Indivisible Goods Research Articles

The (Exact) Price of Cardinality for Indivisible Goods: A Parametric Perspective

We adopt a parametric approach to analyze the worst-case degradation in social welfare when the allocation of indivisible goods is constrained to be fair. Specifically, we are concerned with cardinality-constrained allocations, which require that each agent has at most k items in their allocated bundle. We propose the notion of the price of cardinality, which captures the worst-case multiplicative loss of utilitarian or egalitarian social welfare resulting from imposing the cardinality constraint. We then characterize tight or almost-tight bounds on the price of cardinality as exact functions of the instance parameters, demonstrating how the social welfare improves as k is increased. In particular, one of our main results refines and generalizes the existing asymptotic bound of Θ(√n) on the price of balancedness. We also further extend our analysis to the problem where the items are partitioned into disjoint categories, and each category has its own cardinality constraint. Through a parametric study of the price of cardinality, we provide a framework which aids decision makers in choosing an ideal level of cardinality-based fairness, using their knowledge of the potential loss of utilitarian and egalitarian social welfare.

Fair Division with Market Values

We introduce a model of fair division with market values, where indivisible goods must be partitioned among agents with (additive) subjective valuations, and each good additionally has a market value. The market valuation can be viewed as a separate additive valuation that holds identically across all the agents. We seek allocations that are simultaneously fair with respect to the subjective valuations and under the market valuation. We show that an allocation that satisfies stochastically-dominant envy-freeness up to one good (SD-EF1) with respect to both the subjective valuations and the market valuation does not always exist, but the weaker guarantee of EF1 with respect to the subjective valuations along with SD-EF1 with respect to the market valuation can be guaranteed. We also study a number of other guarantees such as Pareto optimality, EFX, and MMS. In addition, we explore non-additive valuations and extend our model to cake-cutting. Along the way, we identify several tantalizing open questions.

(Almost Full) EFX for Three (and More) Types of Agents

We study the problem of determining an envy-free allocation of indivisible goods among multiple agents with additive valuations. EFX, which stands for envy-freeness up to any good, is a well-studied relaxation of the envy-free allocation problem and has been shown to exist for specific scenarios. EFX is known to exist for three agents, and for any number of agents when there are only two types of valuations. EFX allocations are also known to exist for four agents with at most one good unallocated. In this paper, we show that EFX exists with at most k-2 goods unallocated for any number of agents having k distinct valuations. Additionally, we show that complete EFX allocations exist when all but two agents have identical valuations.

Understanding EFX Allocations: Counting and Variants

Envy-freeness up to any good (EFX) is a popular and important fairness property in the fair allocation of indivisible goods, of which its existence in general is still an open question. In this work, we investigate the problem of determining the minimum number of EFX allocations for a given instance, arguing that this approach may yield valuable insights into the existence and computation of EFX allocations. We focus on restricted instances where the number of goods slightly exceeds the number of agents, and extend our analysis to weighted EFX (WEFX) and a novel variant of EFX for general monotone valuations, termed EFX+. In doing so, we identify the transition threshold for the existence of allocations satisfying these fairness notions. Notably, we resolve open problems regarding WEFX by proving polynomial-time computability under binary additive valuations, and establishing the first constant-factor approximation for two agents.

Achieving Maximin Share and EFX/EF1 Guarantees Simultaneously

We study the problem of computing fair divisions of a set of indivisible goods among agents with additive valuations. For the past many decades, the literature has explored various notions of fairness, that can be primarily seen as either having envy-based or share-based lens. For the discrete setting of resource-allocation problems, envy-free up to any good (EFX) and maximin share (MMS) are widely considered as the flag-bearers of fairness notions in the above two categories, thereby capturing different aspects of fairness herein. Due to lack of existence results of these notions and the fact that a good approximation of EFX or MMS does not imply particularly strong guarantees of the other, it becomes important to understand the compatibility of EFX and MMS allocations with one another. In this work, we identify a novel way to simultaneously achieve MMS guarantees with EFX/EF1 notions of fairness, while beating the best known approximation factors by Chaudhury et al. and Amanatidis et al. Our main contribution is to constructively prove the existence of (i) a partial allocation that is both 2/3-MMS and EFX, and (ii) a complete allocation that is both 2/3-MMS and EF1. Our algorithms run in pseudo-polynomial time if the approximation factor for MMS is relaxed to 2/3 - e for any constant e>0 and in polynomial time if, in addition, the EFX (or EF1) guarantee is relaxed to (1-d)-EFX (or (1-d)-EF1) for any constant d>0. In particular, we improve from the best approximation factor known prior to our work by Chaudhury et al., which computes partial allocations that are 1/2-MMS and EFX in pseudo-polynomial time.

Reducing Leximin Fairness to Utilitarian Optimization

Two prominent objectives in social choice are utilitarian - maximizing the sum of agents' utilities, and leximin - maximizing the smallest agent's utility, then the second-smallest, etc. Utilitarianism is typically computationally easier to attain but is generally viewed as less fair. This paper presents a general reduction scheme that, given a utilitarian solver, produces a distribution over states (deterministic outcomes) that is leximin in expectation. Importantly, the scheme is robust in the sense that, given an approximate utilitarian solver, it produces a lottery that is approximately-leximin (in expectation) - with the same approximation factor. We apply our scheme to several social choice problems: stochastic allocations of indivisible goods, giveaway lotteries, and fair lotteries for participatory budgeting.

Constrained Fair and Efficient Allocations

Fairness and efficiency have become the pillars of modern fair division research, but prior work on achieving both simultaneously is largely limited to the unconstrained setting. We study fair and efficient allocations of indivisible goods under additive valuations and various types of allocation feasibility constraints, and demonstrate the unreasonable effectiveness of the maximum Nash welfare (MNW) solution in this previously uncharted territory. Our main result is that MNW allocations are 1/2-envy-free up to one good (EF1) and Pareto optimal under the broad family of (arbitrary) matroid constraints. We extend these guarantees to complete MNW allocations for base-orderable matroid constraints, and to a family of non-matroid constraints (which includes balancedness). We establish tightness of our results by providing counterexamples for the satisfiability of certain stronger desiderata, but show an improved result for the special case of goods with copies (Gafni et al. 2023). Finally, we also establish novel best-of-both-worlds guarantees for goods with copies and balancedness.

Fair and Efficient Completion of Indivisible Goods

We formulate the problem of fair and efficient completion of indivisible goods, defined as follows: Given a partial allocation of indivisible goods among agents, does there exist an allocation of the remaining goods (i.e., a completion) that satisfies fairness and economic efficiency guarantees of interest? We study the computational complexity of the completion problem for prominent fairness and efficiency notions such as envy-freeness up to one good (EF1), proportionality up to one good (Prop1), maximin share (MMS), and Pareto optimality (PO), and focus on the class of additive valuations as well as its subclasses such as binary additive and lexicographic valuations. We find that while the completion problem is significantly harder than the standard fair division problem (wherein the initial partial allocation is empty), the consideration of restricted preferences facilitates positive algorithmic results for threshold-based fairness notions (Prop1 and MMS). On the other hand, the completion problem remains computationally intractable for envy-based notions such as EF1 and EF1+PO even under restricted preferences.

Fair Division with Social Impact

In this paper, we consider the problem of fair division of indivisible goods, where the allocation of goods impacts society. Specifically, we introduce a second valuation function for each agent, which determines the social impact of allocating a good to the agent. Such impact is considered desirable for the society -- the higher, the better. Our goal is to understand how to allocate goods fairly from the agents' perspective while maintaining society as happy as possible. To this end, we measure the impact on society using the utilitarian social welfare, and provide both possibility and impossibility results. Our findings reveal that achieving good approximations, better than linear in the number of agents, is not possible while ensuring fairness to the agents. These impossibility results can be attributed to the fact that agents are completely unconscious of their social impact. Consequently, we explore scenarios where agents are socially aware, by introducing related fairness notions, and demonstrate that an appropriate definition of fairness is compatible with the social objective.

EF2X Exists for Four Agents

We study the fair allocation of indivisible goods among a group of agents, aiming to limit the envy between any two agents. The central open problem in this literature, which has proven to be extremely challenging, is regarding the existence of an EFX allocation, i.e., an allocation such that any envy from some agent i toward another agent j would vanish if we were to remove any single good from the bundle allocated to j. Prior work has shown that when the agents’ valuations are additive, which has been the main focus of prior works, an EFX allocation is guaranteed to exist for all instances involving up to three agents. Subsequent work extended this guarantee to more general valuations, like nice-cancelable and MMS-feasible. However, the existence of EFX allocations for instances involving four agents remains open, even for additive valuations. We contribute to this literature by focusing on EF2X, a relaxation of EFX which requires that any envy toward some agent would vanish if any two of the goods allocated to that agent were to be removed. Our main result shows that EF2X allocations exist for any instance with four agents, even for the class of cancelable valuations, which is more general than additive. Our proof is constructive, proposing an algorithm that computes such an allocation in pseudo-polynomial time. Furthermore, for instances involving three agents we provide an algorithm that computes an EF2X allocation in polynomial time, in contrast to EFX for which the fastest known algorithm for three agents is only pseudo-polynomial.

The Complexity of Extending Fair Allocations of Indivisible Goods

We initiate the study of computing envy-free allocations of indivisible items in the extension setting, i.e., when some part of the allocation is fixed and the task is to allocate the remaining items. In view of the NP-hardness of the problem, we investigate whether - and under which conditions - one can obtain fixed-parameter algorithms for computing a solution in settings where most of the allocation is already fixed. Our results provide a broad complexity-theoretic classification of the problem which includes: (a) fixed-parameter algorithms tailored to settings with few distinct types of agents or items; (b) lower bounds which exclude the generalization of these positive results to more general settings. We conclude by showing that - unlike when computing allocations from scratch - the non-algorithmic question of whether more relaxed EF1 or EFX allocations exist can be completely resolved in the extension setting.

A General Framework for Fair Allocation under Matroid Rank Valuations

We study the problem of fairly allocating a set of indivisible goods among agents with matroid rank valuations: every good provides a marginal value of 0 or 1 when added to a bundle and valuations are submodular. We present a simple algorithmic framework, called General Yankee Swap , that can efficiently compute allocations that maximize any justice criterion (or fairness objective) satisfying some mild assumptions. Along with maximizing a justice criterion, General Yankee Swap is guaranteed to maximize utilitarian social welfare, ensure strategyproofness and use at most a quadratic number of valuation queries. We show how General Yankee Swap can be used to compute allocations for five different well-studied justice criteria: (a) Prioritized Lorenz dominance, (b) Maximin fairness, (c) Weighted leximin, (d) Max weighted Nash welfare, and (e) Max weighted p -mean welfare. In particular, this framework provides the first polynomial time algorithms to compute weighted leximin, max weighted Nash welfare and max weighted p -mean welfare allocations for agents with matroid rank valuations. We also extend this framework to the setting of binary chores — items with marginal values -1 or 0 — and similarly show that it can be used to maximize any justice criteria satisfying some mild assumptions.

Proportional Allocation of Indivisible Goods up to the Least Valued Good on Average

Proportional Allocation of Indivisible Goods up to the Least Valued Good on Average

Polyhedral Clinching Auctions for Indivisible Goods

Polyhedral Clinching Auctions for Indivisible Goods

On Hill’s Worst-Case Guarantee for Indivisible Bads

We evaluate the fairness of a rule allocating among agents with equal rights by computing their utility for a hypothetical worst-case share, which only depends on their own valuation and the number of agents. For indivisible goods, Budish proposed the maximin share : the least utility of a bundle in the best partition of the objects; unfortunately maximin share is not always satisfiable. Earlier, Hill proposed the worst maximin share over all utilities with the same largest possible single-object value. More conservative than maximin share, it is guaranteed to be satisfiable for any possible profile of utilities and its computation is elementary, whereas it is NP-hard to compute maximin share. We apply Hill’s approach to the allocation of indivisible bads (objects with disutilities) and compute in closed form the worst-case minimax share for a given value of the worst single bad. We show that the worst-case minimax share is close to the original minimax share and that its monotonic closure is the best guaranteed share for all allocation instances with a common upper bound on the value of the worst single bad.

Indivisible goods, equilibrium and convex economies

Indivisible goods, equilibrium and convex economies

Round-Robin Beyond Additive Agents: Existence and Fairness of Approximate Equilibria

Fair allocation of indivisible goods has attracted extensive attention over the last two decades, yielding numerous elegant algorithmic results and producing challenging open questions. The problem becomes much harder in the presence of strategic agents. Ideally, one would want to design truthful mechanisms that produce allocations with fairness guarantees. However, in the standard setting without monetary transfers, it is generally impossible to have truthful mechanisms that provide nontrivial fairness guarantees. Recently, Amanatidis et al. [Amanatidis G, Birmpas G, Fusco F, Lazos P, Leonardi S, Reiffenhäuser R (2023) Allocating indivisible goods to strategic agents: Pure Nash equilibria and fairness. Math. Oper. Res., ePub ahead of print November 30, https://doi.org/10.1287/moor.2022.0058 ] suggested the study of mechanisms that produce fair allocations in their equilibria. Specifically, when the agents have additive valuation functions, the simple Round-Robin algorithm always has pure Nash equilibria, and the corresponding allocations are envy-free up to one good (EF1) with respect to the agents’ true valuation functions. Following this agenda, we show that this outstanding property of the Round-Robin mechanism extends much beyond the above default assumption of additivity. In particular, we prove that for agents with cancelable valuation functions (a natural class that contains, e.g., additive and budget-additive functions), this simple mechanism always has equilibria, and even its approximate equilibria correspond to approximately EF1 allocations with respect to the agents’ true valuation functions. Furthermore, we show that the approximate EF1 fairness of approximate equilibria surprisingly holds for the important class of submodular valuation functions as well, even though exact equilibria fail to exist. Funding: This work was supported by the Horizon 2020 European Research Council Advanced Grant “AMDROMA: Algorithmic and Mechanism Design Research in Online Markets” [Grant 788893], the Ministero dell’Università e della Ricerca Research project of national interest (PRIN) “ALGADIMAR: Algorithms, Games, and Digital Markets,” the Nederlandse Organisatie voor Wetenschappelijk Onderzoek Veni Project “Algorithmic Fair Division in Dynamic, Socially Constrained Environments” [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].

Reachability of Fair Allocations via Sequential Exchanges

In the allocation of indivisible goods, a prominent fairness notion is envy-freeness up to one good (EF1). We initiate the study of reachability problems in fair division by investigating the problem of whether one EF1 allocation can be reached from another EF1 allocation via a sequence of exchanges such that every intermediate allocation is also EF1. We show that two EF1 allocations may not be reachable from each other even in the case of two agents, and deciding their reachability is PSPACE-complete in general. On the other hand, we prove that reachability is guaranteed for two agents with identical or binary utilities as well as for any number of agents with identical binary utilities. We also examine the complexity of deciding whether there is an EF1 exchange sequence that is optimal in the number of exchanges required.

Group incentive compatibility in a market with indivisible goods: A comment

We note that the proofs of Bird (1984), the first to show group strategy-proofness of top trading cycles (TTC), require correction. We provide a counterexample to a critical claim and present corrected proofs in the spirit of the originals. We also present a novel proof of strong group strategy-proofness using the corrected results.

Mixed Fair Division: A Survey

Fair division considers the allocation of scarce resources among agents in such a way that every agent gets a fair share. It is a fundamental problem in society and has received significant attention and rapid developments from the game theory and artificial intelligence communities in recent years. The majority of the fair division literature can be divided along at least two orthogonal directions: goods versus chores, and divisible versus indivisible resources. In this survey, besides describing the state of the art, we outline a number of interesting open questions and future directions in three mixed fair division settings: (i) indivisible goods and chores, (ii) divisible and indivisible goods (mixed goods), and (iii) indivisible goods with subsidy which can be viewed like a divisible good.