When modeling the nuances of fair division, scientists prefer cutting cakes to infants. The classic model is the "you cut, I choose" method: If two people are splitting a cake, one cuts it into two pieces and the other chooses which piece he wants. The person cutting the cake has an incentive to cut the fairest possible way--in half--so that he can still be satisfied with what is left for him. "You cut, I choose" is then free of envy, because each person is served an equivalent piece of cake. But it does not satisfy the property of equitability, wherein each person values their piece of cake as much their counterpart. Imagine two people are sharing a pizza that is covered half in pepperoni and half in anchovies. If one person hates anchovies, there are several possible scenarios in which one diner will be happier than the other.
In the current issue of the Notices of the American Mathematical Society, New York University political scientist Steven J. Brams, Montclair State University mathematician Michael A. Jones and Austrian economist Christian Klamler of the University of Graz outline a new manner of division, called surplus procedure (SP), which partially solves the shortcoming. Rather than dividing the cake so both sharers get an equal quantity, they aim for an equal quality: Each person, A and B, submits to a referee a valuation of his or her ideal piece of cake. Then the cake is cut in a mathematically determined way so that both valuations are partially satisfied, and each eater gets more than half of what they asked for. "Because both A and B receive at least 50 percent of their valuations under SP, the resulting division is not only proportionally equitable but also envy free," the researchers write.
They extended their method to three-person (or more) scenarios, calling this version equitability procedure, or EP. Performed similarly to SP, EP aims to divide the cake so that each person gets more than a third of what they want. (In the case of four people, each would get at least one fourth of their valuation; for five people, they would get at least one fifth.) While SP reconciles envy and equitability, the team admits that in EP, both of these properties are not always satisfied among three or more people. These new cutting methods, however, are "strategy proof," meaning that one player will not be able to manipulate the system by misrepresenting their valuation. This is because each person is trying to maximize their value and they are risk-averse--that is, they won't try to game the system if it means they could get a less valuable piece.
Besides operating in the slightly more complicated realm of pizza, the results are applicable to situations that involve the division of land, too. SP may one day be useful to India and Pakistan in determining how to divvy up Kashmir; EP could help the Shia, Sunnis and Kurds partition Iraq. Although these procedures may not be directly applicable to all sharing problems, "the reasoning that goes into fair-division algorithms is valuable," Brams says. "It shows how mathematics can contribute to making dispute resolution more rigorous and precise."