A Lower Bound for Equitable Cake Cutting

Abstract

We are interested in the problem of dividing a cake -- a heterogeneous divisible good -- among n players, in a way that is ε-equitable: every pair of players must have the same value for their own allocated pieces, up to a difference of at most ε. It is known that such allocations can be computed using O(n ln(1/ε)) operations in the standard Robertson-Webb Model. We establish a lower bound of Ω(ln(1/ε)/lnln(1/ε)) on the complexity of this problem, which is almost tight for a constant number of players. Importantly, our result implies that allocations that are exactly equitable cannot be computed.