Cake-Cutting: Fair Division of Divisible Goods

2016 ◽  
pp. 395-491 ◽  
Author(s):  
Claudia Lindner ◽  
Jörg Rothe
Keyword(s):  
Fair Division ◽  
Cake Cutting ◽  
Divisible Goods

scholarly journals Fair Division with a Secretive Agent

2019 ◽  
Vol 33 ◽  
pp. 1732-1739
Author(s):  
Eshwar Ram Arunachaleswaran ◽  
Siddharth Barman ◽  
Nidhi Rathi
Keyword(s):  
Partial Information ◽  
Fair Division ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Indivisible Goods ◽  
Cake Cutting ◽  
Division Problems ◽  
Divisible Goods ◽  
Arbitrary Selection ◽  
Selection Of

We study classic fair-division problems in a partial information setting. This paper respectively addresses fair division of rent, cake, and indivisible goods among agents with cardinal preferences. We will show that, for all of these settings and under appropriate valuations, a fair (or an approximately fair) division among n agents can be efficiently computed using only the valuations of n − 1 agents. The nth (secretive) agent can make an arbitrary selection after the division has been proposed and, irrespective of her choice, the computed division will admit an overall fair allocation.For the rent-division setting we prove that well-behaved utilities of n − 1 agents suffice to find a rent division among n rooms such that, for every possible room selection of the secretive agent, there exists an allocation (of the remaining n − 1 rooms among the n − 1 agents) which ensures overall envy freeness (fairness). We complement this existential result by developing a polynomial-time algorithm for the case of quasilinear utilities. In this partial information setting, we also develop efficient algorithms to compute allocations that are envy-free up to one good (EF1) and ε-approximate envy free. These two notions of fairness are applicable in the context of indivisible goods and divisible goods (cake cutting), respectively.One of the main technical contributions of this paper is the development of novel connections between different fairdivision paradigms, e.g., we use our existential results for envy-free rent-division to develop an efficient EF1 algorithm.

scholarly journals Truthful fair division without free disposal

2020 ◽  
Vol 55 (3) ◽  
pp. 523-545 ◽  
Author(s):  
Xiaohui Bei ◽  
Guangda Huzhang ◽  
Warut Suksompong
Keyword(s):  
Fair Division ◽  
Multiple Agents ◽  
Two Agents ◽  
Free Disposal ◽  
Cake Cutting ◽  
Strategic Agents ◽  
Additional Constraints

Abstract We study the problem of fairly dividing a heterogeneous resource, commonly known as cake cutting and chore division, in the presence of strategic agents. While a number of results in this setting have been established in previous works, they rely crucially on the free disposal assumption, meaning that the mechanism is allowed to throw away part of the resource at no cost. In the present work, we remove this assumption and focus on mechanisms that always allocate the entire resource. We exhibit a truthful and envy-free mechanism for cake cutting and chore division for two agents with piecewise uniform valuations, and we complement our result by showing that such a mechanism does not exist when certain additional constraints are imposed on the mechanisms. Moreover, we provide bounds on the efficiency of mechanisms satisfying various properties, and give truthful mechanisms for multiple agents with restricted classes of valuations.

scholarly journals Envy-Free Division of Land

2020 ◽  
Vol 45 (3) ◽  
pp. 896-922 ◽  
Author(s):  
Erel Segal-Halevi ◽  
Shmuel Nitzan ◽  
Avinatan Hassidim ◽  
Yonatan Aumann
Keyword(s):  
Aspect Ratio ◽  
Fair Division ◽  
Electronic Media ◽  
Geometric Shape ◽  
Geometric Constraints ◽  
Crucial Importance ◽  
One Dimensional ◽  
Individual Preferences ◽  
Cake Cutting ◽  
Natural Settings

Classic cake-cutting algorithms enable people with different preferences to divide among them a heterogeneous resource (“cake”) such that the resulting division is fair according to each agent’s individual preferences. However, these algorithms either ignore the geometry of the resource altogether or assume it is one-dimensional. In practice, it is often required to divide multidimensional resources, such as land estates or advertisement spaces in print or electronic media. In such cases, the geometric shape of the allotted piece is of crucial importance. For example, when building houses or designing advertisements, in order to be useful, the allotments should be squares or rectangles with bounded aspect ratio. We, thus, introduce the problem of fair land division—fair division of a multidimensional resource wherein the allocated piece must have a prespecified geometric shape. We present constructive division algorithms that satisfy the two most prominent fairness criteria, namely envy-freeness and proportionality. In settings in which proportionality cannot be achieved because of the geometric constraints, our algorithms provide a partially proportional division, guaranteeing that the fraction allocated to each agent be at least a certain positive constant. We prove that, in many natural settings, the envy-freeness requirement is compatible with the best attainable partial-proportionality.

Fair Division: From Cake-Cutting to Dispute Resolution.

1998 ◽  
Vol 105 (9) ◽  
pp. 877
Author(s):  
William F. Lucas ◽  
Steven J. Brams ◽  
Alan D. Taylor
Keyword(s):  
Dispute Resolution ◽  
Fair Division ◽  
Cake Cutting

scholarly journals Fair Division of Mixed Divisible and Indivisible Goods

2020 ◽  
Vol 34 (02) ◽  
pp. 1814-1821
Author(s):  
Xiaohui Bei ◽  
Zihao Li ◽  
Jinyan Liu ◽  
Shengxin Liu ◽  
Xinhang Lu
Keyword(s):  
Piecewise Linear ◽  
Fair Division ◽  
Efficient Algorithms ◽  
Indivisible Goods ◽  
Direct Generalization ◽  
Two Agents ◽  
Divisible Goods

We study the problem of fair division when the resources contain both divisible and indivisible goods. Classic fairness notions such as envy-freeness (EF) and envy-freeness up to one good (EF1) cannot be directly applied to the mixed goods setting. In this work, we propose a new fairness notion envy-freeness for mixed goods (EFM), which is a direct generalization of both EF and EF1 to the mixed goods setting. We prove that an EFM allocation always exists for any number of agents. We also propose efficient algorithms to compute an EFM allocation for two agents and for n agents with piecewise linear valuations over the divisible goods. Finally, we relax the envy-free requirement, instead asking for ϵ-envy-freeness for mixed goods (ϵ-EFM), and present an algorithm that finds an ϵ-EFM allocation in time polynomial in the number of agents, the number of indivisible goods, and 1/ϵ.

scholarly journals Networked Fairness in Cake Cutting

2017 ◽  
Author(s):  
Xiaohui Bei ◽  
Youming Qiao ◽  
Shengyu Zhang
Keyword(s):  
Network Structure ◽  
Transitive Closure ◽  
Rooted Tree ◽  
Fair Division ◽  
Efficient Algorithms ◽  
Research Directions ◽  
Proportional Allocation ◽  
Cake Cutting ◽  
Underlying Network ◽  
New Research

We introduce a graphical framework for fair division in cake cutting, where comparisons between agents are limited by an underlying network structure. We generalize the classical fairness notions of envy-freeness and proportionality in this graphical setting. An allocation is called envy-free on a graph if no agent envies any of her neighbor's share, and is called proportional on a graph if every agent values her own share no less than the average among her neighbors, with respect to her own measure. These generalizations enable new research directions in developing simple and efficient algorithms that can produce fair allocations under specific graph structures. On the algorithmic frontier, we first propose a moving-knife algorithm that outputs an envy-free allocation on trees. The algorithm is significantly simpler than the discrete and bounded envy-free algorithm introduced in [Aziz and Mackenzie, 2016] for compete graphs. Next, we give a discrete and bounded algorithm for computing a proportional allocation on transitive closure of trees, a class of graphs by taking a rooted tree and connecting all its ancestor-descendant pairs.

scholarly journals Fair Division of Time: Multi-layered Cake Cutting

2020 ◽  
Author(s):  
Hadi Hosseini ◽  
Ayumi Igarashi ◽  
Andrew Searns
Keyword(s):  
Real Life ◽  
Fair Division ◽  
Medical Professionals ◽  
Time Interval ◽  
Time Intervals ◽  
Number Of Layers ◽  
Cake Cutting

We initiate the study of multi-layered cake cutting with the goal of fairly allocating multiple divisible resources (layers of a cake) among a set of agents. The key requirement is that each agent can only utilize a single resource at each time interval. Several real-life applications exhibit such restrictions on overlapping pieces, for example, assigning time intervals over multiple facilities and resources or assigning shifts to medical professionals. We investigate the existence and computation of envy-free and proportional allocations. We show that envy-free allocations that are both feasible and contiguous are guaranteed to exist for up to three agents with two types of preferences, when the number of layers is two. We further devise an algorithm for computing proportional allocations for any number of agents when the number of layers is factorable to three and/or some power of two.

Constraints in fair division

2021 ◽  
Vol 19 (2) ◽  
pp. 46-61
Author(s):  
Warut Suksompong
Keyword(s):  
Fundamental Problem ◽  
Fair Division ◽  
Fair Allocation ◽  
Allocation Of Resources ◽  
Open Questions ◽  
Cake Cutting

The fair allocation of resources to interested agents is a fundamental problem in society. While the majority of the fair division literature assumes that all allocations are feasible, in practice there are often constraints on the allocation that can be chosen. In this survey, we discuss fairness guarantees for both divisible (cake cutting) and indivisible resources under several common types of constraints, including connectivity, cardinality, matroid, geometric, separation, budget, and conflict constraints. We also outline a number of open questions and directions.

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