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Cutting a Pie Is Not a Piece of Cake, Researchers Show

June 16, 2009
N-497, 2008-09

A trio of researchers has mathematically determined that it is much easier to equitably cut up a cake than it is to slice up pie. Their work, “Cutting a Pie Is Not a Piece of Cake,” appears in the June-July 2009 issue of the American Mathematical Monthly.

Cutting a cake—whose parts (e.g., the cherry in the middle, the nuts on the side) people may value differently—into fair portions is a challenging problem, but it is one that has largely been solved by mathematicians. By contrast, fair division of a pie into wedge-shaped sectors remains a daunting task.

Steven Brams, a professor in NYU’s Wilf Family Department of Politics, Julius Barbanel, a professor of mathematics at Union College, and Walter Stromquist, a former analyst at the U.S. Department of Treasury, show in their new work that pie-cutting cannot be solved in the same way that cake-cutting has been, raising the possibility that it is not possible to fairly divide a pie.

This is because cake-cutting is more applicable to the division of a rectangular strip of land into lots while pie-cutting is akin to the division of an island into pieces such that everybody gets part of the shoreline.

“While both cakes and pies can be round, we distinguish pie-cutting from cake-cutting by making cuts from the center of a pie versus making parallel cuts across a cake,” explains Brams, author of Mathematics and Democracy: Designing Better Voting and Fair-Division Procedures and co-author of Fair Division: From Cake-Cutting to Dispute Resolution. “If you made parallel cuts when dividing up an island, you might get a slice of land through the middle, but your shoreline would be two disconnected edges rather than a single, and larger, edge that pie-cutting would give you.”

Specifically, unlike cake division, Barbanel, Brams, and Stromquist show that there may be no division of a pie that simultaneously satisfies two important properties of fairness:

  • envy-freeness: each person thinks he or she received a most-valued portion and hence does not envy anybody else;
  • efficiency: there is no other allocation that is better for everybody

In sum, because of the way a pie must be cut, there is not always an envy-free allocation that is equitable—that is, one in which each person values his or her portion the same as every everybody else does.

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Cutting a Pie Is Not a Piece of Cake

Cutting a Pie Is Not a Piece of Cake

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