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 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 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.

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. 

scholarly journals Group Fairness for the Allocation of Indivisible Goods

2019 ◽  
Vol 33 ◽  
pp. 1853-1860 ◽  
Author(s):  
Vincent Conitzer ◽  
Rupert Freeman ◽  
Nisarg Shah ◽  
Jennifer Wortman Vaughan
Keyword(s):  
Local Search ◽  
Polynomial Time ◽  
Fair Division ◽  
Indivisible Goods ◽  
Local Optima ◽  
Pseudo Polynomial Time

We consider the problem of fairly dividing a collection of indivisible goods among a set of players. Much of the existing literature on fair division focuses on notions of individual fairness. For instance, envy-freeness requires that no player prefer the set of goods allocated to another player to her own allocation. We observe that an algorithm satisfying such individual fairness notions can still treat groups of players unfairly, with one group desiring the goods allocated to another. Our main contribution is a notion of group fairness, which implies most existing notions of individual fairness. Group fairness (like individual fairness) cannot be satisfied exactly with indivisible goods. Thus, we introduce two “up to one good” style relaxations. We show that, somewhat surprisingly, certain local optima of the Nash welfare function satisfy both relaxations and can be computed in pseudo-polynomial time by local search. Our experiments reveal faster computation and stronger fairness guarantees in practice.

scholarly journals Geometric Rescaling Algorithms for Submodular Function Minimization

2021 ◽  
Author(s):  
Dan Dadush ◽  
László A. Végh ◽  
Giacomo Zambelli
Keyword(s):  
Iterative Methods ◽  
Polynomial Time ◽  
Black Box ◽  
Submodular Function ◽  
Function Minimization ◽  
Polynomial Algorithms ◽  
Polynomial Time Algorithms ◽  
Wide Range ◽  
Submodular Function Minimization ◽  
Strongly Polynomial

We present a new class of polynomial-time algorithms for submodular function minimization (SFM) as well as a unified framework to obtain strongly polynomial SFM algorithms. Our algorithms are based on simple iterative methods for the minimum-norm problem, such as the conditional gradient and Fujishige–Wolfe algorithms. We exhibit two techniques to turn simple iterative methods into polynomial-time algorithms. First, we adapt the geometric rescaling technique, which has recently gained attention in linear programming, to SFM and obtain a weakly polynomial bound [Formula: see text]. Second, we exhibit a general combinatorial black box approach to turn [Formula: see text]-approximate SFM oracles into strongly polynomial exact SFM algorithms. This framework can be applied to a wide range of combinatorial and continuous algorithms, including pseudo-polynomial ones. In particular, we can obtain strongly polynomial algorithms by a repeated application of the conditional gradient or of the Fujishige–Wolfe algorithm. Combined with the geometric rescaling technique, the black box approach provides an [Formula: see text] algorithm. Finally, we show that one of the techniques we develop in the paper can also be combined with the cutting-plane method of Lee et al., yielding a simplified variant of their [Formula: see text] algorithm.

scholarly journals Multiple Birds with One Stone: Beating 1/2 for EFX and GMMS via Envy Cycle Elimination

2020 ◽  
Vol 34 (02) ◽  
pp. 1790-1797 ◽  
Author(s):  
Georgios Amanatidis ◽  
Evangelos Markakis ◽  
Apostolos Ntokos
Keyword(s):  
Polynomial Time ◽  
Golden Ratio ◽  
Simple Algorithm ◽  
Fair Division ◽  
Existence Results ◽  
Indivisible Goods ◽  
Approximation Factor ◽  
General Existence ◽  
Primary Focus ◽  
Fine Tune

Several relaxations of envy-freeness, tailored to fair division in settings with indivisible goods, have been introduced within the last decade. Due to the lack of general existence results for most of these concepts, great attention has been paid to establishing approximation guarantees. In this work, we propose a simple algorithm that is universally fair in the sense that it returns allocations that have good approximation guarantees with respect to four such fairness notions at once. In particular, this is the first algorithm achieving a (φ−1)-approximation of envy-freeness up to any good (EFX) and a 2/φ+2 -approximation of groupwise maximin share fairness (GMMS), where φ is the golden ratio. The best known approximation factor, in polynomial time, for either one of these fairness notions prior to this work was 1/2. Moreover, the returned allocation achieves envy-freeness up to one good (EF1) and a 2/3-approximation of pairwise maximin share fairness (PMMS). While EFX is our primary focus, we also exhibit how to fine-tune our algorithm and improve further the guarantees for GMMS or PMMS.Finally, we show that GMMS—and thus PMMS and EFX—allocations always exist when the number of goods does not exceed the number of agents by more than two.

scholarly journals Almost Group Envy-free Allocation of Indivisible Goods and Chores

2020 ◽  
Author(s):  
Haris Aziz ◽  
Simon Rey
Keyword(s):  
Resource Allocation ◽  
Polynomial Time ◽  
Fair Allocation ◽  
Indivisible Goods ◽  
Preference Domain ◽  
Polynomial Time Algorithms ◽  
Multi Agent ◽  
Additive Utilities ◽  
Natural Classes ◽  
Allocation Of Indivisible Items

We consider a multi-agent resource allocation setting in which an agent's utility may decrease or increase when an item is allocated. We take the group envy-freeness concept that is well-established in the literature and present stronger and relaxed versions that are especially suitable for the allocation of indivisible items. Of particular interest is a concept called group envy-freeness up to one item (GEF1). We then present a clear taxonomy of the fairness concepts. We study which fairness concepts guarantee the existence of a fair allocation under which preference domain. For two natural classes of additive utilities, we design polynomial-time algorithms to compute a GEF1 allocation. We also prove that checking whether a given allocation satisfies GEF1 is coNP-complete when there are either only goods, only chores or both.

Sign in / Sign up

Export Citation Format

Share Document