Envy-freeness and Relaxed Stability under lower quotas (original) (raw)

2019

We consider the problem of assigning residents to hospitals when hospitals have upper and lower quotas. Apart from this, both residents and hospitals have a preference list which is a strict ordering on a subset of the other side. Stability is a well-known notion of optimality in this setting. Every Hospital-Residents (HR) instance without lower quotas admits at least one stable matching. When hospitals have lower quotas (HRLQ), there exist instances for which no matching that is simultaneously stable and feasible exists. We investigate envy-freeness which is a relaxation of stability for such instances. Yokoi (ISAAC 2017) gave a characterization for HRLQ instances that admit a feasible and envy-free matching. Yokoi's algorithm gives a minimum size feasible envy-free matching, if there exists one. We investigate the complexity of computing a maximum size envy-free matching in an HRLQ instance (MAXEFM problem) which is equivalent to computing an envy-free matching with minimum nu...

Minimal Envy Matchings in the Hospitals/Residents Problem with Lower Quotas

arXiv (Cornell University), 2021

In the Hospitals/Residents problem, every hospital has an upper quota that limits the number of residents assigned to it. While, in some applications, each hospital also has a lower quota for the number of residents it receives. In this setting, a stable matching may not exist. Envy-freeness is introduced as a relaxation of stability that allows blocking pairs involving a resident and an empty position of a hospital. While, envy-free matching might not exist either when lower quotas are introduced. We consider the problem of finding a feasible matching that satisfies lower quotas and upper quotas and minimizes envy in terms of envy-pairs and envyresidents in the Hospitals/Resident problem with Lower Quota. We show that the problem is NP-hard with both envy measurement. We also give a simple exponential-time algorithm for the Minimum-Envy-Pair HRLQ problem.

2021

We consider the hospital-residents problem where both hospitals and residents can have lower quotas. The input is a bipartite graph G = (R ∪ H, E), each vertex in R ∪ H has a strict preference ordering over its neighbors. The sets R and H denote the sets of residents and hospitals respectively. Each hospital has an upper and a lower quota denoting the maximum and minimum number of residents that can be assigned to it. Residents have upper quota equal to one, however, there may be a requirement that some residents must not be left unassigned in the output matching. We call this as the residents’ lower quota. We show that whenever the set of matchings satisfying all the lower and upper quotas is nonempty, there always exists a matching that is popular among the matchings in this set. We give a polynomial-time algorithm to compute such a matching. 2012 ACM Subject Classification Mathematics of computing → Combinatorial algorithms

Popular Matching with Lower Quotas

2017

We consider the well-studied Hospital Residents (HR) problem in the presence of lower quotas (LQ). The input instance consists of a bipartite graph G = (R∪H, E) where R and H denote sets of residents and hospitals respectively. Every vertex has a preference list that imposes a strict ordering on its neighbors. In addition, each hospital h has an associated upper-quota q^+(h) and lower-quota q^-(h). A matching M in G is an assignment of residents to hospitals, and M is said to be feasible if every resident is assigned to at most one hospital and a hospital h is assigned at least q^-(h) and at most q^+(h) residents. Stability is a de-facto notion of optimality in a model where both sets of vertices have preferences. A matching is stable if no unassigned pair has an incentive to deviate from it. It is well-known that an instance of the HRLQ problem need not admit a feasible stable matching. In this paper, we consider the notion of popularity for the HRLQ problem. A matching M is popula...

Maximally Satisfying Lower Quotas in the Hospitals/Residents Problem with Ties

ArXiv, 2022

Motivated by the serious problem that hospitals in rural areas suffer from a shortage of residents, we study the Hospitals/Residents model in which hospitals are associated with lower quotas and the objective is to satisfy them as much as possible. When preference lists are strict, the number of residents assigned to each hospital is the same in any stable matching because of the well-known rural hospitals theorem; thus there is no room for algorithmic interventions. However, when ties are introduced to preference lists, this will no longer apply because the number of residents may vary over stable matchings. In this paper, we formulate an optimization problem to find a stable matching with the maximum total satisfaction ratio for lower quotas. We first investigate how the total satisfaction ratio varies over choices of stable matchings in four natural scenarios and provide the exact values of these maximum gaps. Subsequently, we propose a strategy-proof approximation algorithm for ...

The lattice of envy-free many-to-many matchings with contracts

arXiv (Cornell University), 2022

We study envy-free allocations in a many-to-many matching model with contracts in which agents on one side of the market (doctors) are endowed with substitutable choice functions and agents on the other side of the market (hospitals) are endowed with responsive preferences. Envy-freeness is a weakening of stability that allows blocking contracts involving a hospital with a vacant position and a doctor that does not envy any of the doctors that the hospital currently employs. We show that the set of envy-free allocations has a lattice structure. Furthermore, we define a Tarski operator on this lattice and use it to model a vacancy chain dynamic process by which, starting from any envy-free allocation, a stable one is reached.

Pair-wise envy free and stable matchings for two sided systems with techniques

The two-sided matching model of Gale and Shapley (1962) can be interpreted as one where a non-empty finite set of firms need to employ a non-empty finite set of workers. Further, each firm can employ at most one worker and each worker can be employed by at most one firm. Each worker has preferences over the set of firms and each firm has preferences over the set of workers. An assignment of workers to firms is said to be stable if there does not exist a firm and a worker who prefer each other to the ones they are associated with in the assignment. Gale and Shapley (1962) proved that every two-sided matching problem admits at least one stable matching. In this paper we extend the above model by including a non-empty finite set of techniques. An assignment now comprises disjoint triplets, each triplet consisting of a firm, a worker and a technique. A technique can be likened to a machine that the firm and worker together use for production. Each firm has preferences over the set of ordered pairs of workers and techniques and each worker has preferences over the set of ordered pairs of firms and techniques. We call such models two-sided systems with techniques. There are two kinds of issues we address in the context of this model, now that concerns naturally extend beyond those of pair-wise stability as defined in Gale and Shapley (1962). The first issue is about the possibility of a pair of agents being better off than in their current assignment by perhaps using a different technique. The existence of such a possibility allows for a pair of agents to 'envy' the technique that may have been assigned to a different pair. It is natural to seek an assignment that excludes 'envy' and which may therefore be called 'pair-wise envy free'. The second issue that we address in this paper, pertains to a situation where each firm is initially endowed with a technique. In such a situation we are interested in proving the existence of an assignment such that no coalition can re-allocate the techniques that they have been endowed with, and consequently be better off. A matching which satisfies this property is called stable. Through out the paper, we assume as in (: though in a slightly different context) that the preferences of the workers are lexicographic, with firms enjoying priority over techniques. Modifying the analysis for three-sided systems as in Lahiri , we show that a sufficient condition for a pair-wise envy free matching to exist is the satisfaction of a certain discrimination property. The discrimination property says: given two distinct firm-worker pairs, the technique that is best for the firm in one pair is

Near Feasible Stable Matchings with Complementarities

SSRN Electronic Journal, 2000

The National Resident Matching program strives for a stable matching of medical students to teaching hospitals. With the presence of couples, stable matchings need not exist. For any student preferences, we show that each instance of a stable matching problem has a 'nearby' instance with a stable matching. The nearby instance is obtained by perturbing the capacities of the hospitals. Specifically, given a reported capacity k h for each hospital h, we find a redistribution of the slot capacities k h satisfying |k h − k h | ≤ 4 for all hospitals h and h k h ≤ k h ≤ h k h + 9, such that a stable matching exists with respect to k. Our approach is general and applies to other type of complementarities, as well as matchings with side constraints and contracts.

Socially stable matchings in the Hospitals / Residents problem

In the Hospitals/Residents (HR) problem, agents are partitioned into hospitals and residents. Each agent wishes to be matched to an agent in the other set and has a strict preference over these potential matches. A matching is stable if there are no blocking pairs, i.e., no pair of agents that prefer each other to their assigned matches. Such a situation is undesirable as it could lead to a deviation in which the blocking pair form a private arrangement outside the matching. This however assumes that the blocking pair have social ties or communication channels to facilitate the deviation. Relaxing the stability definition to take account of the potential lack of social ties between agents can yield larger stable matchings. In this paper, we define the Hospitals/Residents problem under Social Stability (HRSS) which takes into account social ties between agents by introducing a social network graph to the HR problem. Edges in the social network graph correspond to resident-hospital pa...

STABLE SOLUTIONS ON MATCHING MODELS WITH QUOTA RESTRICTION

International Game Theory Review, 2011

In this paper, we present a matching market in which an institution has to hire a set of pairs of complementary workers, and has a quota that is the maximum number of candidates pair positions to be …lled. We de…ne a natural stable solution and …rst show that in the unrestricted institution preferences domain, the set of stable solution may be empty and second we obtain a complete characterization of the stable sets under responsive restriction of the institution's preference.

Envy-freeness and Relaxed Stability under lower quotas (original) (raw)

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