New York University Skip to Content Skip to Search Skip to Navigation Skip to Sub Navigation

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:
Research

Type: Press Release

Cutting a Pie Is Not a Piece of Cake

Cutting a Pie Is Not a Piece of Cake


Search News



NYU In the News

Paying It Backward: NYU Alum Funds Scholarships

The Wall Street Journal profiled Trustee Evan Chesler on why he decided to chair the Momentum fund-raising campaign.

A Nobel Prize Party: Cheese, Bubbles, and a Boson

The New Yorker talked to Professor Kyle Cranmer and graduate student Sven Kreiss about NYU’s role in the discovery of the Higgs boson, which resulted in a Nobel prize for the scientists who predicted its existence.

The World as They Knew It

The New York Times reviewed the exhibit at the Institute for the Study of the Ancient World on how ancient Greeks and Romans mapped the known and unknown areas of their world.

Elite Institutions: Far More Diverse Than They Were 20 Years Ago

NYU made stronger gains over the last 20 years in increasing diversity than any other major research university, according to the Chronicle of Higher Education.

Program Seeks to Nurture ‘Data Science Culture’
at Universities

The New York Times reported on the multi-million collaboration among NYU and two other universities to harness the potential of Big Data, including an interview with Professor Yann LeCun, director of NYU’s Center for Data Science.

NYU Footer