The Query Complexity of Cake Cutting

We study the query complexity of cake cutting and give lower bounds for computing approximately envy-free, perfect, and equitable allocations with the minimum number of cuts. These lower bounds are tight for three players in the case of envy-free allocations and for two players in the case of perfect and equitable allocations. We also formalize moving knife procedures and show that a large subclass of this family, which captures all the known moving knife procedures, can be simulated efficiently with arbitrarily small error in the Robertson-Webb query model. Our query complexity upper bounds for envy-free allocations among three players and perfect allocations between two players rely on simulating approximately two moving knife procedures due to Barbanel-Brams and Austin, respectively.