind of a fair cake-cutting problem, in which the fairness criterion is equitability. It is a cake-allocation in which the subjective value of all partners is the same, i.e., each partner is equally happy with his/her share. Mathematically, that means that for all partners i and j:
A proportional cake-cutting is a kind of fair cake-cutting. It is a division of a heterogeneous resource ("cake") that satisfies the proportionality criterion, namely, that every partner feels that his allocated share is worth at least 1/n of the total.
Weller's theorem is a theorem in economics. It says that a heterogeneous resource ("cake") can be divided among n partners with different valuations in a way that is both Pareto-efficient (PE) and envy-free (EF). Thus, it is possible to divide a cake fairly without compromising on economic efficiency.
Utilitarian cake-cutting is a rule for dividing a heterogeneous resource, such as a cake or a land-estate, among several partners with different cardinal utility functions, such that the sum of the utilities of the partners is as large as possible. It is a special case of the utilitarian social choice rule. Utilitarian cake-cutting is often not "fair"; hence, utilitarianism is often in conflict with fair cake-cutting.
The Robertson–Webb rotating-knife procedure is a procedure for envy-free cake-cutting of a two-dimensional cake among three partners. It makes only two cuts, so each partner receives a single connected piece.
The lone divider procedure is a procedure for proportional cake-cutting. It involves a heterogenous and divisible resource, such as a birthday cake, and n partners with different preferences over different parts of the cake. It allows the n people to divide the cake among them such that each person receives a piece with a value of at least 1/n of the total value according to his own subjective valuation.
Strategic fair division is the branch of fair division in which the participants are assumed to hide their preferences and act strategically in order to maximize their own utility, rather than playing sincerely according to their true preferences.
Truthful cake-cutting is the study of algorithms for fair cake-cutting that are also truthful mechanisms, i.e., they incentivize the participants to reveal their true valuations to the various parts of the cake.
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