March 1, 2007 The Surplus Procedure, a formula developed by mathematical political scientist Steven Brams, is a new way of allocating resources to best benefit all parties. In situations in which dividing resources is complicated, such as the division of a country or a divorce, this new method works by numerically taking into account the values people place on the different aspects of what is in dispute. Each party gets at least 50 percent of what they want most, and the rest is divided proportionally.
How do you divide a piece of birthday cake so both kids are happy with what they get? Wise men and women have been trying to answer that question since the time of King Solomon, and it's a problem every parent is familiar with.
"I will ask the girls, 'Which one of you would like to cut, and which would like to choose?'" says mother Rachel Fishman Green.
The method is called divide and choose. Works fine for a plain vanilla cake. The cutter slices it right down the center. Chooser gets one half; cutter the other -- all is fair. But what if the dispute is over something more complex, like a country or the house and kids in a divorce? You can't just slice them down the middle.
How do we divide these things? It's all about fairness.
"I think fairness is one of the most important, if not the most important, problem in the world today," Steven Brams, a mathematical political scientist at New York University, tells DBIS.
Brams carved out a new mathematical formula, called the Surplus Procedure, that promises to make settling disputes a little easier. It works by numerically taking into account the values people place on the different aspects of what's in dispute. Each party first gets at least 50 percent of what they want most. What's left over is then divided proportionally, so both parties get half of what they wanted and then some!
"More than a fair share," Brams says. "They couldn't have done better. There's no allocation that gives them both more."
He says you may one day be able to solve disputes by downloading the Surplus Procedure right from the Internet. And it's also strategy-proof. A person cannot game the system to get more than their fair share. And that's great news for moms.
BACKGROUND: When two people cut a cake, they usually employ a variation of the "I cut, you choose" strategy for dividing the slices. One person cuts and the other person chooses a piece, but the "cutter" holds all the power in this situation. Mathematicians have come up with a new method for cake cutting that puts the cutter and chooser on equal footing -- mathematically speaking -- and happy with their piece of cake. Called the surplus procedure, it is a new, more scientific approach to dispute resolution, and also shows how mathematics can contribute to making dispute resolution more rigorous and precise.
CUT AND CHOOSE: With the "cut and choose" method, the cutter should have great incentive to cut the cake as evenly as possible, knowing that the chooser will most likely choose the biggest piece. The primary advantage to the cut-and-choose approach is that it is "envy free": neither person envies the other's slice because they each know they have received at least half of the cake. But this assumes both parties have identical values -- that is, they are both angling for the largest slice of cake they can get in the negotiation. But values are highly subjective, so equitability is not so easily defined.
ABOUT SP: Using the surplus procedure, the cutter can cut the cake in such a way that the value he places on his piece is approximately the same as the value the chooser places on his piece -- possibly with the result that both might feel they are making out like a bandit and getting 65 percent of their heart's desire. It all comes down to perceived value. It gets more complicated if the cake must be divided between three people. For that problem, Brams devised an extension of the method, called the equitability procedure, which ensures that everyone gets, say, 40 percent of what they want, based on their respective values. Beyond three people, though, the likelihood of achieving both equitability and envy-freeness becomes much less likely.
HOW CAN WE USE IT? There are a broad range of contexts in which fair-division algorithms can be applied, such as the Camp David peace accords and the divorce of Donald and Ivana Trump, as well as the fair division of land. If one person values waterfront property and another values land at the edge of a forest, the surplus procedure will yield a solution that lets them divide the land in such a way that both will ultimately place the same value on their respective parcels of land.
The American Mathematical Society and the Mathematical Association of America contributed to the information contained in the video portion of this report.
Editor's Note: This article is not intended to provide medical advice, diagnosis or treatment.