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 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 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 Pareto-Optimal Allocation of Indivisible Goods with Connectivity Constraints

2019 ◽  
Vol 33 ◽  
pp. 2045-2052 ◽  
Author(s):  
Ayumi Igarashi ◽  
Dominik Peters
Keyword(s):  
Maximum Degree ◽  
Optimal Allocation ◽  
Additional Constraint ◽  
Pareto Optimal ◽  
Np Hard ◽  
Efficient Allocation ◽  
Indivisible Goods ◽  
Underlying Graph

We study the problem of allocating indivisible items to agents with additive valuations, under the additional constraint that bundles must be connected in an underlying item graph. Previous work has considered the existence and complexity of fair allocations. We study the problem of finding an allocation that is Pareto-optimal. While it is easy to find an efficient allocation when the underlying graph is a path or a star, the problem is NP-hard for many other graph topologies, even for trees of bounded pathwidth or of maximum degree 3. We show that on a path, there are instances where no Pareto-optimal allocation satisfies envy-freeness up to one good, and that it is NP-hard to decide whether such an allocation exists, even for binary valuations. We also show that, for a path, it is NP-hard to find a Pareto-optimal allocation that satisfies maximin share, but show that a moving-knife algorithm can find such an allocation when agents have binary valuations that have a non-nested interval structure.

scholarly journals Weighted Envy-freeness in Indivisible Item Allocation

2021 ◽  
Vol 9 (3) ◽  
pp. 1-39
Author(s):  
Mithun Chakraborty ◽  
Ayumi Igarashi ◽  
Warut Suksompong ◽  
Yair Zick
Keyword(s):  
Social Welfare ◽  
Polynomial Time ◽  
Pareto Optimality ◽  
Arbitrary Number ◽  
Fair Division ◽  
Pareto Optimal ◽  
Weak Version ◽  
Strong Version ◽  
Two Agents ◽  
Allocation Of Indivisible Items

We introduce and analyze new envy-based fairness concepts for agents with weights that quantify their entitlements in the allocation of indivisible items. We propose two variants of weighted envy-freeness up to one item (WEF1): strong , where envy can be eliminated by removing an item from the envied agent’s bundle, and weak , where envy can be eliminated either by removing an item (as in the strong version) or by replicating an item from the envied agent’s bundle in the envying agent’s bundle. We show that for additive valuations, an allocation that is both Pareto optimal and strongly WEF1 always exists and can be computed in pseudo-polynomial time; moreover, an allocation that maximizes the weighted Nash social welfare may not be strongly WEF1, but it always satisfies the weak version of the property. Moreover, we establish that a generalization of the round-robin picking sequence algorithm produces in polynomial time a strongly WEF1 allocation for an arbitrary number of agents; for two agents, we can efficiently achieve both strong WEF1 and Pareto optimality by adapting the adjusted winner procedure. Our work highlights several aspects in which weighted fair division is richer and more challenging than its unweighted counterpart.

Optimal Allocation of Risk for Portfolios

2014 ◽  
Vol 644-650 ◽  
pp. 6067-6070
Author(s):  
Hong Wei Liu ◽  
Cai Bo Xiao
Keyword(s):  
Representation Theorem ◽  
Optimal Allocation ◽  
Pareto Optimal ◽  
Risk Allocation ◽  
Equivalent Conditions

In this paper, we propose a framework of the optimal risk allocation, under the pareto optimal we give equivalent conditions and provided its representation theorem under Pareto-optimal allocation, Which is an extension of the ones introduced by Ludger Rüschendorf (2006).

scholarly journals Selling with evidence

2019 ◽  
Vol 14 (2) ◽  
pp. 345-371
Author(s):  
Frédéric Koessler ◽  
Vasiliki Skreta
Keyword(s):  
Asymmetric Information ◽  
Product Quality ◽  
Optimal Allocation ◽  
Pareto Optimal ◽  
Principal Problem ◽  
Equilibrium Allocation ◽  
Interdependent Values ◽  
Ex Ante ◽  
Private Value ◽  
Posted Price

We study the informed‐principal problem in a bilateral asymmetric information trading setting with interdependent values and quasi‐linear utilities. The informed seller proposes a mechanism and voluntarily certifies information about the good's characteristics. When the set of certifiable statements is sufficiently rich, we show that there is an ex ante profit‐maximizing selling procedure that is an equilibrium of the mechanism proposal game. In contrast to posted price settings, the allocation obtained when product characteristics are commonly known (the unravelling outcome) may not be an equilibrium allocation, even when all buyer types agree on the ranking of product quality. Our analysis relies on the concept of strong Pareto optimal allocation, which was originally introduced by Maskin and Tirole (1990) in private value environments.

ALTERNATIVE MONETARY POLICIES IN A TURNPIKE ECONOMY

2010 ◽  
Vol 14 (5) ◽  
pp. 727-762 ◽  
Author(s):  
Rodolfo Manuelli ◽  
Thomas J. Sargent
Keyword(s):  
Money Supply ◽  
Optimal Policy ◽  
Optimal Allocation ◽  
Price Level ◽  
Optimal Rate ◽  
Pareto Optimal ◽  
Monetary Policies

This paper modifies a Townsend turnpike model by letting agents stay at a location long enough to trade some consumption loans, but not long enough to support a Pareto-optimal allocation. Monetary equilibria exist that are nonoptimal in the absence of a scheme to pay interest on currency at a particular rate. Paying interest on currency at the optimal rate delivers a Pareto-optimal allocation, but a different one than the allocation for an associated nonmonetary centralized economy. The price level remains determinate under an optimal policy. We study the response of the model to “helicopter drops” of currency, steady increases in the money supply, and restrictions on private intermediation.

scholarly journals Balancing ecosystem services with energy and food security – Assessing trade-offs from reservoir operation and irrigation investments in Kenya's Tana Basin

2014 ◽  
Vol 18 (8) ◽  
pp. 3259-3277 ◽  
Author(s):  
A. P. Hurford ◽  
J. J. Harou
Keyword(s):  
Ecosystem Services ◽  
Optimal Allocation ◽  
Search Algorithm ◽  
Management Strategies ◽  
Decision Makers ◽  
Pareto Optimal ◽  
Management Problem ◽  
Economic Sectors ◽  
Trade Off ◽  
Trade Offs

Abstract. Competition for water between key economic sectors and the environment means agreeing allocations is challenging. Managing releases from the three major dams in Kenya's Tana River basin with its 4.4 million inhabitants, 567 MW of installed hydropower capacity, 33 000 ha of irrigation and ecologically important wetlands and forests is a pertinent example. This research seeks firstly to identify and help decision-makers visualise reservoir management strategies which result in the best possible (Pareto-optimal) allocation of benefits between sectors. Secondly, it seeks to show how trade-offs between achievable benefits shift with the implementation of proposed new rice, cotton and biofuel irrigation projects. To approximate the Pareto-optimal trade-offs we link a water resources management simulation model to a multi-criteria search algorithm. The decisions or "levers" of the management problem are volume-dependent release rules for the three major dams and extent of investment in new irrigation schemes. These decisions are optimised for eight objectives covering the provision of water supply and irrigation, energy generation and maintenance of ecosystem services. Trade-off plots allow decision-makers to assess multi-reservoir rule-sets and irrigation investment options by visualising their impacts on different beneficiaries. Results quantify how economic gains from proposed irrigation schemes trade-off against the disturbance of ecosystems and local livelihoods that depend on them. Full implementation of the proposed schemes is shown to come at a high environmental and social cost. The clarity and comprehensiveness of "best-case" trade-off analysis is a useful vantage point from which to tackle the interdependence and complexity of "water-energy-food nexus" resource security issues.

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