Fair division
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Fair division, also known as the cake cutting problem, is the problem of dividing a resource in such a way that each recipient believes they have received their fair share. The problem is hard because each recipient may have a different measure of value of the resource: in the "cake cutting" version, one recipient may like marzipan, another like cherries, and so on.
For two people there is a simple solution: the so-called divide and choose method. One person divides the resource into what they believe are equal halves, and the other person chooses the "half" they prefer. Thus, the person making the division has an incentive to divide as fairly as possible: for if they do not, they will likely receive an undesirable portion.
Fair division is usually taught in college math courses and can elude math majors. It was an important open problem during much of the twentieth century, when it was finally solved together by Steven Brams and Alan Taylor in 1995.
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[edit] Many players
The problem can be extended to three or more people, but the method for finding an optimum solution becomes complicated.
One method continues the division to successively smaller "equal" portions. The first person divides the resource into what they believe are equal halves. The second then chooses a half, and each of these two people then divide their portion into thirds. The third person picks two of the resulting portions: one from the first person and one from the second person. If there are four people, each of the first three people divides their portions into fourths, and the process continues.
A problem with this approach is that the portions may become reduced to absurdly small sizes.
Another method begins with the first person portioning off 1 / n of the resource (for n people). Each following person then examines the portion in turn, removing a part for themselves if they believe the portion to be larger than 1 / n. The last person to remove part receives the portion. The process continues until the entire resource has been fairly divided.
The problem was one of the big open problems of the twentieth century. The first cake cutting procedure that produced an envy-free division of cake for any natural number of persons was first published by Steven Brams and Alan Taylor in 1995. But because the procedure is somewhat impractical when the number of persons is large, fair division continues to be an important problem in mathematics and the social sciences.
[edit] Variants
Some cake-cutting procedures are discrete, whereby players make cuts with a knife (usually in a sequence of steps). Moving-knife procedures, on the other hand, allow continuous movement and can let players call "stop" at any point.
A variant of the fair division problem is chore division: this is the "dual" to the cake cutting problem in which an undesirable object is to be distributed amongst the players. The canonical example is a set of chores that the players between them must do. Note that "I cut, you choose" works for chore division.
Other variants include cakes which contain indivisible items (i.e. nuts or berries on the cake) which must be fairly distributed between players (such pieces are referred to as atoms), or the requirement of having connected pieces (i.e. only whole pieces and not fragments are allowed).
[edit] Limitations
The nature of the resource to be divided may affect fair division. The classic example is the tale of 1 Kings 3:15-28 in which King Solomon proposes to settle a dispute between two women who each claim a child by dividing the child in half.
[edit] See also
- Steven Brams
- Game theory
- Nash bargaining game
- Tragedy of the anticommons
- Tragedy of the commons
- Spite
- Weakly additive
- Adjusted Winner procedure
[edit] References
- S J Brams and A D Taylor, "An Envy-Free Cake Division Protocol", American Mathematical Monthly 102, 9-19, 1995.
- Jack Robertson, William Webb, Cake-Cutting Algorithms: Be Fair If You Can, AK Peters Ltd, March 1998. ISBN 1-56881-076-8.
- Skyrms, B. (1996) The Evolution of the Social Contract Cambridge University Press. ISBN-13: 9780521555838
[edit] External links
- Short essay about the cake-cutting problem by S. Abbas Raza of 3 Quarks Daily.
- Fair Division from the Discrete Mathematics Project at the University of Colorado at Boulder.
- Fair Division: Method of Lone Divider
- Fair Division: Method of Markers
- Fair Division: Method of Sealed Bids
