The airport game is a classical and well-known model of fair cost-sharing for a single facility among multiple agents. This paper extends it to the so-called assignment setting, that is, for multiple facilities and agents, each agent chooses a facility to use and shares the cost with the other agents. Such a situation can be often seen in sharing economy, such as sharing fees for office desks among workers, taxis among customers of possibly different destinations on a line, and so on. Our model is regarded as a coalition formation game based on the fair cost-sharing of the airport game; we call our model \emph{a fair ride allocation on a line}. As criteria of solution concepts, we incorporate Nash stability and envy-freeness into our setting. We show that a Nash-stable feasible allocation that minimizes the social cost of agents can be computed efficiently if a feasible allocation exists. For envy-freeness, we provide several structural properties of envy-free allocations. Based on these, we design efficient algorithms for finding an envy-free allocation when at least one of (1) the number of facilities, (2) the capacity of facilities, and (3) the number of agent types, is small. Moreover, we show that a consecutive envy-free allocation can be computed in polynomial time. On the negative front, we show the NP-hardness of determining the existence of an allocation under two relaxed envy-free concepts.
This paper addresses one of the most challenging issues in designing an efficient and sustainable ridesharing service: ridesharing market design. We formulate it as a fair cost allocation problem through the lens of the cooperative game theory. A special property of the cooperative ridesharing game is that its characteristic function values are calculated by solving an optimization problem. Several concepts of fairness are investigated and special attention is paid to a solution concept named nucleolus, which aims to minimize the maximum dissatisfaction in the system. Due to its computational …
Ride-sharing or vehicle-pooling allows commuters to team up spontaneously for transportation cost sharing. This has become a popular trend in the emerging paradigm of sharing economy. One crucial component to support effective ride-sharing is the matching mechanism that pairs up suitable commuters. Traditionally, matching has been performed in a centralized manner, whereby an operator arranges ride-sharing according to a global objective (e.g., total cost of all commuters). However, ride-sharing is a decentralized decision-making paradigm, where commuters are self-interested and only motivated…
Ride-sharing services are gaining popularity and are crucial for a sustainable environment. A special case in which such services are most applicable, is the last mile variant. In this variant it is assumed that all the passengers are positioned at the same origin location (e.g. an airport), and each have a different destination. One of the major issues in a shared ride is fairly splitting of the ride cost among the passengers. In this paper we use the Shapley value, which is one of the most significant solution concepts in cooperative game theory, for fairly splitting the cost of a shared rid…
Bipartite matching markets pair agents on one side of a market with agents, items, or contracts on the opposing side. Prior work addresses online bipartite matching markets, where agents arrive over time and are dynamically matched to a known set of disposable resources. In this paper, we propose a new model, Online Matching with (offline) Reusable Resources under Known Adversarial Distributions (OM-RR-KAD), in which resources on the offline side are reusable instead of disposable; that is, once matched, resources become available again at some point in the future. We show that our model is tr…
Fair division has long been an important problem in the economics literature. In this note, we consider the existence of proportionally fair allocations of indivisible goods, i.e., allocations of indivisible goods in which every agent gets at least her proportionally fair share according to her own utility function. We show that when utilities are additive and utilities for individual goods are drawn independently at random from a distribution, proportionally fair allocations exist with high probability if the number of goods is a multiple of the number of agents or if the number of goods grow…
We consider the age-old problem of allocating items among different agents in a way that is efficient and fair. Two papers, by Dolev et al. and Ghodsi et al., have recently studied this problem in the context of computer systems. Both papers had similar models for agent preferences, but advocated different notions of fairness. We formalize both fairness notions in economic terms, extending them to apply to a larger family of utilities. Noting that in settings with such utilities efficiency is easily achieved in multiple ways, we study notions of fairness as criteria for choosing between differ…
We initiate the study of indivisible chore allocation for agents with asymmetric shares. The fairness concept we focus on is the weighted natural generalization of maxmin share: WMMS fairness and OWMMS fairness. We first highlight the fact that commonly used algorithms that work well for the allocation of goods to asymmetric agents, and even for chores to symmetric agents do not provide good approximations for allocation of chores to asymmetric agents under WMMS. As a consequence, we present a novel polynomial-time constant-approximation algorithm, via linear program, for OWMMS. For two specia…
We study fair allocation of indivisible goods to agents with unequal entitlements. Fair allocation has been the subject of many studies in both divisible and indivisible settings. Our emphasis is on the case where the goods are indivisible and agents have unequal entitlements. This problem is a generalization of the work by Procaccia and Wang wherein the agents are assumed to be symmetric with respect to their entitlements. Although Procaccia and Wang show an almost fair (constant approximation) allocation exists in their setting, our main result is in sharp contrast to their observation. We s…
We study fair and economically efficient allocation of indivisible goods among agents whose valuations are rank functions of matroids. Such valuations constitute a well-studied class of submodular functions (i.e., they exhibit a diminishing returns property) and model preferences in several resource-allocation settings. We prove that, for matroid-rank valuations, a social welfare-maximizing allocation that gives each agent her maximin share always exists. Furthermore, such an allocation can be computed in polynomial time. We establish similar existential and algorithmic results for the pairwis…
We initiate the work on fair and strategyproof allocation of indivisible chores. The fairness concept we consider in this paper is maxmin share (MMS) fairness. We consider three previously studied models of information elicited from the agents: the ordinal model, the cardinal model, and the public ranking model in which the ordinal preferences are publicly known. We present both positive and negative results on the level of MMS approximation that can be guaranteed if we require the algorithm to be strategyproof. Our results uncover some interesting contrasts between the approximation ratios ac…
We consider the problem of fair allocation of indivisible goods to agents with submodular valuation functions, where agents may have either equal entitlements or arbitrary (possibly unequal) entitlements. We focus on share-based fairness notions, specifically, the maximin share (MMS) for equal entitlements and the anyprice share (APS) for arbitrary entitlements, and design allocation algorithms that give each agent a bundle of value at least some constant fraction of her share value. For the equal entitlement case (and submodular valuations), Ghodsi, Hajiaghayi, Seddighin, Seddighin, and Yami …
Most of the existing algorithms for fair division do not consider externalities. Under externalities, the utility an agent obtains depends not only on its allocation but also on the allocation of other agents. An agent has a positive (negative) value for the assigned goods (chores). This work focuses on a special case of externality, i.e., an agent receives positive or negative value for unassigned items independent of which other agent gets it. We show that it is possible to adapt existing algorithms using a transformation to ensure certain fairness and efficiency notions in this setting. Des…
Fairness is well studied in the context of resource allocation. Researchers have proposed various fairness notions like envy-freeness (EF), and its relaxations, proportionality and max-min share (MMS). There is vast literature on the existential and computational aspects of such notions. While computing fair allocations, any algorithm assumes agents' truthful reporting of their valuations towards the resources. Whereas in real-world web-based applications for fair division, the agents involved are strategic and may manipulate for individual utility gain. In this paper, we study strategy-proof …
Given the final ranking of a competition, how should the total prize endowment be allocated among the competitors? We study consistent prize allocation rules satisfying elementary solidarity and fairness principles. In particular, we axiomatically characterize several families of rules satisfying anonymity, order preservation, and endowment monotonicity, which all fall between the Equal Division rule and the Winner-Takes-All rule. Specific characterizations of rules and subfamilies are directly obtained. To our knowledge, this paper is the first that develops and motivates prize allocation rul…
We study fair allocation of indivisible items, where the items are furnished with a set of conflicts, and agents are not permitted to receive conflicting items. This kind of constraint captures, for example, participating in events that overlap in time, or taking on roles in the presence of conflicting interests. We demonstrate, both theoretically and experimentally, that fairness characterizations such as EF1, MMS and MNW still are applicable and useful under item conflicts. Among other existence, non-existence and computability results, we show that a $1/Δ$-approximate MMS allocation for $n$…
Mechanism design for a social utility being the sum of agents' utilities (SoU) is a well-studied problem. There are, however, a number of problems of theoretical and practical interest where a designer may have a different objective than maximization of the SoU. One motivation for this is the desire for more equitable allocation of resources among agents. A second, more subtle, motivation is the fact that a fairer allocation indirectly implies less variation in taxes which can be desirable in a situation where (implicit) individual agent budgetary constraints make payment of large taxes unreal…
We study a fair resource scheduling problem, where a set of interval jobs are to be allocated to heterogeneous machines controlled by agents. Each job is associated with release time, deadline, and processing time such that it can be processed if its complete processing period is between its release time and deadline. The machines gain possibly different utilities by processing different jobs, and all jobs assigned to the same machine should be processed without overlap. We consider two widely studied solution concepts, namely, maximin share fairness and envy-freeness. For both criteria, we di…
Motivated by real-world applications, we study the fair allocation of graphical resources, where the resources are the vertices in a graph. Upon receiving a set of resources, an agent's utility equals the weight of a maximum matching in the induced subgraph. We care about maximin share (MMS) fairness and envy-freeness up to one item (EF1). Regarding MMS fairness, the problem does not admit a finite approximation ratio for heterogeneous agents. For homogeneous agents, we design constant-approximation polynomial-time algorithms, and also note that significant amount of social welfare is sacrific…
We study the problem of allocating indivisible goods among agents with additive valuations. When randomization is allowed, it is possible to achieve compelling notions of fairness such as envy-freeness, which states that no agent should prefer any other agent's allocation to her own. When allocations must be deterministic, achieving exact fairness is impossible but approximate notions such as envy-freeness up to one good can be guaranteed. Our goal in this work is to achieve both simultaneously, by constructing a randomized allocation that is exactly fair ex-ante and approximately fair ex-post…
We consider the problem of fairly dividing a set of items. Much of the fair division literature assumes that the items are `goods' i.e., they yield positive utility for the agents. There is also some work where the items are `chores' that yield negative utility for the agents. In this paper, we consider a more general scenario where an agent may have negative or positive utility for each item. This framework captures, e.g., fair task assignment, where agents can have both positive and negative utilities for each task. We show that whereas some of the positive axiomatic and computational result…
We study the problem of fair and efficient allocation of a set of indivisible chores to agents with additive cost functions. We consider the popular fairness notion of envy-freeness up to one good (EF1) with the efficiency notion of Pareto-optimality (PO). While it is known that an EF1+PO allocation exists and can be computed in pseudo-polynomial time in the case of goods, the same problem is open for chores. Our first result is a strongly polynomial-time algorithm for computing an EF1+PO allocation for bivalued instances, where agents have (at most) two disutility values for the chores. To th…
We consider the problem of approximate maximin share (MMS) allocation of indivisible items among three agents with additive valuation functions. For goods, we show that an $\frac{11}{12}$ - MMS allocation always exists, improving over the previously known bound of $\frac{8}{9}$ . Moreover, in our allocation, we can prespecify an agent that is to receive her full proportional share (PS); we also present examples showing that for such allocations the ratio of $\frac{11}{12}$ is best possible. For chores, we show that a $\frac{19}{18}$-MMS allocation always exists. Also in this case, we can presp…
One of the important yet insufficiently studied subjects in fair allocation is the externality effect among agents. For a resource allocation problem, externalities imply that a bundle allocated to an agent may affect the utilities of other agents. In this paper, we conduct a study of fair allocation of indivisible goods when the externalities are not negligible. We present a simple and natural model, namely \emph{network externalities}, to capture the externalities. To evaluate fairness in the network externalities model, we generalize the idea behind the notion of maximin-share ($\MMS$) to a…
The theory of algorithmic fair allocation is within the center of multi-agent systems and economics in the last decade due to its industrial and social importance. At a high level, the problem is to assign a set of items that are either goods or chores to a set of agents so that every agent is happy with what she obtains. Particularly, in this survey, we focus on indivisible items, for which absolute fairness such as envy-freeness and proportionality cannot be guaranteed. One main theme in the recent research agenda is about designing algorithms that approximately achieve the fairness criteria…
We consider fair allocation of indivisible items in a model with goods, chores, and copies, as a unified framework for studying: (1)~the existence of EFX and other solution concepts for goods with copies; (2)~the existence of EFX and other solution concepts for chores. We establish a tight relation between these issues via two conceptual contributions: First, a refinement of envy-based fairness notions that we term envy \emph{without commons} (denoted $\efxwc$ when applied to EFX). Second, a formal \emph{duality theorem} relating the existence of a host of (refined) fair allocation concepts fo…
We study the problem of dividing indivisible chores among agents whose costs (for the chores) are supermodular set functions with binary marginals. Such functions capture complementarity among chores, i.e., they constitute an expressive class wherein the marginal disutility of each chore is either one or zero, and the marginals increase with respect to supersets. In this setting, we study the broad landscape of finding fair and efficient chore allocations. In particular, we establish the existence of $(i)$ EF1 and Pareto efficient chore allocations, $(ii)$ MMS-fair and Pareto efficient allocat…
The rapid growth of wireless and mobile Internet has led to wide applications of exchanging resources over network, in which how to fairly allocate resources has become a critical challenge. To motivate sharing, a BD Mechanism is proposed for resource allocation, which is based on a combinatorial structure called bottleneck decomposition. The mechanism has been shown with properties of fairness, economic efficiency [17], and truthfulness against two kinds of strategic behaviors [2,3]. Unfortunately, the crux on how to compute a bottleneck decomposition of any graph is remain untouched. In this…
In this paper, we study the classic problem of fairly allocating indivisible items with the extra feature that the items lie on a line. Our goal is to find a fair allocation that is contiguous, meaning that the bundle of each agent forms a contiguous block on the line. While allocations satisfying the classical fairness notions of proportionality, envy-freeness, and equitability are not guaranteed to exist even without the contiguity requirement, we show the existence of contiguous allocations satisfying approximate versions of these notions that do not degrade as the number of agents or items…
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 protoco…
Ranking alternatives is a natural way for humans to explain their preferences. It is being used in many settings, such as school choice, course allocations and residency matches. In some cases, several `items' are given to each participant. Without having any information on the underlying cardinal utilities, arguing about fairness of allocation requires extending the ordinal item ranking to ordinal bundle ranking. The most commonly used such extension is stochastic dominance (SD), where a bundle X is preferred over a bundle Y if its score is better according to all additive score functions. SD…
We study the fair division problem of allocating multiple resources among a set of agents with Leontief preferences that are each required to complete a finite amount of work, which we term "limited demands". We examine the behavior of the classic Dominant Resource Fairness (DRF) mechanism in this setting and show it is fair but only weakly Pareto optimal and inefficient in many natural examples. We propose as an alternative the Least Cost Product (LCP) mechanism, a natural adaptation of Maximum Nash Welfare to this setting. We characterize the structure of allocations of the LCP mechanism in …
Fairness is becoming an increasingly important concern when designing markets, allocation procedures, and computer systems. I survey some recent developments in the field of multi-agent fair allocation.
We study fair allocation of indivisible public goods subject to cardinality (budget) constraints. In this model, we have n agents and m available public goods, and we want to select $k \leq m$ goods in a fair and efficient manner. We first establish fundamental connections between the models of private goods, public goods, and public decision making by presenting polynomial-time reductions for the popular solution concepts of maximum Nash welfare (MNW) and leximin. These mechanisms are known to provide remarkable fairness and efficiency guarantees in private goods and public decision making se…
We study the problem of allocating multiple types of resources to agents with Leontief preferences. The classic Dominant Resource Fairness (DRF) mechanism satisfies several desired fairness and incentive properties, but is known to have poor performance in terms of social welfare approximation ratio. In this work, we propose a new approximation ratio measure, called \emph{\fratio}, which is defined as the worst-case ratio between the optimal social welfare (resp. utilization) among all \emph{fair} allocations and that by the mechanism, allowing us to break the lower bound barrier under the cla…
We consider item allocation to individual agents who have additive valuations, in settings in which there are protected groups, and the allocation needs to give each protected group its "fair" share of the total welfare. Informally, within each protected group we consider the total welfare that the allocation gives the members of the group, and compare it to the maximum possible welfare that an allocation can give to the group members. An allocation is fair towards the group if the ratio between these two values is no worse then the relative size of the group. For divisible items, our formal d…
In networking and computing, resource allocation is typically addressed using classical sharing protocols as, for instance, the proportional division rule, the max-min fair allocation , or other solutions inspired by cooperative game theory. In this paper, we argue that, describing the resource allocation problem as a cooperative game, such classical resource allocation approaches, as well as associated notions of fairness, show important limitations. We identify in the individual satisfaction rate the key aspect of the challenge of defining a new notion of fairness and, consequently, a resour…
Erasmus University Rotterdam（伊拉斯姆斯大学鹿特丹分校）；Eindhoven University of Technology（埃因霍温理工大学）
Task allocation problems have traditionally focused on cost optimization. However, more and more attention is being given to cases in which cost should not always be the sole or major consideration. In this paper we study a fair task allocation problem in transportation where an optimal allocation not only has low cost but more importantly, it distributes tasks as even as possible among heterogeneous participants who have different capacities and costs to execute tasks. To tackle this fair minimum cost allocation problem we analyze and solve it in two parts using two novel polynomial-time algo…
We consider a multi-agent resource allocation setting that models the assignment of papers to reviewers. A recurring issue in allocation problems is the compatibility of welfare/efficiency and fairness. Given an oracle to find a welfare-achieving allocation, we embed such an oracle into a flexible algorithm called the Constrained Round Robin (CRR) algorithm, that achieves the required welfare level. Our algorithm also allows the system designer to lower the welfare requirements in order to achieve a higher degree of fairness. If the welfare requirement is lowered enough, a strengthening of env…
Dominant resource fairness (DRF) is a popular mechanism for multi-resource allocation in cloud computing systems. In this paper, we consider a problem of multi-resource fair allocation with bounded number of tasks. Firstly, we propose the lexicographically max-min normalized share (LMMNS) fair allocation mechanism, which is a natural generalization of DRF, and design a non-trivial optimal algorithm to find a LMMNS fair allocation, whose running time is linear in the number of users. Secondly, we prove that LMMNS satisfies envy-freeness (EF) and group strategy-proofness (GSP), and analysis the …
We study fairness in house allocation, where $m$ houses are to be allocated among $n$ agents so that every agent receives one house. We show that maximizing the number of envy-free agents is hard to approximate to within a factor of $n^{1-γ}$ for any constant $γ>0$, and that the exact version is NP-hard even for binary utilities. Moreover, we prove that deciding whether a proportional allocation exists is computationally hard, whereas the corresponding problem for equitability can be solved efficiently.
Indian Institute of Science（印度科学理工学院）；Chennai Mathematical Institute（金奈数学研究所）
We study the problem of allocating indivisible goods among n agents in a fair manner. For this problem, maximin share (MMS) is a well-studied solution concept which provides a fairness threshold. Specifically, maximin share is defined as the minimum utility that an agent can guarantee for herself when asked to partition the set of goods into n bundles such that the remaining (n-1) agents pick their bundles adversarially. An allocation is deemed to be fair if every agent gets a bundle whose valuation is at least her maximin share. Even though maximin shares provide a natural benchmark for fairn…
Fair allocation of indivisible goods is a well-explored problem. Traditionally, research focused on individual fairness - are individual agents satisfied with their allotted share? - and group fairness - are groups of agents treated fairly? In this paper, we explore the coexistence of individual envy-freeness (i-EF) and its group counterpart, group weighted envy-freeness (g-WEF), in the allocation of indivisible goods. We propose several polynomial-time algorithms that provably achieve i-EF and g-WEF simultaneously in various degrees of approximation under three different conditions on the age…
We propose a notion of fairness for allocation problems in which different agents may have different reservation utilities, stemming from different outside options, or property rights. Fairness is usually understood as the absence of envy, but this can be incompatible with reservation utilities. It is possible that Alice's envy of Bob's assignment cannot be remedied without violating Bob's participation constraint. Instead, we seek to rule out {\em justified envy}, defined as envy for which a remedy would not violate any agent's participation constraint. We show that fairness, meaning the abse…
We study the existence of fair distributions when we have more guests than pieces to allocate, focusing on envy-free distributions among those who receive a piece. The conditions on the demand from the guests can be weakened from those of classic cake-cutting and rent-splitting results of Stromquist, Woodall, and Su. We extend existing variations of the cake-cutting problem with secretive guests and those that resist the removal of any sufficiently small set of guests.
We study the problem of fairly allocating a multiset $M$ of $m$ indivisible items among $n$ agents with additive valuations. Specifically, we introduce a parameter $t$ for the number of distinct types of items and study fair allocations of multisets that contain only items of these $t$ types, under two standard notions of fairness: 1. Envy-freeness (EF): For arbitrary $n$, $t$, we show that a complete EF allocation exists when at least one agent has a unique valuation and the number of items of each type exceeds a particular finite threshold. We give explicit upper and lower bounds on this thr…
