American and Austrian researchers have devised mathematical algorithms that could be used in determining the fair division of land or other resources. One application would be to settle border disputes, which might include the partition of countries like Iraq into Shiite, Sunni, and Kurdish regions. The methods, Surplus Procedure (SP) and Equitability Procedure (EP), are described in the December 2006 issue of The Notices of the American Mathematical Society.

Traditionally, fair division of property has used the “you cut, I choose” method. As with a birthday cake divided between two people, one person cuts the cake while the other chooses the piece he or she prefers. However, “cut-and-choose” lacks equitability—that is, the subjective value that the two people place on the pieces they get may not be the same. For example, suppose one half of the cake is frosted with vanilla icing and the other with chocolate icing, and suppose the cutter values chocolate icing twice as much as vanilla. It is possible that the cutter’s valuation of the piece she gets will be less than the chooser’s valuation of the piece he gets, making the division inequitable.

Professors Steven Brams (New York University), Michael Jones (Montclair State University), and Christian Klamler (Austria’s University of Graz) have developed new methods that make such divisions equitable. Using SP, a mathematical algorithm, two parties can divide property in a way that leaves each with a 50 percent share plus some part of the surplus that remains. The algorithm takes into account which features of a commodity are valued most by each party. For a division among three or more parties, the EP method gives all parties more-than proportional shares. Both SP and EP are “strategy-proof”: the players cannot assuredly manipulate these procedures to their advantage.

Brams and his colleagues offer division of land as an example of how their algorithm works. For example, land bordering water might be more valuable to one party, while land bordering a forest might be more valuable to another. The SP method shows how to divide the land in such a way that the both parties receive approximately the same value from the parcels of land they get.

Brams is well known for his work on fair-division algorithms. In his 1999 book, The Win-Win Solution: Guaranteeing Fair Shares to Everybody, written jointly with Alan D. Taylor, he describes a host of contexts in which such algorithms apply, from the Camp David peace accords to the divorce of Donald and Ivana Trump. One of the algorithms described in the book, called “adjusted winner,” has been patented by NYU, where Brams is a professor of politics.

An advance copy of “Better Ways to Cut a Cake” is available at: http://www.ams.org/staff/jackson/fea-brams.pdf.