Formulas for fairness: applying the math of cake cutting to conflict resolution - includes related articles on cake division and fair allocation of assets in a divorce

Science News, May 4, 1996 by Ivars Peterson

Applying the math of cake cutting to conflict resolution

The marriage had lasted more than 30 years. The wife had given up a promising career as an actress and singer to raise the couple's four sons and manage the household. The husband's business success, in which the wife also played a significant part, had enabled the family to maintain a grand and gracious lifestyle centered in New York City and Paris.

Then, the husband left his wife to live with another woman. In 1981, after a lengthy proceeding, the Supreme Court of New York State granted a divorce. But it took another 2 years of bitter and costly legal wrangling to determine how to divide the couple's joint assets.

According to state guidelines, this case met the criteria for an equal division of marital property, which included a very expensive Paris apartment.

To achieve an equitable split, the court granted the husband all the real estate except for the apartment and ordered him to compensate his former wife for her share of that property. She was obliged to sell the Paris apartment within 3 years of the settlement and divide the proceeds with her former husband.

Stunned by the outcome, she appealed the decision. She prized the Paris apartment and would, at the age of 70, have to seek another home after having lived there for more than 25 years.

She lost her case. However, one of the appeal panel judges did protest that the court-imposed settlement, so meticulously formulated and delicately balanced, was nonetheless unfair to the wife.

Political scientist Steven J. Brams of New York University agrees strongly with the dissenting judge's view. He can also point to potentially fairer methods of handling contentious issues such as the division of marital property.

Brams and mathematician Alan D. Taylor of Union College in Schenectady, N.Y., have worked out mathematical procedures that they claim can be used to settle disputes in ways that both parties see as fair and equitable.

"There are about 1.2 million divorce cases in the United States each year," Brams says. Seldom are both parties satisfied with the provisions of a settlement.

"We have specific procedures that offer very practical solutions in such situations," he asserts.

Nor is divorce the only arena in which these conflict resolution schemes could play a role. Inheritance squabbles, international border disputes, and treaty and contract negotiations could benefit from strategies that promise fair outcomes.

The key to these new methods is the recognition that people generally have different opinions about the values of the items to be shared or the importance of the issues to be settled. These differences make it possible to work out agreements in which all of the parties feel as if they've gotten the best deal. Researchers working on fairness term such allocations "envyfree." The new methods put together in a practical framework some notions of fairness, equity, and justice that have developed over the last 50 years in philosophy, theoretical economics, and mathematics.

"From the viewpoint of economics, what Brams and Taylor are doing is one case of a more general fair division problem," says economist Herve Moulin of Duke University in Durham, N.C.

Economists interested in fair division emphasize general principles that underlie the fair allocation of resources, going beyond such precepts as "no envy." They also consider such specific issues as allocating the cost of constructing a shared road or a computer network among potential users, introducing incentives to modify undesirable human behavior, and distributing risk among communities faced with environmental hazards.

"The central issue is how we can get the participants in a scheme of fair division to behave in the right way and not manipulate the system to their own advantage," Moulin says.

Brams and Taylor offer mathematical recipes for solving a particular subset of these problems. "For divorce settlements and things like that, their methods have a lot of potential," Moulin remarks.

When mathematicians ponder fair division, they usually start with a cake.

Suppose a thickly frosted, elaborately decorated birthday cake must be divided among several people. Different people may prefer different parts of the cake-the thickest pink icing for one, strawberry slices for another.

Is there a step-by-step procedure for cutting the cake into pieces so that each participant can guarantee his or her own satisfaction? There's a familiar strategy for two persons: "I cut, you choose." The first person divides the cake into two pieces that appear equally desirable to him.

The pieces may not seem equally desirable to the second person, so she picks the one she prefers. They both automatically end up with a piece that they think is at least as good as the piece they didn't get.

The Convention of the Law of the Sea, which went into effect in 1994, incorporates such a scheme to protect the interests of developing countries when a highly industrialized nation wants to mine a portion of the seabed underlying international waters. The country seeking to mine would divide that area into two portions. An independent agency representing the developing countries would then choose one of the two tracts, reserving it for future use.