Cutting a Pie is Not a Piece of Cake

American Mathematical Monthly, Forthcoming

22 Pages Posted: 2 Jan 2009

See all articles by Julius B. Barbanel

Julius B. Barbanel

Union College

Steven J. Brams

New York University (NYU) - Wilf Family Department of Politics

Walter Stromquist

Bryn Mawr College

Date Written: December 23, 2008


Cutting a cake -- or any heterogeneous, divisible good -- fairly has received much attention in recent years, but fair division of a pie into wedge-shaped sectors has received far less. Whereas cake-cutting is applicable to land division, pie-cutting is more applicable to the division of an island's shoreline into connected lots, or a daily cycle into on-call periods. Unlike cake division, we show that there may be no envy-free and efficient division of a pie among n players using n radial cuts (the minimum number), nor may there be an envy-free allocation that is equitable (each player receives the same value in its measure), though an envy-free and equitable allocation is always possible. Thus, pie-cutting is harder than cake-cutting - not a piece of cake.

Keywords: Fair division, cake-cutting, pie-cutting, envy-freeness, efficiency, equitability, divisible good

JEL Classification: C72, C78, D61, D63, D74

Suggested Citation

Barbanel, Julius B. and Brams, Steven and Stromquist, Walter Rees, Cutting a Pie is Not a Piece of Cake (December 23, 2008). American Mathematical Monthly, Forthcoming. Available at SSRN:

Julius B. Barbanel

Union College ( email )

Schenectady, NY 12308-3151
United States

Steven Brams (Contact Author)

New York University (NYU) - Wilf Family Department of Politics ( email )

Dept. of Politics
19 West 4th St., 2nd Fl.
New York, NY 10012
United States
212-998-8510 (Phone)
212-995-4184 (Fax)


Walter Rees Stromquist

Bryn Mawr College ( email )

Bryn Mawr, PA
United States

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