9 research outputs found
Review: Cake-Cutting Algorithms: Be Fair if You Can
No abstract provided in this article
A Discrete and Bounded Envy-free Cake Cutting Protocol for Four Agents
We consider the well-studied cake cutting problem in which the goal is to
identify a fair allocation based on a minimal number of queries from the
agents. The problem has attracted considerable attention within various
branches of computer science, mathematics, and economics. Although, the elegant
Selfridge-Conway envy-free protocol for three agents has been known since 1960,
it has been a major open problem for the last fifty years to obtain a bounded
envy-free protocol for more than three agents. We propose a discrete and
bounded envy-free protocol for four agents
Cake Division with Minimal Cuts: Envy-Free Procedures for 3 Person, 4 Persons, and Beyond
The minimal number of parallel cuts required to divide a cake into n pieces is n-1. A new 3-person procedure, requiring 2 parallel cuts, is given that produces an envy- free division, whereby each person thinks he or she receives at least a tied- for- largest piece. An extension of this procedure leads to a 4-person division, us ing 3 parallel cuts, that makes at most one player envious. Finally, a 4-person envy-free procedure is given, but it requires up to 5 parallel cuts, and some pieces may be disconnected. All these procedures improve on extant procedures by using fewer moving knives, making fewer people envious, or using fewer cuts. While the 4-person, 5-cut procedure is complex, endowing people with more information about others' preferences, or allowing them to do things beyond stopping moving knives, may yield simpler procedures for making envy- free divisions with minimal cuts, which are known always to existFAIR DIVISION; CAKE CUTTING; ENVY-FREENESS; MAXIMIN
Cutting a Cake Is Not Always a 'Piece of Cake': A Closer Look at the Foundations of Cake-Cutting Through the Lens of Measure Theory
Cake-cutting is a playful name for the fair division of a heterogeneous,
divisible good among agents, a well-studied problem at the intersection of
mathematics, economics, and artificial intelligence. The cake-cutting
literature is rich and edifying. However, different model assumptions are made
in its many papers, in particular regarding the set of allowed pieces of cake
that are to be distributed among the agents and regarding the agents' valuation
functions by which they measure these pieces. We survey the commonly used
definitions in the cake-cutting literature, highlight their strengths and
weaknesses, and make some recommendations on what definitions could be most
reasonably used when looking through the lens of measure theory
Fair allocation of a multiset of indivisible items
We study the problem of fairly allocating a multiset of indivisible
items among agents with additive valuations. Specifically, we introduce a
parameter for the number of distinct types of items and study fair
allocations of multisets that contain only items of these types, under two
standard notions of fairness:
1. Envy-freeness (EF): For arbitrary , , we show that a complete EF
allocation exists when at least one agent has a unique valuation and the number
of items of each type exceeds a particular finite threshold. We give explicit
upper and lower bounds on this threshold in some special cases.
2. Envy-freeness up to any good (EFX): For arbitrary , , and for , we show that a complete EFX allocation always exists. We give two different
proofs of this result. One proof is constructive and runs in polynomial time;
the other is geometrically inspired.Comment: 34 pages, 6 figures, 1 table, 1 algorith
An Improved Envy-Free Cake Cutting Protocol for Four Agents
We consider the classic cake-cutting problem of producing envy-free allocations, restricted to the case of four agents. The problem asks for a partition of the cake to four agents, so that every agent finds her piece at least as valuable as every other agent's piece. The problem has had an interesting history so far. Although the case of three agents is solvable with less than 15 queries, for four agents no bounded procedure was known until the recent breakthroughs of Aziz and Mackenzie (STOC 2016, FOCS 2016). The main drawback of these new algorithms, however, is that they are quite complicated and with a very high query complexity. With four agents, the number of queries required is close to 600. In this work we provide an improved algorithm for four agents, which reduces the current complexity by a factor of 3.4. Our algorithm builds on the approach of Aziz and Mackenzie (STOC 2016) by incorporating new insights and simplifying several steps. Overall, this yields an easier to grasp procedure with lower complexity