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 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 Pareto optimal allocation under uncertain preferences: uncertainty models, algorithms, and complexity

2019 ◽  
Vol 276 ◽  
pp. 57-78 ◽  
Author(s):  
Haris Aziz ◽  
Péter Biró ◽  
Ronald de Haan ◽  
Baharak Rastegari
Keyword(s):  
Optimal Allocation ◽  
Pareto Optimal ◽  
Algorithms And Complexity ◽  
Uncertain Preferences ◽  
Uncertainty Models

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 The Complexity of Computing Maximin Share Allocations on Graphs

2020 ◽  
Vol 34 (02) ◽  
pp. 2006-2013
Author(s):  
Gianluigi Greco ◽  
Francesco Scarcello
Keyword(s):  
A Priori ◽  
Utility Functions ◽  
Indivisible Goods ◽  
Bounded Treewidth ◽  
Underlying Graph ◽  
Low Degree

Maximin share is a compelling notion of fairness proposed by Buddish as a relaxation of more traditional concepts for fair allocations of indivisible goods. In this paper we consider this notion within a setting where bundles of goods must induce connected subsets over an underlying graph. This setting received much attention in earlier literature, and our study answers a number of questions that were left open. First, we show that computing maximin share allocations is FΔ2P-complete, even when focusing on consistent scenarios, that is, where such allocations are a-priori guaranteed to exist. Moreover, the problem remains intractable if all agents have the same type, i.e., have the same utility functions, and if either the values returned by the utility functions are polynomially bounded, or the underlying graphs have a low degree of cyclicity (more precisely, have bounded treewidth). However, if these conditions hold all together, then computing maximin share allocations (or checking that none exists) becomes tractable. The result is established via machineries based on logspace alternating machines that use partial representations of connected bundles, which are interesting in their own.

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.

scholarly journals ON THE ROMAN BONDAGE NUMBER OF A GRAPH

2013 ◽  
Vol 05 (01) ◽  
pp. 1350001 ◽  
Author(s):  
A. BAHREMANDPOUR ◽  
FU-TAO HU ◽  
S. M. SHEIKHOLESLAMI ◽  
JUN-MING XU
Keyword(s):  
Decision Problem ◽  
Maximum Degree ◽  
Minimum Weight ◽  
Domination Number ◽  
Np Hard ◽  
Roman Domination ◽  
Minimum Cardinality ◽  
Bondage Number ◽  
Roman Domination Number ◽  
Roman Dominating Function

A Roman dominating function (RDF) on a graph G = (V, E) is a function f : V → {0, 1, 2} such that every vertex v ∈ V with f(v) = 0 has at least one neighbor u ∈ V with f(u) = 2. The weight of a RDF is the value f(V(G)) = Σu∈V(G) f(u). The minimum weight of a RDF on a graph G is called the Roman domination number, denoted by γR(G). The Roman bondage number bR(G) of a graph G with maximum degree at least two is the minimum cardinality of all sets E′ ⊆ E(G) for which γR(G - E′) > γR(G). In this paper, we first show that the decision problem for determining bR(G) is NP-hard even for bipartite graphs and then we establish some sharp bounds for bR(G) and characterizes all graphs attaining some of these bounds.

scholarly journals Finding Fair and Efficient Allocations for Matroid Rank Valuations

2021 ◽  
Vol 9 (4) ◽  
pp. 1-41
Author(s):  
Nawal Benabbou ◽  
Mithun Chakraborty ◽  
Ayumi Igarashi ◽  
Yair Zick
Keyword(s):  
Convex Function ◽  
Real World ◽  
Function Class ◽  
Efficient Allocation ◽  
Indivisible Goods ◽  
Valuation Function ◽  
Bipartite Matching ◽  
Optimal Outcomes ◽  
Efficient Allocations ◽  
Rank Functions

In this article, we present new results on the fair and efficient allocation of indivisible goods to agents whose preferences correspond to matroid rank functions . This is a versatile valuation class with several desirable properties (such as monotonicity and submodularity), which naturally lends itself to a number of real-world domains. We use these properties to our advantage; first, we show that when agent valuations are matroid rank functions, a socially optimal (i.e., utilitarian social welfare-maximizing) allocation that achieves envy-freeness up to one item (EF1) exists and is computationally tractable. We also prove that the Nash welfare-maximizing and the leximin allocations both exhibit this fairness/efficiency combination by showing that they can be achieved by minimizing any symmetric strictly convex function over utilitarian optimal outcomes. To the best of our knowledge, this is the first valuation function class not subsumed by additive valuations for which it has been established that an allocation maximizing Nash welfare is EF1. Moreover, for a subclass of these valuation functions based on maximum (unweighted) bipartite matching, we show that a leximin allocation can be computed in polynomial time. Additionally, we explore possible extensions of our results to fairness criteria other than EF1 as well as to generalizations of the above valuation classes.

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