Minimizing envy and maximizing average Nash social welfare in the allocation of indivisible goods

Nguyen, Trung Thanh; Rothe, Jörg

doi:10.1016/j.dam.2014.09.010

Cited by 34 publications

(19 citation statements)

References 12 publications

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“…The best previously known approximation guarantee was ρ ∈ O(m) [27]; in this work we provide an algorithm that guarantees a factor of at most ρ ≈ 2.889.…”

Section: Approximation Guaranteementioning

confidence: 98%

“…For the types of valuations that we consider here, i.e., for additive valuations, this problem was recently shown to be NPhard [34,25]. Following up on this hardness result, Nguyen and Rothe [27] studied the approximability of the Nash social welfare objective, which led to the first approximation algorithm. In particular, they show that their algorithm approximates the geometric mean objective within a factor of (m−n+1).…”

Section: Computational Complexitymentioning

confidence: 99%

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Approximating the Nash Social Welfare with Indivisible Items

Cole,

Gkatzelis

2015

Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing

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We study the problem of allocating a set of indivisible items among agents with additive valuations, with the goal of maximizing the geometric mean of the agents' valuations, i.e., the Nash social welfare. This problem is known to be NPhard, and our main result is the first efficient constant-factor approximation algorithm for this objective. We first observe that the integrality gap of the natural fractional relaxation is exponential, so we propose a different fractional allocation which implies a tighter upper bound and, after appropriate rounding, yields a good integral allocation.An interesting contribution of this work is the fractional allocation that we use. The relaxation of our problem can be solved efficiently using the Eisenberg-Gale program, whose optimal solution can be interpreted as a market equilibrium with the dual variables playing the role of item prices. Using this market-based interpretation, we define an alternative equilibrium allocation where the amount of spending that can go into any given item is bounded, thus keeping the highly priced items under-allocated, and forcing the agents to spend on lower priced items. The resulting equilibrium prices reveal more information regarding how to assign items so as to obtain a good integral allocation.

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“…The best previously known approximation guarantee was ρ ∈ O(m) [27]; in this work we provide an algorithm that guarantees a factor of at most ρ ≈ 2.889.…”

Section: Approximation Guaranteementioning

confidence: 98%

Section: Computational Complexitymentioning

confidence: 99%

Approximating the Nash Social Welfare with Indivisible Items

Cole,

Gkatzelis

2015

Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing

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“…In addition, it has been recently proved that there is a constant factor for which one cannot have an approximation algorithm of a factor better than [21]. A first approximation algorithm for the problem with an approximation factor of () Om is given in [18], where m denotes the number of goods. This result was improved by Cole and Gkatzelis in [6] where the authors obtained an approximation algorithm with a constant factor 2.89.…”

Section: A Related Workmentioning

confidence: 99%

“…Very recently, Cole et al [4] have achieved a further improved factor of 2 for the problem. For the special case when all agents have the same utility functions, there is a polynomial time approximation scheme (PTAS) for the problem [18].…”

Section: A Related Workmentioning

confidence: 99%

“…Budish [9] takes this idea into account and introduce the notion of the max-min share, which relies on the determining of an allocation of maximum egalitarian welfare. Since maximizing the Nash-product welfare is also considered to be a fair way for allocating item (see, e.g., [3,4,5,6,18,19]), it is very natural to define the notion of Nash-product share. Secondly, although checking the existence of proportionally fair allocation is NP-complete, we do not know yet if checking the existence of max-min fair allocation is also NPcomplete.…”

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confidence: 99%

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On the Nash Product Share Criterion for Fair Allocation Problems

Thành¹,

Nguyen²

2017

Fair - Nghiên Cứu Cơ Bản Và Ứng Dụng Công Nghệ Thông Tin - 2017

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Allocating indivisible items between agents in a fair manner is a fundamental problem that has attracted a lot of interests in the last decades due to the wide range of its applications. There are several common criteria for defining what a fair allocation is, including: max-min share, proportional share, min-max share, envy-freeness, CEEI. In this paper, we introduce a new notion of fairness, called Nash-product share, which can be determined through computing an allocation of maximum Nash-product welfare, assuming that all agents have the same additive utilities. An allocation satisfies this fairness criterion is called a Nashproduct fair allocation. We first show that computing the Nash-product share of every agent is NP-hard, even with two agents only. In addition, we prove that the problem of testing the existence of a Nash-product fair allocation is an NP-hard problem when the number of agents is part of the input. Finally, since a Nash-product fair allocation does not always exist, we investigate the problem that, given a problem instance, asks for the largest value c for which there is an allocation such that every agent receives a subset of items of values of at least c times her Nash-product share. For the case where the number of agents is constant, we present a polynomial-time approximation scheme (PTAS). I. GIỚI THIỆUFair allocation is a fundamental problem which is of interest in both computer science and economics due to the wide range of its applications (see a book chapter by Bouveret [8] and Lang and Rothe [15]). This problem concerns with allocating a finite set of goods (that may also be called items or objects) among a group of agents having different preferences over the subsets of goods. We will assume that the preferences of agents are presented by additive (or linear) utility functions. Our goal is to find an allocation that satisfies a certain notion of fairness. Among others, maxmin share, proportional share, and envy-freeness are the three notions that have been studied intensively in the literature. The max-min share of an agent is defined as the bundle that the agent can guarantee for herself when partitioning the items into bundles but choosing last. In a proportionally fair allocation between n agents, each agent receives a bundle of value at least 1/ n of the whole. An allocation is said to be envy-free if no agent wants to exchange her bundle of goods with that of any other agent. These three fairness criteria have attractive theoretical and practical properties (see the references cited above and the references therein). In [7], it was shown that envy-freeness is the strongest among the above fairness notions and implies proportional fairness, which implies max-min fairnessthe weakest one. On the other hand, as argued in the literature (see, for example, [14,16,17]), an allocation satisfying any of the these fairness notions is not guaranteed to exist in general. Furthermore, checking the existence of an envyfree (proportional fair) allocation is NP-complete (see [7]) even in th...

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Enabling assessment of distributive justice through models for climate change planning: A review of recent advances and a research agenda

Jafino

Kwakkel

Taebi

2021

WIREs Climate Change

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Models for supporting climate adaptation and mitigation planning, mostly in the form of Integrated Assessment Models, are poorly equipped for aiding questions related to fairness of adaptation and mitigation strategies, because they often disregard distributional outcomes. When evaluating policies using such models, the costs and benefits are typically aggregated across all actors in the system, and over the entire planning horizon. While a policy may be beneficial when considering the aggregate outcome, it can be harmful to some people, somewhere, at some point in time. The practice of aggregating over all actors and over time thus gives rise to problems of justice; it could also exacerbate existing injustices. While the literature discusses some of these injustices in ad‐hoc and case specific manner, a systematic approach for considering distributive justice in model‐based climate change planning is lacking. This study aims to fill this gap by proposing 11 requirements that an Integrated Assessment Model should meet in order to enable the assessment of distributive justice in climate mitigation and adaptation policies. We derive the requirements from various ethical imperatives stemming from the theory of distributive justice. More specifically, we consider both intra‐generational (among people within one generation) and intergenerational (between generations) distributive justice. We investigate to what extent the 11 requirements are being met in recent model‐based climate planning studies, and highlight several directions for future research to advance the accounting for distributive justice in model‐based support for climate change planning. This article is categorized under: Climate, Nature, and Ethics > Climate Change and Global Justice

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Minimizing envy and maximizing average Nash social welfare in the allocation of indivisible goods

Cited by 34 publications

References 12 publications

Approximating the Nash Social Welfare with Indivisible Items

Approximating the Nash Social Welfare with Indivisible Items

On the Nash Product Share Criterion for Fair Allocation Problems

Enabling assessment of distributive justice through models for climate change planning: A review of recent advances and a research agenda

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