Cake cutting really is not a piece of cake

Abstract

We consider the well-known cake cutting problem in which a protocol wants to divide a cake among n ≥ 2 players in such a way that each player believes that they got a fair share. The standard RobertsonWebb model allows the protocol to make two types of queries, Evaluation and Cut, to the players. A deterministic divide-and-conquer protocol with complexity O(n log n) is known. Improving on previous lower bounds, we provide an Ω(n log n) lower bound on the complexity of any deterministic protocol, even if the protocol is allowed to assign to a player a piece that is a union of intervals and only guarantee approximate fairness. We accomplish this by lower bounding the complexity to find for a single player, a piece of cake that is both rich in value, and thin in width. We then introduce a version of cake cutting in which the players are able to cut with only finite precision. In this case, we can extend the Ω(n log n) lower bound to include randomized protocols.

Mind the gap: Cake cutting with separation

“…A particularly interesting question concerns Theorem 3.1, where we presented an algorithm that computes an MMS-fair allocation using O (n 2 ) queries given the agents' maximin shares. Without separation, it is well-known that a proportional allocation can be found using O (n log n) queries via a divide-and-conquer approach [35], and that this is tight [31,58]. What is the optimal query complexity of computing an MMS-fair allocation in our setting?…”

Waste Makes Haste

“…The algorithm of Steinhaus [1948] generates a proportional division with connected pieces in O(n 2 ) queries, and an improved algorithm by Even and Paz [1984] requires only O(n log n) queries. Later results proved that this runtime is asymptotically optimal even if disconnected pieces are allowed [Woeginger and Sgall 2007;Edmonds and Pruhs 2011].…”

Ordinal Maximin Share Approximation for Goods

“…Aziz and Ye (2014) define an allocation robust-fair if it remains fair even when the valuation of an agent changes, as long as its ordinal information remains unchanged. Edmonds and Pruhs (2011) study cake-cutting settings in which agents can only cut the cake with a finite precision.…”