July 21, 2014
NYU Politics Professor Steven Brams and his colleague Julius Barbanel, a mathematician at Union College, have developed a new method for sharing items desired by two parties—ranging from cake to land—in such a way that neither feels shortchanged.
Their work, which appears in the journal the Mathematical Intelligencer, is based on the cut-and-choose method that dates back to Biblical times: Abraham divided land equally, and Lot could choose the part he wanted.
However, disputes are often more complicated than those recounted in the Old Testament. What happens, for instance, when more than two cuts can be made, or when people prefer different, specific sections of whatever is to be divided?
To address these matters, Barbanel and Brams include a “giveback procedure” that is designed to be efficient, equitable, and void of jealousy.
Their approach relies on a “referee” (a computer or even a parent), with the parties telling the referee which parts of item (e.g., parcel of land, portion of cake) they value the most. In mathematical terms, these are called each party’s probability density functions, or pdfs. The referee then marks out, for instance, the cake at all points where the pdfs of the disgruntled would-be cake eaters cross and assigns portions. If at this point the two parties receive the same size of cake, the task is over.
If not, the giveback process starts.
The party who received the more valued part of the cake during the first round must give a part of it back to the other person, starting with those parts in which the ratio of their pdfs is the smallest. This goes on until the parties value their portions equally and so are on a par. This method only works with a finite number of cuts if the players’ pdfs are straight-lined, or are so-called “piecewise” linear sections.
The researchers see notable potential in the division of land. In the case of beachfront property co-owned by two developers, for example, it can help to determine who gets what strips of land to build on based on the pieces of land they value most.
Brams, a professor in NYU’s Wilf Family Department of Politics, is the author of Game Theory and the Humanities: Bridging Two Worlds (2011) and Mathematics and Democracy: Designing Better Voting and Fair-Division Procedures (2008), among other works.