A Little Charity Guarantees Almost Envy-Freeness

Fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute m goods to n agents in a "fair" manner, where every agent has a valuation for each subset of goods. We assume general valuations. Envy-freeness is the most extensively studied notion of fairness. However, envy-free allocations do not always exist when goods are indivisible. The notion of fairness we consider here is "envy-freeness up to any good" (EFX) where no agent envies another agent after the removal of any single good from the other agent's bundle. It is not known if such an allocation always exists even when n = 3. We show there is always a partition of the set of goods into n + 1 subsets (X1,..., Xn, P) where for i ∈ [n], Xi is the bundle allocated to agent i and the set P is unallocated (or donated to charity) such that we have: (1) envy-freeness up to any good, (2) no agent values P higher than her own bundle, and (3) fewer than n goods go to charity, i.e., |P| n). Our proof is constructive. When agents have additive valuations and |P| is large (i.e., when |P| is close to n), our allocation also has a good maximin share (MMS) guarantee. Moreover, a minor variant of our algorithm also shows the existence of an allocation which is 4/7 groupwise maximin share (GMMS): this is a notion of fairness stronger than MMS. This improves upon the current best bound of 1/2 known for an approximate GMMS allocation

A Little Charity Guarantees Almost Envy-Freeness

Fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute $m$ goods to $n$ agents in a "fair" manner, where every agent has a valuation for each subset of goods. We assume general valuations. Envy-freeness is the most extensively studied notion of fairness. However, envy-free allocations do not always exist when goods are indivisible. The notion of fairness we consider here is "envy-freeness up to any good" (EFX) where no agent envies another agent after the removal of any single good from the other agent's bundle. It is not known if such an allocation always exists even when $n=3$. We show there is always a partition of the set of goods into $n+1$ subsets $(X_1,\ldots,X_n,P)$ where for $i \in [n]$, $X_i$ is the bundle allocated to agent $i$ and the set $P$ is unallocated (or donated to charity) such that we have$\colon$ 1) envy-freeness up to any good, 2) no agent values $P$ higher than her own bundle, and 3) fewer than $n$ goods go to charity, i.e., $|P| < n$ (typically $m \gg n$). Our proof is constructive. When agents have additive valuations and $\lvert P \rvert$ is large (i.e., when $|P|$ is close to $n$), our allocation also has a good maximin share (MMS) guarantee. Moreover, a minor variant of our algorithm also shows the existence of an allocation which is $4/7$ groupwise maximin share (GMMS): this is a notion of fairness stronger than MMS. This improves upon the current best bound of $1/2$ known for an approximate GMMS allocation

Fair and Efficient Allocations under Subadditive Valuations

We study the problem of allocating a set of indivisible goods among agents with subadditive valuations in a fair and efficient manner. Envy-Freeness up to any good (EFX) is the most compelling notion of fairness in the context of indivisible goods. Although the existence of EFX is not known beyond the simple case of two agents with subadditive valuations, some good approximations of EFX are known to exist, namely $\tfrac{1}{2}$-EFX allocation and EFX allocations with bounded charity. Nash welfare (the geometric mean of agents' valuations) is one of the most commonly used measures of efficiency. In case of additive valuations, an allocation that maximizes Nash welfare also satisfies fairness properties like Envy-Free up to one good (EF1). Although there is substantial work on approximating Nash welfare when agents have additive valuations, very little is known when agents have subadditive valuations. In this paper, we design a polynomial-time algorithm that outputs an allocation that satisfies either of the two approximations of EFX as well as achieves an $\mathcal{O}(n)$ approximation to the Nash welfare. Our result also improves the current best-known approximation of $\mathcal{O}(n \log n)$ and $\mathcal{O}(m)$ to Nash welfare when agents have submodular and subadditive valuations, respectively. Furthermore, our technique also gives an $\mathcal{O}(n)$ approximation to a family of welfare measures, $p$-mean of valuations for $p\in (-\infty, 1]$, thereby also matching asymptotically the current best known approximation ratio for special cases like $p =-\infty$ while also retaining the fairness properties

Guaranteeing Envy-Freeness under Generalized Assignment Constraints

We study fair division of goods under the broad class of generalized assignment constraints. In this constraint framework, the sizes and values of the goods are agent-specific, and one needs to allocate the goods among the agents fairly while further ensuring that each agent receives a bundle of total size at most the corresponding budget of the agent. Since, in such a constraint setting, it may not always be feasible to partition all the goods among the agents, we conform -- as in recent works -- to the construct of charity to designate the set of unassigned goods. For this allocation framework, we obtain existential and computational guarantees for envy-free (appropriately defined) allocation of divisible and indivisible goods, respectively, among agents with individual, additive valuations for the goods. We deem allocations to be fair by evaluating envy only with respect to feasible subsets. In particular, an allocation is said to be feasibly envy-free (FEF) iff each agent prefers its bundle over every (budget) feasible subset within any other agent's bundle (and within the charity). The current work establishes that, for divisible goods, FEF allocations are guaranteed to exist and can be computed efficiently under generalized assignment constraints. In the context of indivisible goods, FEF allocations do not necessarily exist, and hence, we consider the fairness notion of feasible envy-freeness up to any good (FEFx). We show that, under generalized assignment constraints, an FEFx allocation of indivisible goods always exists. In fact, our FEFx result resolves open problems posed in prior works. Further, for indivisible goods and under generalized assignment constraints, we provide a pseudo-polynomial time algorithm for computing FEFx allocations, and a fully polynomial-time approximation scheme (FPTAS) for computing approximate FEFx allocations.Comment: 29 page

EFX Allocations: Simplifications and Improvements

The existence of EFX allocations is a fundamental open problem in discretefair division. Given a set of agents and indivisible goods, the goal is todetermine the existence of an allocation where no agent envies anotherfollowing the removal of any single good from the other agent's bundle. Sincethe general problem has been illusive, progress is made on two fronts: $(i)$proving existence when the number of agents is small, $(ii)$ proving existenceof relaxations of EFX. In this paper, we improve results on both fronts (andsimplify in one of the cases). We prove the existence of EFX allocations with three agents, restricting onlyone agent to have an MMS-feasible valuation function (a strict generalizationof nice-cancelable valuation functions introduced by Berger et al. whichsubsumes additive, budget-additive and unit demand valuation functions). Theother agents may have any monotone valuation functions. Our proof technique issignificantly simpler and shorter than the proof by Chaudhury et al. onexistence of EFX allocations when there are three agents with additivevaluation functions and therefore more accessible. Secondly, we consider relaxations of EFX allocations, namely, approximate-EFXallocations and EFX allocations with few unallocated goods (charity). Chaudhuryet al. showed the existence of $(1-\epsilon)$-EFX allocation with$O((n/\epsilon)^{\frac{4}{5}})$ charity by establishing a connection to aproblem in extremal combinatorics. We improve their result and prove theexistence of $(1-\epsilon)$-EFX allocations with $\tilde{O}((n/\epsilon)^{\frac{1}{2}})$ charity. In fact, some of our techniques can be usedto prove improved upper-bounds on a problem in zero-sum combinatoricsintroduced by Alon and Krivelevich.<br

Fair Allocation of goods and chores -- Tutorial and Survey of Recent Results

Fair resource allocation is an important problem in many real-world scenarios, where resources such as goods and chores must be allocated among agents. In this survey, we delve into the intricacies of fair allocation, focusing specifically on the challenges associated with indivisible resources. We define fairness and efficiency within this context and thoroughly survey existential results, algorithms, and approximations that satisfy various fairness criteria, including envyfreeness, proportionality, MMS, and their relaxations. Additionally, we discuss algorithms that achieve fairness and efficiency, such as Pareto Optimality and Utilitarian Welfare. We also study the computational complexity of these algorithms, the likelihood of finding fair allocations, and the price of fairness for each fairness notion. We also cover mixed instances of indivisible and divisible items and investigate different valuation and allocation settings. By summarizing the state-of-the-art research, this survey provides valuable insights into fair resource allocation of indivisible goods and chores, highlighting computational complexities, fairness guarantees, and trade-offs between fairness and efficiency. It serves as a foundation for future advancements in this vital field

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