Indivisible Goods Research Articles

On Pareto optimal balanced exchanges

We investigate a market without money in which every agent offers indivisible goods in multiple copies, in exchange for goods of other agents. The exchange must be balanced in the sense that each agent should receive a quantity of good(s) equal to the one she transfers to others. We describe the market in graph-theoretic terms hence we use the notion of circulations to describe a balanced exchange of goods. Each agent has strict preferences over the agents from which she will receive goods and an upper bound on the quantity of each transaction, while a positive integer weight reflects the social importance of each unit exchanged. In this paper, we propose a simple variant of the Top Trading Cycles mechanism that finds a Pareto optimal circulation. We then offer necessary and sufficient conditions for a circulation to be Pareto optimal and, as a consequence, a easy recognition procedure. Last, we show that finding a maximum weight Pareto optimal circulation is NP-hard but becomes polynomial if weights are concordant with preferences.

EFX Exists for Three Agents

We study the problem of distributing a set of indivisible goods among agents with additive valuations in a fair manner. The fairness notion under consideration is envy-freeness up to any good (EFX). Despite significant efforts by many researchers for several years, the existence of EFX allocations has not been settled beyond the simple case of two agents. In this article, we show constructively that an EFX allocation always exists for three agents. Furthermore, we falsify the conjecture of Caragiannis et al. by showing an instance with three agents for which there is a partial EFX allocation (some goods are not allocated) with higher Nash welfare than that of any complete EFX allocation.

A fair and truthful mechanism with limited subsidy

The notion of envy-freeness is a natural and intuitive fairness requirement in resource allocation. With indivisible goods, such fair allocations are not guaranteed to exist. Classical works have avoided this issue by introducing an additional divisible resource, i.e., money. In this paper, we aim to design a truthful allocation mechanism of indivisible goods to achieve fairness and efficiency criteria with a limited amount of subsidy. Following the work of Halpern and Shah, our central question is as follows: to what extent do we need to rely on the power of money to accomplish these objectives? We show that, when agents have matroidal valuations, there is a truthful allocation mechanism that achieves envy-freeness and utilitarian optimality by subsidizing each agent with at most 1, the maximum marginal contribution of each item for each agent. The design of the mechanism rests crucially on the underlying matroidal M-convexity of the Lorenz dominating allocations.

Improved maximin guarantees for subadditive and fractionally subadditive fair allocation problem

In this work, we study the maximin share fairness notion (MMS) for allocation of indivisible goods in the subadditive and fractionally subadditive settings. While previous work refutes the possibility of obtaining an allocation which is better than 1/2-MMS, the only positive result for the subadditive setting states that when the number of items is equal to m, there always exists an Ω(1/log⁡m)-MMS allocation. Since the number of items may be larger than the number of agents (n), such a bound can only imply a weak bound of Ω(1nlog⁡n)-MMS allocation in general.In this work, we improve this bound exponentially to Ω(1log⁡nlog⁡log⁡n)-MMS guarantee. In addition to this, we prove that when the valuation functions are fractionally subadditive, a 0.2192235-MMS allocation is guaranteed to exist. This also improves upon the previous bound of 1/5-MMS guarantee for the fractionally subadditive setting.

Allocating Indivisible Goods to Strategic Agents: Pure Nash Equilibria and Fairness

We consider the problem of fairly allocating a set of indivisible goods to a set of strategic agents with additive valuation functions. We assume no monetary transfers, and therefore, a mechanism in our setting is an algorithm that takes as input the reported—rather than the true—values of the agents. Our main goal is to explore whether there exist mechanisms that have pure Nash equilibria for every instance and, at the same time, provide fairness guarantees for the allocations that correspond to these equilibria. We focus on two relaxations of envy-freeness, namely, envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX), and we positively answer the preceding question. In particular, we study two algorithms that are known to produce such allocations in the nonstrategic setting: round-robin (EF1 allocations for any number of agents) and a cut-and-choose algorithm of Plaut and Roughgarden (EFX allocations for two agents). For round-robin, we show that all of its pure Nash equilibria induce allocations that are EF1 with respect to the underlying true values, whereas for the algorithm of Plaut and Roughgarden, we show that the corresponding allocations not only are EFX, but also satisfy maximin share fairness, something that is not true for this algorithm in the nonstrategic setting! Further, we show that a weaker version of the latter result holds for any mechanism for two agents that always has pure Nash equilibria, which all induce EFX allocations. Funding: This work was supported by the Horizon 2020 European Research Council Advanced “Algorithmic and Mechanism Design Research in Online Markets” [Grant 788893], the Ministero dell’Università e della Ricerca Research project of national interest (PRIN) “Algorithms, Games, and Digital Markets,” the Future Artificial Intelligence Research project funded by the NextGenerationEU program within the National Recovery and Resilience Plan (PNRR-PE-AI) scheme [M4C2, investment 1.3, line on Artificial Intelligence], the National Recovery and Resilience Plan-Ministero dell’Università e della Ricerca (PNRR-MUR) project IR0000013-SoBigData.it, the Nederlandse Organisatie voor Wetenschappelijk Onderzoek Veni Project [Grant VI.Veni.192.153], and the National Recovery and Resilience Plan Greece 2.0 funded by the European Union under the NextGenerationEU Program [Grant MIS 5154714].

Improving Envy Freeness up to Any Good Guarantees Through Rainbow Cycle Number

We study the problem of fairly allocating a set of indivisible goods among n agents with additive valuations. Envy freeness up to any good (EFX) is arguably the most compelling fairness notion in this context. However, the existence of an EFX allocation has not been settled and is one of the most important problems in fair division. Toward resolving this question, many impressive results show the existence of its relaxations. In particular, it is known that 0.618-EFX allocations exist and that EFX allocation exists if we do not allocate at most (n-1) goods. Reducing the number of unallocated goods has emerged as a systematic way to tackle the main question. For example, follow-up works on three- and four-agents cases, respectively, allocated two more unallocated goods through an involved procedure. In this paper, we study the general case and achieve sublinear numbers of unallocated goods. Through a new approach, we show that for every [Formula: see text], there always exists a [Formula: see text]-EFX allocation with sublinear number of unallocated goods and high Nash welfare. For this, we reduce the EFX problem to a novel problem in extremal graph theory. We define the notion of rainbow cycle number [Formula: see text] in directed graphs. For all [Formula: see text] is the largest k such that there exists a k-partite graph [Formula: see text], in which each part has at most d vertices (i.e., [Formula: see text] for all [Formula: see text]); for any two parts Vi and Vj, each vertex in Vi has an incoming edge from some vertex in Vj and vice versa; and there exists no cycle in G that contains at most one vertex from each part. We show that any upper bound on [Formula: see text] directly translates to a sublinear bound on the number of unallocated goods. We establish a polynomial upper bound on [Formula: see text], yielding our main result. Furthermore, our approach is constructive, which also gives a polynomial-time algorithm for finding such an allocation. Funding: J. Garg was supported by the Directorate for Computer and Information Science and Engineering [Grant CCF-1942321]. R. Mehta was supported by the Directorate for Computer and Information Science and Engineering [Grant CCF-1750436].

Fair-Share Allocations for Agents with Arbitrary Entitlements

We consider the problem of fair allocation of indivisible goods to n agents with no transfers. When agents have equal entitlements, the well-established notion of the maximin share (MMS) serves as an attractive fairness criterion for which, to qualify as fair, an allocation needs to give every agent at least a substantial fraction of the agent’s MMS. In this paper, we consider the case of arbitrary (unequal) entitlements. We explain shortcomings in previous attempts that extend the MMS to unequal entitlements. Our conceptual contribution is the introduction of a new notion of a share, the AnyPrice share (APS), that is appropriate for settings with arbitrary entitlements. Even for the equal entitlements case, this notion is new and satisfies [Formula: see text], for which the inequality is sometimes strict. We present two equivalent definitions for the APS (one as a minimization problem, the other as a maximization problem) and provide comparisons between the APS and previous notions of fairness. Our main result concerns additive valuations and arbitrary entitlements, for which we provide a polynomial-time algorithm that gives every agent at least a [Formula: see text] - fraction of the agent’s APS. This algorithm can also be viewed as providing strategies in a certain natural bidding game, and these strategies secure each agent at least a [Formula: see text] - fraction of the agent’s APS. Funding: T. Ezra’s research is partially supported by the European Research Council Advanced [Grant 788893] AMDROMA “Algorithmic and Mechanism Design Research in Online Markets” and MIUR PRIN project ALGADIMAR “Algorithms, Games, and Digital Markets.” U. Feige’s research is supported in part by the Israel Science Foundation [Grant 1122/22].

Exchange of indivisible goods under matroid constraints

We study the problem of reallocating indivisible goods among a set of strategic agents by generalizing the original Sharpley-Scarf market to the setting where an object can be allocated to multiple agents but subject to an associated matroid constraint. We refer to this modified market as the matroid Sharpley-Scarf market. For this general market, we present a Pareto-optimal, individually rational and group strategy-proof mechanism. Our mechanism is simple and natural generalization of the Top Trading Cycle mechanism and the Serial Dictator mechanism. In our analysis, we demonstrate that the weak core may be absent in the matroid Sharpley-Scarf market, while a relaxed version called the constrained core does exist. Furthermore, we extend the concept called competitive equilibrium to the matroid Sharpley-Scarf market and show that an allocation is a competitive equilibrium if and only if it can be produced by our mechanism.

Weighted fair division with matroid-rank valuations: Monotonicity and strategyproofness

We study the problem of fairly allocating indivisible goods to agents with weights corresponding to their entitlements. Previous work has shown that, when agents have binary additive valuations, the maximum weighted Nash welfare rule is resource-, population-, and weight-monotone, satisfies group-strategyproofness, and can be implemented in polynomial time. We generalize these results to the class of weighted additive welfarist rules with concave functions and agents with matroid-rank (also known as binary submodular) valuations.

Simultaneous eating algorithm and greedy algorithm in assignment problems

The simultaneous eating algorithm (SEA) and probabilistic serial (PS) mechanism are well known for allocating a set of divisible or indivisible goods to agents with ordinal preferences. The PS mechanism is SEA at the same eating speed. The prominent property of SEA is ordinal efficiency. Recently, we extended the PS mechanism (EPS) from a fixed quota of each good to a variable varying in a polytope constrained by submodular functions. In this article, we further generalized some results on SEA. After formalizing the extended ESA (ESEA), we show that it can be characterized by ordinal efficiency. We established a stronger summation optimization than the Pareto type of ordinal efficiency by an introduced weight vector. The weights in the summation optimization coincide with agents’ preferences at the acyclic positive values of an allocation. Hence, the order of goods selected to eat in ESEA is exactly the one chosen in the execution of the well-known greedy algorithm.

Finding Fair Allocations under Budget Constraints

We study the fair allocation of indivisible goods among agents with identical, additive valuations but individual budget constraints. Here, the indivisible goods--each with a specific size and value--need to be allocated such that the bundle assigned to each agent is of total size at most the agent's budget. Since envy-free allocations do not necessarily exist in the indivisible goods context, compelling relaxations--in particular, the notion of envy-freeness up to k goods (EFk)--have received significant attention in recent years. In an EFk allocation, each agent prefers its own bundle over that of any other agent, up to the removal of k goods, and the agents have similarly bounded envy against the charity (which corresponds to the set of all unallocated goods). It has been shown in prior work that an allocation that satisfies the budget constraints and maximizes the Nash social welfare is 1/4-approximately EF1. However, the computation (or even existence) of exact EFk allocations remained an intriguing open problem. We make notable progress towards this by proposing a simple, greedy, polynomial-time algorithm that computes EF2 allocations under budget constraints. Our algorithmic result implies the universal existence of EF2 allocations in this fair division context. The analysis of the algorithm exploits intricate structural properties of envy-freeness. Interestingly, the same algorithm also provides EF1 guarantees for important special cases. Specifically, we settle the existence of EF1 allocations for instances in which: (i) the value of each good is proportional to its size, (ii) all the goods have the same size, or (iii) all the goods have the same value. Our EF2 result even extends to the setting wherein the goods' sizes are agent specific.

How to Cut a Discrete Cake Fairly

Cake-cutting is a fundamental model of dividing a heterogeneous resource, such as land, broadcast time, and advertisement space. In this study, we consider the problem of dividing indivisible goods fairly under the connectivity constraints of a path. We prove that a connected division of indivisible items satisfying a discrete counterpart of envy-freeness, called envy-freeness up to one good (EF1), always exists for any number of agents n with monotone valuations. Our result settles an open question raised by Bilò et al. (2019), who proved that an EF1 connected division always exists for four agents with monotone valuations. Moreover, the proof can be extended to show the following (1) ``secretive" and (2) ``extra" versions: (1) for n agents with monotone valuations, the path can be divided into n connected bundles such that an EF1 assignment of the remaining bundles can be made to the other agents for any selection made by the “secretive agent”; (2) for n+1 agents with monotone valuations, the path can be divided into n connected bundles such that when any ``extra agent” leaves, an EF1 assignment of the bundles can be made to the remaining agents.

Computing fair and efficient allocations with few utility values

We study the problem of allocating indivisible goods among agents with additive valuations in a fair and efficient manner, when agents have few utility values for the goods. We consider the compelling fairness notion of envy-freeness up to any good (EFX) in conjunction with Pareto-optimality (PO). Amanatidis et al. [1] showed that when there are at most two utility values, an EFX allocation can be computed in polynomial-time. We improve this result by showing that for such instances an allocation that is EFX and PO can be computed in polynomial-time. This is the first class apart from identical or binary valuations, for which EFX+PO allocations are shown to exist and are polynomial-time computable. In contrast, we show that when there are three utility values, EFX+PO allocations need not exist, and even deciding if EFX+PO allocations exist is NP-hard.Our techniques allow us to obtain similar results for the fairness notion of equitability up to any good (EQX) together with PO. We show that for instances with two positive values an EQX+PO allocation can be computed in polynomial-time, and deciding if an EQX+PO allocation exists is NP-hard when there are three utility values.We also study the problem of maximizing Nash welfare (MNW), and show that our EFX+PO algorithm returns an allocation that approximates the MNW to a factor of 1.061 for two valued instances, in addition to being EFX+PO. In contrast, we show that for three valued instances, computing an MNW allocation is APX-hard. Finally, we give a polynomial-time algorithm for computing an MNW allocation for two-valued instances where the ratio of the two values is greater than a certain threshold.

On Fair Division under Heterogeneous Matroid Constraints

We study fair allocation of indivisible goods among additive agents with feasibility constraints. In these settings, every agent is restricted to get a bundle among a specified set of feasible bundles. Such scenarios have been of great interest to the AI community due to their applicability to real-world problems. Following some impossibility results, we restrict attention to matroid feasibility constraints that capture natural scenarios, such as the allocation of shifts to medical doctors and the allocation of conference papers to referees. We focus on the common fairness notion of envy-freeness up to one good (EF1). Previous algorithms for finding EF1 allocations are either restricted to agents with identical feasibility constraints or allow free disposal of items. An open problem is the existence of EF1 complete allocations among agents who differ both in their valuations and in their feasibility constraints. In this work, we make progress on this problem by providing positive and negative results for several matroid and valuation types. Among other results, we devise polynomial-time algorithms for finding EF1 allocations in the following settings: (i) n agents with heterogeneous (non-identical) binary valuations and partition matroids with heterogeneous capacities; (ii) two agents with heterogeneous additive valuations and partition matroids with heterogeneous capacities; and (iii) three agents with heterogeneous binary valuations and identical base-orderable matroid constraints.

Extending the characterization of maximum Nash welfare

In the allocation of indivisible goods, the maximum Nash welfare rule has recently been characterized as the only rule within the class of additive welfarist rules that satisfies envy-freeness up to one good. We extend this characterization to the class of all welfarist rules.

Sales-Based Rebate Design

In this paper, we study a family of sales-based rebate mechanisms as an effective tool to implement price discrimination across a population of buyers with correlated heterogeneous valuations on indivisible goods and services. In order to implement such sales-based rebate mechanisms, the seller charges each buyer a fixed price at the time of purchase contingent on a rebate that is a function of the ex post sales volume to be realized at the end of the sales period. The seller declares both a price and a menu of rebates as a function of sales. We show that, when there is a common component of uncertainty in consumers’ valuations (to which we refer as the quality of the product), such rebates enable a seller to effectively induce different expected net prices at different valuations. Importantly, this effective price discrimination over valuations is achieved keeping both the base price and the rebate uniform across all buyers. This uniformity of price and rebate across buyers is a key advantage of our proposed rebate mechanism, thereby providing a new mechanism for price discrimination in crowd-based markets. We use tools and techniques from game theory and variational optimization to provide insight into the economics of such mechanisms. In particular, we identify two mechanisms that are monotone functions of the sales volume that are easy to implement in practice and perform well when compared with the much more complex optimal mechanism. We provide a rigorous analysis of the optimal mechanism and discuss practical limitations in implementing the globally optimal design, further demonstrating the efficacy of our proposed monotone mechanisms. This paper was accepted by Gabriel Weintraub, revenue management and market analytics. Funding: This research was supported by a Vannevar Bush Fellowship from the Office of Secretary of State and Army Research Office [MURI Project W911NF-19-1-0217]. Supplemental Material: The data files are available at https://doi.org/10.1287/mnsc.2023.4691 .

Best of Both Worlds: Ex Ante and Ex Post Fairness in Resource Allocation

Consider the problem of allocating indivisible goods among agents with additive valuations, where monetary payments are not allowed. When randomization is allowed, it is possible to achieve compelling notions of fairness such as EV, which states that no agent should prefer any other agent's allocation to their own. When allocations must be deterministic, achieving exact fairness is impossible but approximate notions such as EV up to one good can be guaranteed. In “Best of Both Worlds: Ex Ante and Ex Post Fairness in Resource Allocation,” H. Aziz, R. Freeman, N. Shah, and R. Vaish ask whether it is possible to achieve both types of guarantees simultaneously. More specifically, they ask whether there exists a probability distribution over deterministic allocations such that every deterministic allocation is envy-free up to one good and the distribution is exactly envy-free in expectation. The main result of the paper answers this question in the affirmative, showing that ex ante and ex post fairness need not be in conflict.

A characterization of maximum Nash welfare for indivisible goods

In the allocation of indivisible goods, the maximum Nash welfare (MNW) rule, which chooses an allocation maximizing the product of the agents' utilities, has received substantial attention for its fairness. We characterize MNW as the only additive welfarist rule that satisfies envy-freeness up to one good. Our characterization holds even in the simplest setting of two agents.

Equitability and welfare maximization for allocating indivisible items

We study fair allocations of indivisible goods and chores in conjunction with system efficiency, measured by two social welfare functions, namely utilitarian and egalitarian welfare. To model preference, each agent is associated with a cardinal and additive valuation function. The fairness criteria we are concerned with are equitability up to any item (EQX) and equitability up to one item (EQ1). For the trade-off between fairness and efficiency, we investigate efficiency loss under these fairness constraints and establish the price of fairness. From the computational perspective, we provide a complete picture of the computational complexity of (i) deciding the existence of an EQX/EQ1 and welfare-maximizing allocation; (ii) computing a welfare maximizer among all EQX/EQ1 allocations.

Outside options in neutral allocation of discrete resources

Serial dictatorships have emerged as the canonical simple mechanisms in the literature on the allocation of indivisible goods without transfers. They are the only neutral and group-strategy-proof mechanisms in environments in which agents have no outside options and hence no individual rationality constraints (Svensson in Soc Choice Welfare 16:557–567, 1999). Accounting for outside options and individual rationality constraints, our main result constructs the class of group-strategy-proof, neutral, and non-wasteful mechanisms. These mechanisms are also Pareto efficient and we call them binary serial dictatorships. The abundance of the outside option—anybody who wants can opt out to get it—is crucial for our result.