239 research outputs found
Pareto-Optimal Allocation of Indivisible Goods with Connectivity Constraints
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.Comment: 21 pages, full version of paper at AAAI-201
Competitive division of a mixed manna
A mixed manna contains goods (that everyone likes) and bads (that everyone dislikes),
as well as items that are goods to some agents, but bads or satiated to others.
If all items are goods and utility functions are homogeneous of degree 1 and concave
(and monotone), the competitive division maximizes the Nash product of utilities
(Gale–Eisenberg): hence it is welfarist (determined by the set of feasible utility profiles),
unique, continuous, and easy to compute.
We show that the competitive division of a mixed manna is still welfarist. If the zero
utility profile is Pareto dominated, the competitive profile is strictly positive and still
uniquely maximizes the product of utilities. If the zero profile is unfeasible (for instance,
if all items are bads), the competitive profiles are strictly negative and are the
critical points of the product of disutilities on the efficiency frontier. The latter allows
for multiple competitive utility profiles, from which no single-valued selection can be
continuous or resource monotonic.
Thus the implementation of competitive fairness under linear preferences in interactive
platforms like SPLIDDIT will be more difficult when the manna contains bads
that overwhelm the goods
Pareto optimality in many-to-many matching problems
Consider a many-to-many matching market that involves two finite disjoint sets, a set A of applicants and a set C of courses. Each applicant has preferences on the different sets of courses she can attend, while each course has a quota of applicants that it can admit. In this paper, we examine Pareto optimal matchings (briefly POM) in the context of such markets, that can also incorporate additional constraints, e.g., each course bearing some cost and each applicant having a limited budget available. We provide necessary and sufficient conditions for a many-to-many matching to be Pareto optimal and show that checking whether a given matching is Pareto optimal can be accomplished in 0(1 A 12 I C 12) time. Moreover, we provide a generalized version of serial dictatorship, which can be used to obtain any many-to-many POM. We also study some structural questions related to POM. We show that, unlike in the one-to-one case, finding a maximum cardinality POM is NP-hard for many-to-many markets. (C) 2014 Elsevier B.V. All rights reserved
Efficient Fair Division with Minimal Sharing
A collection of objects, some of which are good and some are bad, is to be
divided fairly among agents with different tastes, modeled by additive
utility-functions. If the objects cannot be shared, so that each of them must
be entirely allocated to a single agent, then a fair division may not exist.
What is the smallest number of objects that must be shared between two or more
agents in order to attain a fair and efficient division? We focus on
Pareto-optimal, envy-free and/or proportional allocations. We show that, for a
generic instance of the problem -- all instances except of a zero-measure set
of degenerate problems -- a fair Pareto-optimal division with the smallest
possible number of shared objects can be found in polynomial time, assuming
that the number of agents is fixed. The problem becomes computationally hard
for degenerate instances, where agents' valuations are aligned for many
objects.Comment: Add experiments with Spliddit.org dat
Democratic Fair Allocation of Indivisible Goods
We study the problem of fairly allocating indivisible goods to groups of
agents. Agents in the same group share the same set of goods even though they
may have different preferences. Previous work has focused on unanimous
fairness, in which all agents in each group must agree that their group's share
is fair. Under this strict requirement, fair allocations exist only for small
groups. We introduce the concept of democratic fairness, which aims to satisfy
a certain fraction of the agents in each group. This concept is better suited
to large groups such as cities or countries. We present protocols for
democratic fair allocation among two or more arbitrarily large groups of agents
with monotonic, additive, or binary valuations. For two groups with arbitrary
monotonic valuations, we give an efficient protocol that guarantees
envy-freeness up to one good for at least of the agents in each group,
and prove that the fraction is optimal. We also present other protocols
that make weaker fairness guarantees to more agents in each group, or to more
groups. Our protocols combine techniques from different fields, including
combinatorial game theory, cake cutting, and voting.Comment: Appears in the 27th International Joint Conference on Artificial
Intelligence and the 23rd European Conference on Artificial Intelligence
(IJCAI-ECAI), 201
Finding fair and efficient allocations
We study the problem of fair division, where the goal is to allocate a set of items among a set of agents in a ``fair" manner. In particular, we focus on settings in which the items to be divided are either indivisible goods or divisible bads. Despite their practical significance, both these settings have been much less investigated than the divisible goods setting. In the first part of the dissertation, we focus on the fair division of indivisible goods. Our fairness criterion is envy-freeness up to any good (EFX). An allocation is EFX if no agent envies another agent following the removal of a single good from the other agent's bundle. Despite significant investment by the research community, the existence of EFX allocations remains open and is considered one of the most important open problems in fair division. In this thesis, we make significant progress on this question. First, we show that when agents have general valuations, we can determine an EFX allocation with a small number of unallocated goods (almost EFX allocation). Second, we demonstrate that when agents have structured valuations, we can determine an almost EFX allocation that is also efficient in terms of Nash welfare. Third, we prove that EFX allocations exist when there are three agents with additive valuations. Finally, we reduce the problem of finding improved guarantees on EFX allocations to a novel problem in extremal graph theory. In the second part of this dissertation, we turn to the fair division of divisible bads. Like in the setting of divisible goods, competitive equilibrium with equal incomes (CEEI) has emerged as the best mechanism for allocating divisible bads. However, neither a polynomial time algorithm nor any hardness result is known for the computation of CEEI with bads. We study the problem of dividing bads in the classic Arrow-Debreu setting (a setting that generalizes CEEI). We show that in sharp contrast to the Arrow-Debreu setting with goods, determining whether a competitive equilibrium exists, is NP-hard in the case of divisible bads. Furthermore, we prove the existence of equilibrium under a simple and natural sufficiency condition. Finally, we show that even on instances that satisfy this sufficiency condition, determining a competitive equilibrium is PPAD-hard. Thus, we settle the complexity of finding a competitive equilibrium in the Arrow-Debreu setting with divisible bads.Die Arbeit untersucht das Problem der gerechten Verteilung (fair division), welches zum Ziel hat, eine Menge von Gegenständen (items) einer Menge von Akteuren (agents) \zuzuordnen". Dabei liegt der Schwerpunkt der Arbeit auf Szenarien, in denen die zu verteilenden Gegenstände entweder unteilbare Güter (indivisible goods) oder teilbare Pflichten (divisible bads) sind. Trotz ihrer praktischen Relevanz haben diese Szenarien in der Forschung bislang bedeutend weniger Aufmerksamkeit erfahren als das Szenario mit teilbaren Gütern (divisible goods). Der erste Teil der Arbeit konzentriert sich auf die gerechte Verteilung unteilbarer Güter. Unser Gerechtigkeitskriterium ist Neid-Freiheit bis auf irgendein Gut (envy- freeness up to any good, EFX). Eine Zuordnung ist EFX, wenn kein Akteur einen anderen Akteur beneidet, nachdem ein einzelnes Gut aus dem Bündel des anderen Akteurs entfernt wurde. Die Existenz von EFX-Zuordnungen ist trotz ausgeprägter Bemühungen der Forschungsgemeinschaft ungeklärt und wird gemeinhin als eine der wichtigsten offenen Fragen des Feldes angesehen. Wir unternehmen wesentliche Schritte hin zu einer Klärung dieser Frage. Erstens zeigen wir, dass wir für Akteure mit allgemeinen Bewertungsfunktionen stets eine EFX-Zuordnung finden können, bei der nur eine kleine Anzahl von Gütern unallokiert bleibt (partielle EFX-Zuordnung, almost EFX allocation). Zweitens demonstrieren wir, dass wir für Akteure mit strukturierten Bewertungsfunktionen eine partielle EFX-Zuordnung bestimmen können, die zusätzlich effizient im Sinne der Nash-Wohlfahrtsfunktion ist. Drittens beweisen wir, dass EFX-Zuordnungen für drei Akteure mit additiven Bewertungsfunktionen immer existieren. Schließlich reduzieren wir das Problem, verbesserte Garantien für EFX-Zuordnungen zu finden, auf ein neuartiges Problem in der extremalen Graphentheorie. Der zweite Teil der Arbeit widmet sich der gerechten Verteilung teilbarer Pflichten. Wie im Szenario mit teilbaren Gütern hat sich auch hier das Wettbewerbsgleichgewicht bei gleichem Einkommen (competitive equilibrium with equal incomes, CEEI) als der beste Allokationsmechanismus zur Verteilung teilbarer Pflichten erwiesen. Gleichzeitig sind weder polynomielle Algorithmen noch Schwere-Resultate für die Berechnung von CEEI mit Pflichten bekannt. Die Arbeit untersucht das Problem der Verteilung von Pflichten im klassischen Arrow-Debreu-Modell (einer Generalisierung von CEEI). Wir zeigen, dass es NP-hart ist, zu entscheiden, ob es im Arrow-Debreu-Modell mit Pflichten ein Wettbewerbsgleichgewicht gibt { im scharfen Gegensatz zum Arrow-Debreu-Modell mit Gütern. Ferner beweisen wir die Existenz eines Gleichgewichts unter der Annahme einer einfachen und natürlichen hinreichenden Bedingung. Schließlich zeigen wir, dass die Bestimmung eines Wettbewerbsgleichgewichts sogar für Eingaben, die unsere hinreichende Bedingung erfüllen, PPAD-hart ist. Damit klären wir die Komplexität des Auffindens eines Wettbewerbsgleichgewichts im Arrow-Debreu-Modell mit teilbaren Pflichten
Truthful and Fair Resource Allocation
How should we divide a good or set of goods among a set of agents? There are various constraints that we can consider. We consider two particular constraints. The first is fairness - how can we find fair allocations? The second is truthfulness - what if we do not know agents valuations for the goods being allocated? What if these valuations need to be elicited, and agents will misreport their valuations if it is beneficial? Can we design procedures that elicit agents' true valuations while preserving the quality of the allocation? We consider truthful and fair resource allocation procedures through a computational lens. We first study fair division of a heterogeneous, divisible good, colloquially known as the cake cutting problem. We depart from the existing literature and assume that agents have restricted valuations that can be succinctly communicated. We consider the problems of welfare-maximization, expressiveness, and truthfulness in cake cutting under this model. In the second part of this dissertation we consider truthfulness in settings where payments can be used to incentivize agents to truthfully reveal their private information. A mechanism asks agents to report their private preference information and computes an allocation and payments based on these reports. The mechanism design problem is to find incentive compatible mechanisms which incentivize agents to truthfully reveal their private information and simultaneously compute allocations with desirable properties. The traditional approach to mechanism design specifies mechanisms by hand and proves that certain desirable properties are satisfied. This limits the design space to mechanisms that can be written down and analyzed. We take a computational approach, giving computational procedures that produce mechanisms with desirable properties. Our first contribution designs a procedure that modifies heuristic branch and bound search and makes it usable as the allocation algorithm in an incentive compatible mechanism. Our second contribution draws a novel connection between incentive compatible mechanisms and machine learning. We use this connection to learn payment rules to pair with provided allocation rules. Our payment rules are not exactly incentive compatibility, but they minimize a measure of how much agents can gain by misreporting.Engineering and Applied Science
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On the Trade-offs between Modeling Power and Algorithmic Complexity
Mathematical modeling is a central component of operations research. Most of the academic research in our field focuses on developing algorithmic tools for solving various mathematical problems arising from our models. However, our procedure for selecting the best model to use in any particular application is ad hoc. This dissertation seeks to rigorously quantify the trade-offs between various design criteria in model construction through a series of case studies. The hope is that a better understanding of the pros and cons of different models (for the same application) can guide and improve the model selection process.
In this dissertation, we focus on two broad types of trade-offs. The first type arises naturally in mechanism or market design, a discipline that focuses on developing optimization models for complex multi-agent systems. Such systems may require satisfying multiple objectives that are potentially in conflict with one another. Hence, finding a solution that simultaneously satisfies several design requirements is challenging. The second type addresses the dynamics between model complexity and computational tractability in the context of approximation algorithms for some discrete optimization problems. The need to study this type of trade-offs is motivated by certain industry problems where the goal is to obtain the best solution within a reasonable time frame. Hence, being able to quantify and compare the degree of sub-optimality of the solution obtained under different models is helpful. Chapters 2-5 of the dissertation focus on trade-offs of the first type and Chapters 6-7 the second type
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