Finding Fair and Efficient Allocations

We study the problem of allocating a set of indivisible goods among a set of agents in a fair and efficient manner. An allocation is said to be fair if it is envy-free up to one good (EF1), which means that each agent prefers its own bundle over the bundle of any other agent up to the removal of one good. In addition, an allocation is deemed efficient if it satisfies Pareto optimality (PO). While each of these well-studied properties is easy to achieve separately, achieving them together is far from obvious. Recently, Caragiannis et al. (2016) established the surprising result that when agents have additive valuations for the goods, there always exists an allocation that simultaneously satisfies these two seemingly incompatible properties. Specifically, they showed that an allocation that maximizes the Nash social welfare (NSW) objective is both EF1 and PO. However, the problem of maximizing NSW is NP-hard. As a result, this approach does not provide an efficient algorithm for finding a fair and efficient allocation. In this paper, we bypass this barrier, and develop a pseudopolynomial time algorithm for finding allocations that are EF1 and PO; in particular, when the valuations are bounded, our algorithm finds such an allocation in polynomial time. Furthermore, we establish a stronger existence result compared to Caragiannis et al. (2016): For additive valuations, there always exists an allocation that is EF1 and fractionally PO. Another contribution of our work is to show that our algorithm provides a polynomial-time 1.45-approximation to the NSW objective. This improves upon the best known approximation ratio for this problem (namely, the 2-approximation algorithm of Cole et al. (2017)). Unlike many of the existing approaches, our algorithm is completely combinatorial.Comment: 40 pages. Updated versio

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

Fair Division of a Graph

We consider fair allocation of indivisible items under an additional constraint: there is an undirected graph describing the relationship between the items, and each agent's share must form a connected subgraph of this graph. This framework captures, e.g., fair allocation of land plots, where the graph describes the accessibility relation among the plots. We focus on agents that have additive utilities for the items, and consider several common fair division solution concepts, such as proportionality, envy-freeness and maximin share guarantee. While finding good allocations according to these solution concepts is computationally hard in general, we design efficient algorithms for special cases where the underlying graph has simple structure, and/or the number of agents -or, less restrictively, the number of agent types- is small. In particular, despite non-existence results in the general case, we prove that for acyclic graphs a maximin share allocation always exists and can be found efficiently.Comment: 9 pages, long version of accepted IJCAI-17 pape

High-Multiplicity Fair Allocation Using Parametric Integer Linear Programming

Using insights from parametric integer linear programming, we significantly improve on our previous work [Proc. ACM EC 2019] on high-multiplicity fair allocation. Therein, answering an open question from previous work, we proved that the problem of finding envy-free Pareto-efficient allocations of indivisible items is fixed-parameter tractable with respect to the combined parameter "number of agents" plus "number of item types." Our central improvement, compared to this result, is to break the condition that the corresponding utility and multiplicity values have to be encoded in unary required there. Concretely, we show that, while preserving fixed-parameter tractability, these values can be encoded in binary, thus greatly expanding the range of feasible values.Comment: 15 pages; Published in the Proceedings of ECAI-202

On the Proximity of Markets with Integral Equilibria

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, 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; here fairness refers to envy-freeness 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.Comment: 17 page

Weighted Proportional Allocations of Indivisible Goods and Chores: Insights via Matchings

We study the fair allocation of indivisible goods and chores under ordinal valuations for agents with unequal entitlements. We show the existence and polynomial time computation of weighted necessarily proportional up to one item (WSD-PROP1) allocations for both goods and chores, by reducing it to a problem of finding perfect matchings in a bipartite graph. We give a complete characterization of these allocations as corner points of a perfect matching polytope. Using this polytope, we can optimize over all allocations to find a min-cost WSD-PROP1 allocation of goods or most efficient WSD-PROP1 allocation of chores. Additionally, we show the existence and computation of sequencible (SEQ) WSD-PROP1 allocations by using rank-maximal perfect matching algorithms and show incompatibility of Pareto optimality under all valuations and WSD-PROP1. We also consider the Best-of-Both-Worlds (BoBW) fairness notion. By using our characterization, we show the existence and polynomial time computation of Ex-ante envy free (WSD-EF) and Ex-post WSD-PROP1 allocations under ordinal valuations for both chores and goods.Comment: Accepted at AAMAS 202

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