Fairness Properties Research Articles

Short trading cycles: Paired kidney exchange with strict ordinal preferences

I study kidney exchange with strict ordinal preferences and with constraints on the lengths of the exchange cycles. Efficient deterministic mechanisms have poor fairness properties in this environment. Instead, I propose an individually rational, ordinally efficient and anonymous random mechanism for two-way kidney exchange based on Bogomolnaia and Moulin’s (2001) Probabilistic Serial mechanism. Individual rationality incentivizes patient–donor pairs who are compatible with each other to participate in the exchange, thus increasing the overall transplantation rate. Finally, individual rationality, ex-post efficiency and weak strategyproofness are incompatible for any mechanism.

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 (Eisenberg and Gale 1959; Orlin 2010), 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 (Caragiannis et al. 2016); here fairness refers to envyfreeness 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.

Walrasian Dynamics in Multi-Unit Markets

In a multi-unit market, a seller brings multiple units of a good and tries to sell them to a set of buyers that have monetary endowments. While a Walrasian equilibrium does not always exist in this model, natural relaxations of the concept that retain its desirable fairness properties do exist. We study the dynamics of (Walrasian) envy-free pricing mechanisms in this environment, showing that for any such pricing mechanism, the best response dynamic starting from truth-telling converges to a pure Nash equilibrium with small loss in revenue and welfare. Moreover, we generalize these bounds to capture all the (reasonable) Nash equilibria for a large class of (monotone) pricing mechanisms. We also identify a natural mechanism, which selects the minimum Walrasian envy-free price, in which for n=2 buyers the best response dynamic converges from any starting profile. We conjecture convergence of the mechanism for any number of buyers and provide simulation results to support our conjecture.

Fairness GAN: Generating datasets with fairness properties using a generative adversarial network

We introduce the Fairness GAN (generative adversarial network), an approach for generating a dataset that is plausibly similar to a given multimedia dataset, but is more fair with respect to protected attributes in decision making. We propose a novel auxiliary classifier GAN that strives for demographic parity or equality of opportunity and show empirical results on several datasets, including the CelebFaces Attributes (CelebA) dataset, the Quick, Draw! dataset, and a dataset of soccer player images and the offenses for which they were called. The proposed formulation is well suited to absorbing unlabeled data; we leverage this to augment the soccer dataset with the much larger CelebA dataset. The methodology tends to improve demographic parity and equality of opportunity while generating plausible images.

Constrained Stochastic Cost-Sharing

This paper presents a model of a multi-divisional firm to share the joint yet uncertain and fixed cost of running a central operational unit. A firm aims at allocating this cost ex ante, subject to constraints imposed by the asymmetric and limited liabilities of the different divisions. We study solutions that are made up of a vector of ex ante payments which are allocated in absence of costs, and a remaining solution that is contingent on the cost. Under a mild continuity condition we find different classes of egalitarian solutions. The class of egalitarian proportional solutions is characterized by dependency on the disutility of the total cost instead of details of the distribution. In this class, there is a unique proportional solution which systematically minimizes the maximal transfer. A fundamentally different egalitarian solution is the stochastic egalitarian constrained equal costs solution. It is characterized using a local symmetry property which states that incremental costs should be distributed equally among those divisions with sufficient liability. This egalitarian solution has a smaller largest transfer than any egalitarian proportional solution. We conclude by showing how our results generalize when egalitarianism is replaced by a more general fairness property.

Constrained stochastic cost allocation

This paper presents a model of a multi-divisional firm to share the joint yet uncertain and fixed cost of running a central operational unit. A firm aims at allocating this cost ex ante, subject to constraints imposed by the asymmetric and limited liabilities of the different divisions. We study solutions that are made up of a vector of ex ante payments which are allocated in absence of costs, and a remaining solution that is contingent on the cost. Under a mild continuity condition we find different classes of egalitarian solutions. The class of egalitarian proportional solutions is characterized by dependency on the disutility of the total cost instead of details of the distribution. In this class, there is a unique proportional solution which systematically minimizes the maximal transfer. A fundamentally different egalitarian solution is the stochastic egalitarian constrained equal costs solution. It is characterized using a local symmetry property which states that incremental costs should be distributed equally among those divisions with sufficient liability. This egalitarian solution has a smaller largest transfer than any egalitarian proportional solution. We conclude by showing how our results generalize when egalitarianism is replaced by a more general fairness property.

A Market-Based Framework for Multi-Resource Allocation in Fog Computing

Fog computing is transforming the network edge into an intelligent platform by bringing storage, computing, control, and networking functions closer to end users, things, and sensors. How to allocate multiple resource types (e.g., CPU, memory, bandwidth) of capacity-limited heterogeneous fog nodes to competing services with diverse requirements and preferences in a fair and efficient manner is a challenging task. To this end, we propose a novel market-based resource allocation framework in which the services act as buyers and fog resources act as divisible goods in the market. The proposed framework aims to compute a market equilibrium (ME) solution at which every service obtains its favorite resource bundle under the budget constraint, while the system achieves high resource utilization. This paper extends the general equilibrium literature by considering a practical case of satiated utility functions. In addition, we introduce the notions of non-wastefulness and frugality for equilibrium selection and rigorously demonstrate that all the non-wasteful and frugal ME are the optimal solutions to a convex program. Furthermore, the proposed equilibrium is shown to possess salient fairness properties, including envy-freeness, sharing-incentive, and proportionality. Another major contribution of this paper is to develop a privacy-preserving distributed algorithm, which is of independent interest, for computing an ME while allowing market participants to obfuscate their private information. Finally, extensive performance evaluation is conducted to verify our theoretical analyses.

Matching with Stochastic Arrival

This paper examines efficiency and fairness properties in a dynamic allocation problem. The model applies to situations in which objects of different types arrive stochastically over time and must be assigned to agents in a queue, such as the allocation of public housing units. The main result demonstrates the impossibility of designing an allocation mechanism in an environment with stochastic arrival that can guarantee resulting assignments that are efficient or fair ex post.

Random matching under priorities: stability and no envy concepts

We consider stability concepts for random matchings where agents have preferences over objects and objects have priorities for the agents. When matchings are deterministic, the standard stability concept also captures the fairness property of no (justified) envy. When matchings can be random, there are a number of natural stability and fairness concepts that coincide with stability and no envy whenever matchings are deterministic. We formalize known stability concepts for random matchings for a general setting that allows weak preferences and weak priorities, unacceptability, and an unequal number of agents and objects. We then present a clear taxonomy of the stability concepts and identify logical relations between them. Finally, we present a transformation from the most general setting to the most restricted setting, and show how almost all our stability concepts are preserved by that transformation.

Iterative Local Voting for Collective Decision-making in Continuous Spaces

Many societal decision problems lie in high-dimensional continuous spaces not amenable to the voting techniques common for their discrete or single-dimensional counterparts. These problems are typically discretized before running an election or decided upon through negotiation by representatives. We propose a algorithm called Iterative Local Voting for collective decision-making in this setting. In this algorithm, voters are sequentially sampled and asked to modify a candidate solution within some local neighborhood of its current value, as defined by a ball in some chosen norm, with the size of the ball shrinking at a specified rate. We first prove the convergence of this algorithm under appropriate choices of neighborhoods to Pareto optimal solutions with desirable fairness properties in certain natural settings: when the voters' utilities can be expressed in terms of some form of distance from their ideal solution, and when these utilities are additively decomposable across dimensions. In many of these cases, we obtain convergence to the societal welfare maximizing solution.We then describe an experiment in which we test our algorithm for the decision of the U.S. Federal Budget on Mechanical Turk with over 2,000 workers, employing neighborhoods defined by various L-Norm balls. We make several observations that inform future implementations of such a procedure.

TACD: A throughput allocation method based on variant of Cobb–Douglas for hybrid storage system

The development of the cloud and data centers alike has caused users to increasingly share storage systems, and storage systems have adopted SSDs to improve performance. The number of users is growing sharply and causing resource competition, so the throughput allocation of a hybrid storage system has an important effect on users’ performance. A hybrid storage architect faces a challenge: sharing I/O throughput resources between SSDs and HDDs fairly with high resource utilization. Drawing on economic game-theory, many methods based on Leontief preference have been proposed to determine fair throughput allocation in a hybrid storage system with having the properties of Sharing Incentive (SI), Envy Freeness (EF) and Pareto Optimality (PO). However, users’ workload characteristics vary because of the adoption of SSDs, so there is heterogeneity among the storage media. The heterogeneity of storage media and the diversity of user workloads together create more constraints on fairness, which imposes restrictions to improve the resource utilization. To improve the resource utilization, some researchers have proposed relaxing the limitation of fairness properties by adjusting users’ allocations. However, when adjusting the allocation between users, some users’ allocations are reduced but others’ are improved, with the goal of increasing the resource utilization of the whole system, this will pose a problem because of the Leontief preference restriction: some users’ performance is degraded, while other users’ performance is improved.In this paper, we first propose one variant of the Cobb–Douglas preference based on users’ workloads and allocations, and then present a throughput allocation model based on this variant named TACD that can capture the effects of diminishing returns and substitutability in throughput allocation for hybrid storage. By making use of these effects, TACD can improve some users’ performance while providing the same performance for other users and achieving higher resource utilization in the whole system, and it can provide a fair allocation while enjoying SI, EF and PO properties. The variant of the Cobb–Douglas preference proposed in this paper has taken consideration of users’ workloads, so TACD can conduct the throughput allocation based on these workloads for meeting the performance requirements of users reasonably. Extensive experiments are conducted to prove the effectiveness of TACD. The simulation results show that TACD can fit the throughput allocation of hybrid storage system very well. The Linux results show that compared with the allocation method based on Leontief preference, TACD can provide a higher resource utilization and performance for users throughout the system, it can improve some users’ performance when maintaining comparable performance for other users, in contrast to the Leontief method based on weaker fairness under the same resource utilization, and its allocation result enjoys SI, EF and PO properties.

Strengthening Competitive Forces in Russian Mandatory Health Insurance

The Russian healthcare system has experienced significant reforms since the early 1990s. The traditional Semashko system of the Soviet Union was replaced by a more decentralised and insurance-based system, which in turn was augmented and modified several times with the goal to strengthen competitive forces for better quality of healthcare, to open equal access to quality medical care, in particular. Nevertheless, despite these serious attempts, the actual situation is still far from this goal. A serious lack of financial means, and framework conditions, which are not in favour of strong competitive forces, have to be blamed in the first place. In this context, a rigorous formal analysis is applied to provide insight into the competitive forces in Russian mandatory health insurance. The paper investigates first the current system. Thereafter, a modified system with more possibilities of competition will be proposed. Besides an increase in funding, this modified system requires structural changes in comparison to the current system. The results obtained refer to existence of equilibria and their structural, efficiency and fairness properties, which are of relevance for the further development of the Russian healthcare system.

Stable Matching in Large Economies

We study stability of two‐sided many‐to‐one matching in which firms' preferences for workers may exhibit complementarities. Although such preferences are known to jeopardize stability in a finite market, we show that a stable matching exists in a large market with a continuum of workers, provided that each firm's choice is convex and changes continuously as the set of available workers changes. We also study the existence and structure of stable matchings under preferences exhibiting substitutability and indifferences in a large market. Building on these results, we show that an approximately stable matching exists in large finite economies. We extend our framework to ensure a stable matching with desirable incentive and fairness properties in the presence of indifferences in firms' preferences.

Obvious Manipulations in Cake-Cutting

In cake-cutting, strategy-proofness is a very costly requirement in terms of fairness: for n = 2 it implies a dictatorial allocation, whereas for n≥3 it requires that one agent receives no cake. We show that a weaker version of this property recently suggested by Troyan and Morril, called not-obvious manipulability, is compatible with the strong fairness property of proportionality, which guarantees that each agent receives 1/n of the cake. Both properties are satisfied by the leftmost leaves mechanism, an adaptation of the Dubins - Spanier moving knife procedure. Most other classical proportional mechanisms in literature are obviously manipulable, including the original moving knife mechanism. Not-obvious manipulability explains why leftmost leaves is manipulated less often in practice than other proportional mechanisms.

The Social Cost Problem, Rights and the (Non)Empty Core

The “Coase theorem” is made of the efficiency and the neutrality theses. Using cooperative game theory, we show that these two theses are not compatible: there exist only two types of rights assignments that guarantee a nonempty core. Thus, the efficiency thesis holds – there exist (two types of) rights assignments under which the core of the game is nonempty – but the neutrality thesis does not hold – for all other rights assignments the core is empty. This complements the results found in the literature on Coase and the core. Our paper also adds two more results. Indeed, we add two principles that are not discussed in the literature about the Coase theorem: a democratic principle for rights assignments and a fairness principle for the monetary payoffs distributed to the agents, that we translate into two natural properties for a solution. We show, and this is our second main result, that the nonempty core requirement is not compatible with our democratic and fairness properties. Thus, under democratic and fairness criteria, the efficiency thesis does not hold.

Strict fairness of equilibria in asymmetric information economies and mixed markets

We investigate the fairness property of equal-division competitive market equilibria (CME) in asymmetric information economies with a space of agents that may contain non-negligible (large) traders. We first propose an extension to our framework of the notion of strict fairness due to Zhou (J Econ Theory 57:158–175, 1992). We prove that once agents are asymmetrically informed, any equal-division CME allocation is strictly fair, but a strictly fair allocation might not be supported by an equilibrium price. Then, we investigate the role of large traders and we provide two sufficient conditions under which, in the case of complete information economies, a redistribution of resources is strictly fair if and only if it results from a competitive mechanism.

Necessary players, Myerson fairness and the equal treatment of equals

This article addresses linear sharing rules on transferable utility games (TU-games) with various structures, namely communication structures and conference structures as defined by Myerson in two papers (Myerson in Mathematics of Operations Research 2:225–229, 1977; Myerson in International Journal of Game Theory 9:169–182, 1980). Here, using matrix expressions, we rewrite those sharing rules. With this presentation we identify the close relationship between the fairness property and an equal treatment of necessary players axiom. Moreover, we show that the latter is implied by the equal treatment of equals, linking the fairness property to the notion of equality.

Envy-Free Matchings with Lower Quotas

While every instance of the Hospitals/Residents problem admits a stable matching, the problem with lower quotas (HR-LQ) has instances with no stable matching. For such an instance, we expect the existence of an envy-free matching, which is a relaxation of a stable matching preserving a kind of fairness property. In this paper, we investigate the existence of an envy-free matching in several settings, in which hospitals have lower quotas and not all doctor–hospital pairs are acceptable. We first provide an algorithm that decides whether a given HR-LQ instance has an envy-free matching or not. Then, we consider envy-freeness in the Classified Stable Matching model due to Huang (in: Procedings of 21st annual ACM-SIAM symposium on discrete algorithms (SODA2010), SIAM, Philadelphia, pp 1235–1253, 2010), i.e., each hospital has lower and upper quotas on subsets of doctors. We show that, for this model, deciding the existence of an envy-free matching is NP-hard in general, but solvable in polynomial time if quotas are paramodular.

The Shapley value for the probability game

The main goal of this paper is to introduce the probability game. On one hand, we analyze the Shapley value by providing an axiomatic characterization. We propose the so-called independent fairness property, meaning that for any two players, the player with larger individual value gets a larger portion of the total benefit. On the other, we use the Shapley value for studying the profitability of merging two agents.

Fairness in group identification

We study the problem of classifying individuals into groups, using agents’ opinions on who belong to which group as input. Our focus is on the rules that satisfy equal treatment of equals, a minimal fairness property, in addition to independence of irrelevant opinions and non-degeneracy. We show that a rule satisfies the three axioms if and only if it is the liberal rule, a strong one-vote rule, a one-row rule, or a one-column rule. The last three families of rules can be ruled out by simple, intuitive properties. Thus, invoking equal treatment of equals, which is substantially weaker than symmetry, we obtain a characterization of the liberal rule.