Cutting a Pie is Not a Piece of Cake

Abstract

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