Saturday, 30 January 2010
Fair Division......A Piece of Cake...
Many really complex problems of mathematics are founded in simple social practices, such as the fair division of assets. A company dissolves and the partners need to divide up the assets. The last surviving parent dies without a will and the children have to decide how to distribute the home, boat, car, golf-club membership...etc. Such issues keep lawyers in business and lead to terrible fighting between the parties because of the different ways they value the things to be divided.
One of the most commonly talked about is the idea of dividing a cake fairly between two people. Most people think they know how to do that. Here is a quote from Alan D. Taylor's introduction to The Geometry of Efficient Fair Division, by Julius B. Barbanel
Discussions of cake cutting almost always begin with the procedure known as divide-and-choose. Historically, this two-person scheme traces its origins back 5000 years to the Bible’s account of land division between Abram (later to be called Abraham) and Lot, and it resurfaces more explicitly two-and-a-half millennia ago as Hesiod, in his Theogony, describes the division of meat into two piles by Prometheus, with Zeus then choosing the pile that he preferred.
Mathematical investigations of fair division date from the early 1940s. The constructive vein was first opened by the Polish mathematician Hugo Steinhaus (see ) and his colleagues Stefan Banach and Bronislaw Knaster. Steinhaus appears to have been the first to ask if there is an obvious extension of divide-and-choose to the case wherein there are three participants instead of two, and he derived the scheme referred to in a number of mathematical texts for non-majors (see  and ) as “the lone-divider method.” But extending this procedure to four or more participants is somewhat complicated, and was not actually achieved until Harold Kuhn  did so in 1967. Banach and Knaster, however, took an entirely different tack and devised a fair-division scheme for any number of participants that is known today as the “last-diminisher method.
The idea that the "cut and choose" solution is "fair" assumes that the object to be divided is uniform and that both people approach the division with the same value system. Consider, for instance, a cake baked in a square pan with frosting on the top and sides. Perhaps the cake has one half that is chocolate and one half is vanilla. Now what if one of the people really likes frosting, or really LOVES chocolate, and other is indifferent to the frosting amount or flavoring(you can play with the implications of levels of these factors). Is there an advantage to being the one who does the cutting or does the choosing? To further complicate the process, consider the different ways you might cut the cake if you were one of these people and you knew the other person's preferences. What if the information access was not equal? How would it effect the choice of who cuts if you knew they had more information about your preferences than you had about theirs? What if you had the information advantage? And what if there were three people.. how would you extend the cut-and-choose method (it can be done)?
And what if the cake is not a cake, but the remains of the family estate, A car, a house, a boat, membership in the golf club...? Now what would be "Fair"?..
Ok, maybe "Piece of Cake" is not such a piece-of-cake after all. Maybe I'll take time to talk soon about the more complicate situation when an estate contains non-divisible items, a family home, the boat in which your dad took you out fishing when you were a kid... which half of your mom's antique Ming Dynasty vase would be more valuable...perhaps not simple when it is your family.