Fair Division of Indivisible Goods

2016 ◽  
pp. 493-550 ◽  
Author(s):  
Jérôme Lang ◽  
Jörg Rothe
Keyword(s):  
Fair Division ◽  
Indivisible Goods

scholarly journals Equitable Allocations of Indivisible Goods

2019 ◽  
Author(s):  
Rupert Freeman ◽  
Sujoy Sikdar ◽  
Rohit Vaish ◽  
Lirong Xia
Keyword(s):  
Economic Efficiency ◽  
Pareto Optimality ◽  
Real World ◽  
Fair Division ◽  
Prior Work ◽  
Indivisible Goods ◽  
Preference Data ◽  
Novel Algorithm

In fair division, equitability dictates that each participant receives the same level of utility. In this work, we study equitable allocations of indivisible goods among agents with additive valuations. While prior work has studied (approximate) equitability in isolation, we consider equitability in conjunction with other well-studied notions of fairness and economic efficiency. We show that the Leximin algorithm produces an allocation that satisfies equitability up to any good and Pareto optimality. We also give a novel algorithm that guarantees Pareto optimality and equitability up to one good in pseudopolynomial time.  Our experiments on real-world preference data reveal that approximate envy-freeness, approximate equitability, and Pareto optimality can often be achieved simultaneously.

scholarly journals On the Proximity of Markets with Integral Equilibria

2019 ◽  
Vol 33 ◽  
pp. 1748-1755 ◽  
Author(s):  
Siddharth Barman ◽  
Sanath Kumar Krishnamurthy
Keyword(s):  
Polynomial Time ◽  
Fair Division ◽  
Prior Work ◽  
Indivisible Goods ◽  
Polynomial Time Algorithms ◽  
Fairness And Efficiency ◽  
Pareto Efficient ◽  
Efficient Allocations ◽  
Strong Existence ◽  
Strongly Polynomial

We study Fisher markets that admit equilibria wherein each good is integrally assigned to some agent. While strong existence and computational guarantees are known for equilibria of Fisher markets with additive valuations (Eisenberg and Gale 1959; Orlin 2010), such equilibria, in general, assign goods fractionally to agents. Hence, Fisher markets are not directly applicable in the context of indivisible goods. In this work we show that one can always bypass this hurdle and, up to a bounded change in agents’ budgets, obtain markets that admit an integral equilibrium. We refer to such markets as pure markets and show that, for any given Fisher market (with additive valuations), one can efficiently compute a “near-by,” pure market with an accompanying integral equilibrium.Our work on pure markets leads to novel algorithmic results for fair division of indivisible goods. Prior work in discrete fair division has shown that, under additive valuations, there always exist allocations that simultaneously achieve the seemingly incompatible properties of fairness and efficiency (Caragiannis et al. 2016); here fairness refers to envyfreeness up to one good (EF1) and efficiency corresponds to Pareto efficiency. However, polynomial-time algorithms are not known for finding such allocations. Considering relaxations of proportionality and EF1, respectively, as our notions of fairness, we show that fair and Pareto efficient allocations can be computed in strongly polynomial time.

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 Fair Division Under Cardinality Constraints

2018 ◽  
Author(s):  
Arpita Biswas ◽  
Siddharth Barman
Keyword(s):  
Resource Allocation ◽  
Fair Division ◽  
Division Problem ◽  
Fair Allocation ◽  
Indivisible Goods ◽  
Solution Concepts ◽  
Cardinality Constraints ◽  
The Given

We consider the problem of fairly allocating indivisible goods, among agents, under cardinality constraints and additive valuations. In this setting, we are given a partition of the entire set of goods---i.e., the goods are categorized---and a limit is specified on the number of goods that can be allocated from each category to any agent. The objective here is to find a fair allocation in which the subset of goods assigned to any agent satisfies the given cardinality constraints. This problem naturally captures a number of resource-allocation applications, and is a generalization of the well-studied unconstrained fair division problem.  The two central notions of fairness, in the context of fair division of indivisible goods, are envy freeness up to one good (EF1) and the (approximate) maximin share guarantee (MMS). We show that the existence and algorithmic guarantees established for these solution concepts in the unconstrained setting can essentially be achieved under cardinality constraints. Furthermore, focusing on the case wherein all the agents have the same additive valuation, we establish that EF1 allocations exist even under matroid constraints.

scholarly journals Maximin Share Allocations on Cycles

2020 ◽  
Vol 69 ◽  
pp. 613-655
Author(s):  
Miroslaw Truszczynski ◽  
Zbigniew Lonc
Keyword(s):  
Resource Allocation ◽  
Social Choice ◽  
Fundamental Problem ◽  
Fair Division ◽  
Connected Subgraph ◽  
Indivisible Goods ◽  
Multi Agent Systems ◽  
Agent Systems ◽  
Multi Agent ◽  
Share Value

The problem of fair division of indivisible goods is a fundamental problem of resource allocation in multi-agent systems, also studied extensively in social choice. Recently, the problem was generalized to the case when goods form a graph and the goal is to allocate goods to agents so that each agent’s bundle forms a connected subgraph. For the maximin share fairness criterion, researchers proved that if goods form a tree, an allocation offering each agent a bundle of at least her maximin share value always exists. Moreover, it can be found in polynomial time. In this paper we consider the problem of maximin share allocations of goods on a cycle. Despite the simplicity of the graph, the problem turns out to be significantly harder than its tree version. We present cases when maximin share allocations of goods on cycles exist and provide in this case results on allocations guaranteeing each agent a certain fraction of her maximin share. We also study algorithms for computing maximin share allocations of goods on cycles.

scholarly journals Improving Nash Social Welfare Approximations

2020 ◽  
Vol 68 ◽  
pp. 225-245
Author(s):  
Peter McGlaughlin ◽  
Jugal Garg
Keyword(s):  
Social Welfare ◽  
Optimal Allocation ◽  
Focal Point ◽  
Market Equilibrium ◽  
Fair Division ◽  
Pareto Optimal ◽  
Indivisible Goods ◽  
Market Prices ◽  
Economic Concept ◽  
Strongly Polynomial

We consider the problem of fairly allocating a set of indivisible goods among n agents. Various fairness notions have been proposed within the rapidly growing field of fair division, but the Nash social welfare (NSW) serves as a focal point. In part, this follows from the ‘unreasonable’ fairness guarantees provided, in the sense that a max NSW allocation meets multiple other fairness metrics simultaneously, all while satisfying a standard economic concept of efficiency, Pareto optimality. However, existing approximation algorithms fail to satisfy all of the remarkable fairness guarantees offered by a max NSW allocation, instead targeting only the specific NSW objective. We address this issue by presenting a 2 max NSW, Prop-1, 1/(2n) MMS, and Pareto optimal allocation in strongly polynomial time. Our techniques are based on a market interpretation of a fractional max NSW allocation. We present novel definitions of fairness concepts in terms of market prices, and design a new scheme to round a market equilibrium into an integral allocation in a way that provides most of the fairness properties of an integral max NSW allocation.

Fair Division in Allocating Cabinet Ministries

2019 ◽  
pp. 886-898
Author(s):  
Steven J. Brams
Keyword(s):  
Northern Ireland ◽  
Political Parties ◽  
Fair Division ◽  
Indivisible Goods ◽  
Parliamentary Government

This chapter describes a mechanism for allocating cabinet ministries, considered to be indivisible goods, to political parties in a parliamentary government according to the proportion of seats that each party holds in a parliament. It precludes the usual bargaining and horse trading over how many, and which, ministries each party will get by using one of two apportionment schemes that prescribe the order of choice in which parties, based on their size, choose ministries. It was used, in modified form, to allocate ministries to each of the major Catholic and Protestant political parties in Northern Ireland in 1999.

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 Improving Nash Social Welfare Approximations

2019 ◽  
Author(s):  
Jugal Garg ◽  
Peter McGlaughlin
Keyword(s):  
Social Welfare ◽  
Optimal Allocation ◽  
Focal Point ◽  
Market Equilibrium ◽  
Fair Division ◽  
Pareto Optimal ◽  
Indivisible Goods ◽  
Market Prices ◽  
Economic Concept ◽  
Strongly Polynomial

We consider the problem of fairly allocating a set of indivisible goods among n agents. Various fairness notions have been proposed within the rapidly growing field of fair division, but the Nash social welfare (NSW) serves as a focal point. In part, this follows from the 'unreasonable' fairness guarantees provided, in the sense that a max NSW allocation meets multiple other fairness metrics simultaneously, all while satisfying a standard economic concept of efficiency, Pareto optimality. However, existing approximation algorithms fail to satisfy all of the remarkable fairness guarantees offered by a max NSW allocation, instead targeting only the specific NSW objective. We address this issue by presenting a 2 max NSW, Prop-1, 1/(2n) MMS, and Pareto optimal allocation in strongly polynomial time. Our techniques are based on a market interpretation of a fractional max NSW allocation. We present novel definitions of fairness concepts in terms of market prices, and design a new scheme to round a market equilibrium into an integral allocation that provides most of the fairness properties of an integral max NSW allocation. 

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