The problem of fairly dividing a cake (as a metaphor for a heterogeneous divisible good) has been the subject of much interest since the 1940's, and is of importance in multiagent resource allocation. Two fairness criteria are usually considered: proportionality, in the sense that each of the n agents receives at least 1/n of the cake; and the stronger property of envy-freeness, namely that each agent prefers its own piece of cake to the others' pieces. For proportional division, there are algorithms that require O(nlogn) steps, and recent lower bounds imply that one cannot do better. In stark contrast, known (discrete) algorithms for envy-free division require an unbounded number of steps, even when there are only four agents.In this paper, we give an Omega(n<sup>2</sup>) lower bound for the number of steps required by envy-free cake-cutting algorithms. This result provides, for the first time, a true separation between envy-free and proportional division, thus giving a partial explanation for the notorious difficulty of the former problem.

Ariel D. Procaccia