Testing substitutability

2014

Abstract. This paper studies the structure of stable multipartner matchings in two-sided markets where choice functions are quotafilling in the sense that they satisfy the substitutability axiom and, in addition, fill a quota whenever possible. It is shown that (i) the set of stable matchings is a lattice under the common revealed preference orderings of all agents on the same side, (ii) the supremum (infimum) operation of the lattice for each side consists componentwise of the join (meet) operation in the revealed preference ordering of the agents on that side, and (iii) the lattice has the polarity, distributivity, complementariness and full-quota properties. JEL classification: C78

Boletin De La Sociedad Espanola De Ceramica Y Vidrio, 2016

Essays on matching markets This thesis is on matching theory and its applications to the analysis and design of matching markets. A predominant concept in this thesis and in the matching literature is stability. Stable matchings are desirable both from a practical and a theoretical point of view and have played a critical role in the success of clearinghouses for medical residency matching and school choice. In what follows we provide a brief summary of each chapter. In the first chapter, entitled Stability of Equilibrium Outcomes under Deferred Acceptance: Acyclicity and Dropping Strategies, we study the preference revelation game induced by the deferred acceptance (DA) mechanism. Under DA no student can improve by misrepresenting his preferences. However, hospitals typically can. Furthermore, Nash equilibrium outcomes may be unstable with respect to agents' true preferences. We focus on Nash equilibria in which each hospital plays a so-called dropping strategy. A dropping stra...

2008

The scientific community has also been enthusiastic. In its fiftieth anniversary self-review, the U.S. National Science Foundation reported that "from a financial standpoint, the big payoff for NSF's longstanding support [of auction theory research] came in 1995 ... [when t]he Federal Communications Commission established a system for using auctions." See Milgrom (2004). 913 914 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005 to fill its remaining openings. This procedure is precisely the hospital-offering version of the Gale-Shapley matching algorithm. Hence, the Gale-Shapley matching algorithm is a special case of the Kelso-Crawford procedure. The possibility of extending the National Resident Matching Program (the "Match") to permit wage competition is an important consideration in assessing public policy toward the Match, particularly because work by Jeremy Bulow and Jonathan Levin (2003) lends some theoretical support for the position that the Match may compress and reduce doctors' wages relative to a perfectly competitive standard. The practical possibility of such an extension depends on many details, including, importantly, the form in which doctors and hospitals would have to report their preferences for use in the Match. In its current incarnation, the Match can accommodate hospital preferences that encompass affirmative action constraints and a subtle relationship between internal medicine and its subspecialties, so it will be important for any replacement algorithm to allow similar preferences to be expressed. In Section IV, we introduce a parameterized family of valuations that accomplishes that. A second important similarity is between the Gale-Shapley doctor-offering algorithm and the Ausubel-Milgrom proxy auction. Explaining this relationship requires restating the algorithm in a different form from the one used for the preceding comparison. We show that if the hospitals in the Match consider doctors to be substitutes, then the doctor-offering algorithm is equivalent to a certain cumulative offer process in which the hospitals at each round can choose from all the offers they have received at any round, current or past. In a different environment, where there is but a single "hospital" or auctioneer with unrestricted preferences and general contract terms, this cumulative offer process coincides exactly with the Ausubel-Milgrom proxy auction. Despite the close connections among these mechanisms, previous analyses have treated them separately. In particular, analyses of auctions typically assume that bidders' payoffs are quasi-linear. No corresponding assumption is made in analyzing the medical match or the college admissions problem; indeed, the very

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.

2021

We explore the possibility of designing matching mechanisms that can accommodate non-standard choice behavior. We pin down the necessary and sufficient conditions on participants’ choice behavior for the existence of stable and incentive compatible mechanisms. Our results imply that well-functioning matching markets can be designed to adequately accommodate a plethora of choice behaviors, including the standard behavior that is consistent with preference maximization. To illustrate the significance of our results in practice, we show that a simple modification in a commonly used matching mechanism enables it to accommodate non-standard choice behavior.

Review of Economic Design, 2001

This paper studies the structure of stable multipartner matchings in two-sided markets where choice functions are quotafilling in the sense that they satisfy the substitutability axiom and, in addition, fill a quota whenever possible. It is shown that (i) the set of stable matchings is a lattice under the common revealed preference orderings of all agents on the same side, (ii) the supremum (infimum) operation of the lattice for each side consists componentwise of the join (meet) operation in the revealed preference ordering of the agents on that side, and (iii) the lattice has the polarity, distributivity, complementariness and full-quota properties.

American Economic Journal: Microeconomics, 2012

We introduce a model in which firms trade goods via bilateral contracts which specify a buyer, a seller, and the terms of the exchange. This setting subsumes (many-to-many) matching with contracts, as well as supply chain matching. When firms' relationships do not exhibit a supply chain structure, stable allocations need not exist. By contrast, in the presence of supply chain structure, a natural substitutability condition characterizes the maximal domain of firm preferences for which stable allocations are guaranteed to exist. Furthermore, the classical lattice structure, rural hospitals theorem, and one-sided strategy-proofness results all generalize to this setting. (JEL C78, D85, D86, L14)

Games and Economic Behavior, 2009

Hatfield and Milgrom (2005) present a unified model of matching with contracts, which includes the standard two-sided matching and some package auction models as special cases. They show that the doctor-optimal stable mechanism is strategy-proof for doctors if hospitals' preferences satisfy substitutes and the law of aggregate demand. We show that the doctor-optimal stable mechanism is group strategy-proof for doctors under these same conditions. That is, no group of doctors can make each of its members strictly better off by jointly misreporting their preferences. We derive as a corollary of this result that no individually rational matching is preferred by all the doctors to the doctor-optimal stable match.

Proceedings of the 22nd ACM Conference on Economics and Computation, 2021

It is well known that every stable matching instance I has a rotation poset R(I) that can be computed efficiently and the downsets of R(I) are in one-to-one correspondence with the stable matchings of I. Furthermore, for every poset P, an instance I(P) can be constructed efficiently so that the rotation poset of I(P) is isomorphic to P. In this case, we say that I(P) realizes P. Many researchers exploit the rotation poset of an instance to develop fast algorithms or to establish the hardness of stable matching problems. In order to gain a parameterized understanding of the complexity of sampling stable matchings, Bhatnagar et al.[1] introduced stable matching instances whose preference lists are restricted but nevertheless model situations that arise in practice. In this paper, we study four such parameterized restrictions. Our goal is to characterize the rotation posets that arise from these models: k-bounded, where each agent has at most k acceptable partners; k-attribute, where e...

International Journal of Game Theory, 2008

For the many-to-one matching model we give a procedure to partition the set of substitutable preference profiles into equivalence classes with the property that all profiles in the same class have the same set of stable matchings. This partition allows to reduce the amount of information required by centralized stable mechanisms.