Divide and select (also Cut and select or I cut, you choose ) is a cake-cutting procedure without the jealousy of two partners. It involves either a heterogeneous (or "cake") resource and two partners who have different preferences over the portion of the cake. The protocol goes as follows: one person ("cutter") cuts the cake in half; other people ("voters") choose one part; the cutter receives the remaining pieces.
Video Divide and choose
Histori
Divide-and-choose is mentioned in the Bible, in Genesis (chapter 13). When Abraham and Lot came to the land of Canaan, Abraham showed that they divided him among them. Then Abraham, who came from the south, divided the land into "left" (west) and "right" (east) parts, and allowed Lot to choose. Lot chose the eastern section containing Sodom and Gomorrah, and Abraham was left with a western part containing Beer Sheva, Hebron, Beit El and Shechem.
Maps Divide and choose
Analysis
Dividing-and-choose is free of envy in the following sense: each of the two partners can act in a way that ensures that, according to their own subjective tastes, the portion they allocate is at least as valuable as the other, regardless of what the couple does other. Here's how each partner can act:
- Cutters can cut the cake into two halves that they think is the same. Then, regardless of what the voters do, they will be left with as precious pieces as other pieces.
- Voters can select parts they consider more valuable. Then, even if the cutter divides the cake into very unbalanced pieces (in the eyes of the electorate), the electorate has no reason to complain because they choose a more valuable piece in their own eyes.
For external viewers, the division may seem unfair, but for the two partners involved, the division is fair - no partner is jealous of the other.
If the value function of the partner is an additional function, the divide-and-select is also proportional in the following sense: each partner may act in a way that ensures that the allocated allocation has a value of at least 1/2 of the total value of the cake. This is because, with the additive rating, each division is enviously proportional.
The protocol works well to share the desired resources (as in fair cake deductions) and to split unwanted resources (as in the division of tasks).
Divide and choose to assume that the parties have the same rights and want to decide on the distribution itself or use mediation rather than arbitration. Items are assumed to be divided in any way, but each side can rate bits differently.
Cutters have an incentive to divide as equally as possible: because otherwise they will likely receive unwanted sections. This rule is a concrete application of the concept of the abyss of ignorance.
Sharing and choosing methods does not guarantee everyone gets half the cake with their own judgment, and so it's not the right division. There is no limited procedure for proper division but can be done using two moving blades; see the procedure of moving the Austin knife.
Efficiency issues
Sharing-and-select may result in inefficient allocations.
One commonly used example is a cake that is half vanilla and half brown. Suppose Bob likes only chocolate, and Carol is just vanilla. If Bob is a cutter and he is unaware of Carol's likes, his safe strategy is to divide the cake so that every half contains the same amount of chocolate. But then, regardless of Carol's choice, Bob only gets half the chocolate and Pareto's inefficiently clear allocation. It is likely that Bob, in his ignorance, will put all the vanilla (and some chocolates) in a larger portion, so Carol gets everything he wants while he will receive less than what he can get by negotiating.
Alternative
If Bob knew what Carol liked and liked, he could cut the cake into a whole piece of chocolate, and the pieces were all made from vanilla, Carol would choose a vanilla slice, and Bob would get all the chocolates. On the other hand, if she does not like Carol, she can cut the cake to a little more than half a vanilla in one serving and the rest of vanilla and all the chocolate on the other. Carol might also be motivated to take portions with chocolate to avoid Bob. There are procedures to resolve this but are very unstable in the face of small errors in the assessment. A more practical solution that can not guarantee optimality but is much better than split and choose which has been designed by Steven Brams and Alan Taylor, in particular the customized winning procedure (AW).
In 2006, Steven J. Brams, Michael A. Jones, and Christian Klamler detailed a new way to cut a cake called the surplus procedure (SP) that meets the eligibility and solves the above problem. Neither a person's subjective judgment of their work as the whole proportion is the same.
Split between more than two parties
Martin Gardner popularized the problem of designing equally fair procedures for the larger group, in his "Mathematics" column in Scientific American (anthologized in). One procedure begins with one person cutting off what they consider to be a fair part. Anyone can cut it smaller. But whoever last cuts it, must take it.
More recent methods are reported. It was developed by Aziz and Mackenzie. Although it is faster in principle than the previous method, it is still potentially very slow: O (n 3 (n 2 ) n ), where n is the number of divisions desired.
See also
- Fair share
- Resource allocation
- Market makers, players in financial markets who offer to buy or sell at a certain price (plus spread)
Notes and references
Source of the article : Wikipedia
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