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See Inside February 2007

The Kindest Cut

Always do the math before you divvy
Steve Mirsky



FRANK VERONSKY
In Mel Brooks's The Producers--also known as the Enron business plan, as noted in this space in October 2002--Max Bialystock and Leo Bloom conspire to sell 25,000 percent of the Broadway show Springtime for Hitler. In a moment of introspection, Max asks, "How much percentage of a play can there be altogether?" To which Leo gently responds, "You can only sell 100 percent of anything." Until recently, I accepted this statement as fact.

A mathematician, a political scientist and an economist walk into a bar. Wait, that's not right. A mathematician, a political scientist and an economist recently wrote a paper--amazingly, that is right--in which they point out that, under special circumstances, two people can split something up and both feel like they got more than half. No, the entire mirror factory isn't filled with smoke.

The paper, which appeared in the December issue of Notices of the American Mathematical Society, is entitled "Better Ways to Cut a Cake." The report does not deal with knife-sharpening technology. What it does deal with is the theory and method behind slicing up an object to maximize the satisfaction of those parties, possibly at a party, who will then receive the slices. "We use cake as a metaphor for dividing a heterogeneous divisible good, an item that people may have different preferences for," explains the mathematician, Michael Jones of Montclair State University. So the cake could be a stand-in for a tract of land that's part forest and part seashore. Or the cake could be an apartment that has small rooms with views and large windowless rooms. Or the cake could be a chicken with white meat and dark meat.

One thing the cake can't be is a pie. In fact, mathematics has a rich (but still light and moist) history of cake-cutting theory, which looks at ways to divide a space with a cut that goes completely across the space. And there is a literature of pie-cutting theory, in which cuts begin at the center and travel radially outward. So it takes two cuts to completely sever a pie. Although, mathematically, as Jones points out, "In a sense, you can think of a cake as a pie in which one cut has already been made." Which sounds like something a wise man on a mountaintop would tell Betty Crocker.

Anyway, back to the Faber College homecoming parade. In other words, cut the cake. Traditionally, if two people are splitting a cake, the method is simple and was already ancient when Marie Antoinette was providing cake counseling to the masses: one slices, the other chooses. The slicer therefore wants to make the division equal, knowing he'll get stuck with the littler piece if he botches the job.

But this system can break down with certain cakes. "For example," Jones says, "if a cake is half chocolate and half vanilla, and one person likes chocolate a lot and the other person is indifferent, then there's a way to have both people, in their opinions, receive more than half the cake." The procedure, which involves equations that you are welcome to review at home over coffee and Danish, gets fairly complex, especially if more than two people are involved in the cutting. But you can see, in the chocolate-vanilla two-person example, that the chocolate lover will feel more than half-satisfied if he gets, say, 80 percent of the chocolate half, despite it being only 40 percent of the entire cake. Meanwhile his flavor-impartial buddy will be more than half-satisfied by getting the remaining 60 percent of the entire cake. And the cake maker will have an economic motivation to complicate his cakes and hike his prices.

The cake-cutting theory could find application in, for example, land division negotiations. It also indirectly pinpoints exactly where the Enron boys' scheme was half-baked. Because it turns out that you really can have more than 100 percent of the cake. But you have to at least have a cake.

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