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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.

This Press Release is in the following Topics:

Type: Press Release

Cutting a Pie Is Not a Piece of Cake

Cutting a Pie Is Not a Piece of Cake

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