Chapter 17-Matching, Allocation, and Exchange of Discrete Resources

Economic Theory, 2008

American Economic Review, 2002

This paper investigates a class of matching problems-the assignment of indivisible items to agents where some agents have prior claims to some of the items. As a running example, we will refer to the indivisible items as houses. House allocation problems are not only of theoretical interest, but also of practical importance. A house allocation mechanism assigns a set of houses (or offices, tasks, etc.) to prospective tenants, allotting at most one house to each tenant. Rents are exogenously given and there is no medium of exchange, such as money. In general some houses will have existing tenants, some houses will be empty, and some applicants for housing will be new (e.g., freshmen). The canonical examples are assignment of college students to dormitory rooms and public housing units. Other examples are assignment of offices and tasks to individuals. Many universities in the U.S. employ some variant of a mechanism called the random serial dictatorship with squatting rights (RSD) to allocate dormitory rooms. Each existing tenant can either keep her house or enter the applicant pool. Each applicant is randomly given a (possibly seniority-weighted or GPA-weighted) priority and each is assigned, in priority order, her top choice among the houses that remain. This mechanism is strategy-proof (i.e., dominant strategy incentive

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

Allocation and exchange of discrete resources such as kidneys, school seats, and many other resources for which agents have single-unit demand is conducted via direct mechanisms without monetary transfers. Incentive compatibility and efficiency are primary concerns in designing such mechanisms. We show that a mechanism is individually strategy-proof and always selects the efficient outcome with respect to some Arrovian social welfare function if and only if the mechanism is group strategy-proof and Pareto efficient. We construct the full class of these mechanisms and show that each of them can be implemented by endowing agents with control rights over resources.

SSRN Electronic Journal, 2017

In the school choice market, where scarce public school seats are assigned to students, a key operational issue is how to reassign seats that are vacated after an initial round of centralized assignment. Practical solutions to the reassignment problem must be simple to implement, truthful and efficient while also alleviating costly student movement between schools. We propose and axiomatically justify a class of reassignment mechanisms, the Permuted Lottery Deferred Acceptance (PLDA) mechanisms. Our mechanisms generalize the commonly used Deferred Acceptance (DA) school choice mechanism to a two-round setting and retain its desirable incentive and efficiency properties. School choice systems typically run DA with a lottery number assigned to each student to break ties in school priorities. We show that under natural conditions on demand, the second round tie-breaking lottery can be correlated arbitrarily with that of the first round without affecting allocative welfare, and reversing the lottery order between rounds minimizes reassignment among all PLDA mechanisms. Empirical investigations based on data from NYC high school admissions support our theoretical findings.

International Journal of Game Theory, 2006

We give a simple and concise proof that so-called generalized median stable matchings are well-defined for college admissions problems. Furthermore, we discuss the fairness properties of median stable matchings and conclude with two illustrative examples of college admissions markets, the lattices of stable matchings, and the corresponding generalized median stable matchings.

2019

Real-world matching scenarios, like the matching of students to courses in a university setting, involve complex downward-feasible constraints like credit limits, time-slot constraints for courses, basket constraints (say, at most one humanities elective for a student), in addition to the preferences of students over courses and vice versa, and class capacities. We model this problem as a many-to-many bipartite matching problem where both students and courses specify preferences over each other and students have a set of downward-feasible constraints. We propose an Iterative Algorithm Framework that uses a many-to-one matching algorithm and outputs a many-to-many matching that satisfies all the constraints. We prove that the output of such an algorithm is Pareto-optimal from the student-side if the many-to-one algorithm used is Pareto-optimal from the student side. For a given matching, we propose a new metric called the Mean Effective Average Rank (MEAR), which quantifies the goodn...

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...

Games and Economic Behavior

This paper provindes three simple mechanisms to implement allocations in the core of matching markets. We analyse some sequential mechanisms which mimic matching procedures for many-to-one real life matching markets.

SSRN Electronic Journal, 2017

We consider school choice problems (Abdulkadiroglu and Sönmez, 2003) where students are assigned to public schools through a centralized assignment mechanism. We study the family of so-called rank-priority mechanisms, each of which is induced by an order of rank-priority pairs. Following the corresponding order of pairs, at each step a rank-priority mechanism considers a rank-priority pair and matches an available student to an unfilled school if the student and the school rank and prioritize each other in accordance with the rank-priority pair. The Boston or immediate acceptance mechanism is a particular rank-priority mechanism. Our first main result is a characterization of the subfamily of rank-priority mechanisms that Nash implement the set of stable (i.e., fair) matchings (Theorem 1). We show that our characterization also holds for "sub-implementation" and "sup-implementation" (Corollaries 3 and 4). Our second main result is a strong impossibility result: under incomplete information, no rank-priority mechanism implements the set of stable matchings (Theorem 2).

We study the problem of matching student to courses. The main problem with current system is its inability to account for student preferences. Furthermore, the system is logistically inefficient. We look specifically at cases of simultaneous enrollment. Namely, we provide an auction style mechanism for assignment of courses to students in the same enrollment time which does not suffer from the same flaws. We show that, with some natural assumptions, produces more expected utility per course. Lastly, we propose some additions to alleviate for unequal strategizing abilities among students .

Proceedings of the AAAI Conference on Artificial Intelligence

Applications such as employees sharing office spaces over a workweek can be modeled as problems where agents are matched to resources over multiple rounds. Agents' requirements limit the set of compatible resources and the rounds in which they want to be matched. Viewing such an application as a multi-round matching problem on a bipartite compatibility graph between agents and resources, we show that a solution (i.e., a set of matchings, with one matching per round) can be found efficiently if one exists. To cope with situations where a solution does not exist, we consider two extensions. In the first extension, a benefit function is defined for each agent and the objective is to find a multi-round matching to maximize the total benefit. For a general class of benefit functions satisfying certain properties (including diminishing returns), we show that this multi-round matching problem is efficiently solvable. This class includes utilitarian and Rawlsian welfare functions. For a...

Discrete Optimization, 2014

Consider a many-to-many matching market that involves two finite disjoint sets, a set of applicants A and a set of courses C. Each applicant has preferences on the different sets of courses she can attend, while each course has a quota of applicants that it can admit. In this paper, we examine Pareto optimal matchings (briefly POM) in the context of such markets, that can also incorporate additional constraints, e.g., each course bearing some cost and each applicant having an available budget. We provide necessary and sufficient conditions for a many-to-many matching to be Pareto optimal and show that checking whether a given matching is Pareto optimal requires O(|A| 2 • |C| 2) time. Moreover, we provide a generalized version of serial dictatorship, which can be used to obtain any many-to-many POM. We also study the problems of finding a minimum cardinality and a maximum cardinality POM. We show that the former is NP-complete even in one-to-one markets with the preference list of each applicant containing at most two entries. For the latter problem we show that, although it is polynomially solvable in the special one-to-one case, it is NP-complete for many-to-many markets.

International Economic Review, 2012

We consider a common indivisible good allocation problem in which agents have both social and private endowments. Popular applications include student assignment to on-campus housing, kidney exchange, and particular school choice problems. In a series of experiments Chen and Sönmez (American Economic Review 92: [1669][1670][1671][1672][1673][1674][1675][1676][1677][1678][1679][1680][1681][1682][1683][1684][1685][1686] 2002) have shown that a popular mechanism from recent theory, the Top Trading Cycles (TTC) mechanism, induces a signi…cantly higher participation rate by agents with private endowments and leads to signi…cantly more e¢ cient outcomes than the most commonly used real-life mechanism, the Random Serial Dictatorship with Squatting Rights.

Journal of Dynamics and Games, 2015

The assignment game is a two-sided market, say buyers and sellers, where demand and supply are unitary and utility is transferable by means of prices. This survey is structured in three parts: a first part, from the introduction of the assignment game by Shapley and Shubik (1972) until the publication of the book of Roth and Sotomayor (1990), focused on the notion of core; the subsequent investigations that broaden the scope to other notions of solution for these markets; and its extensions to assignment markets with multiple sides or multiple partnership. These extended two-sided assignment markets, that allow for multiple partnership, better represent the situation in a labour market or an auction.