Guest editorial: Special issue on matching under preferences

No abstract available

The stability of the roommate problem revisited

The lack of stability in some matching problems suggests that alternative solution concepts to the core might be a step towards furthering our understanding of matching market performance. We propose absorbing sets as a solution for the class of roommate problems with strict preferences. This solution, which always exists, either gives the matchings in the core or predicts other matchings when the core is empty. Furthermore, it satisfies the interesting property of outer stability. We also determine the matchings in absorbing sets and find that in the case of multiple absorbing sets a similar structure is shared by all.roommate problem, core, absorbing sets

Finding all stable matchings with assignment constraints

In this paper we consider stable matchings that are subject to assignment constraints. These are matchings that require certain assigned pairs to be included, insist that some other assigned pairs are not, and, importantly, are stable. Our main contribution is an algorithm that determines when assignment constraints are compatible with stability. Whenever a stable matching consistent with the assignment constraints exists, our algorithm will output all of them (each in polynomial time per solution). This provides market designers with (i) a tool to test the feasibility of stable matchings with assignment constraints, and (ii) a separate tool to implement them

Stabil halmazok hozzárendelési játékokban = Stable sets in assignment games

A dolgozatban a kooperatív játékelmélet egyik legrégebbi megoldáskoncepcióját a (Neumann-Morgenstern) stabil halmazokat vizsgáljuk egy speciális játékosztályon a hozzárendelési játékokon. A stabil halmaz az elosztásoknak egy olyan részhalmaza, amelynek elemei egymást nem dominálják (belső stabilitás), viszont minden a halmazon kívüli elosztást dominál valamelyik (külső stabilitás). Ez így egy elég egyszerű és természetes megoldáskoncepció, azonban később kiderült, hogy matematikailag elég rosszul kezelhető. Sokáig az is kérdés volt, hogy egyáltalán létezik-e minden játékban stabil halmaz. Később megmutatták, hogy minden 4 személyes játékban létezik, és találtak 10 személyes játékot, amelyben nem, a kettő közti esetekről azonban továbbra sem tudunk semmit. Ezt a megoldáskoncepciót vizsgáljuk a hozzárendelési játékok osztályán. Shapley és Shubik is több cikkben foglalkozott ezzel a témával, megadtak egy halmazt, amelyről az volt a sejtésük, hogy stabil, ezáltal minden hozzárendelési játékban létezik stabil halmaz, de ezt a sejtésüket nem tudták bizonyítani. Később egy 2013-as cikkükben Núñez és Rafels bebizonyították ennek a halmaznak a stabilitását. A dolgozat legfontosabb eredménye a stabil halmazoknak egy új karakterizációja ezen a játékosztályon. Megmutatjuk, hogy az elosztáshalmaznak egy részhalmaza pontosan akkor stabil, ha belső stabil (ez ugyanaz mint az eredeti definícióban), összefüggő, van egy olyan pontja, ahol minden eladó- és egy olyan ahol minden vevő kifizetése 0, és tartalmazza bizonyos redukált játékok magjait. A karakterizációt felhasználva szinte egyből megkapjuk, hogy a Shapley és Shubik által javasolt halmaz stabil, sőt ennél többet is megmutatunk. Abban az esetben, ha a játék magja nem stabil megadunk végtelen sok stabil halmazt. A létezésen kívül több, a stabil halmazok szerkezetére vonatkozó állítást is megfogalmazunk. A karakterizációnak egy nemkooperatív játékelméleti felhasználását is bemutatjuk. Harsányi János megfogalmazott egy kritikát a stabil halmazokkal kapcsolatban. Szerinte az eredeti definíció nem szerencsés, mert nem veszi figyelembe a közvetett, több lépésen keresztüli dominanciát, ezért javasolt egy alkujátékot, amelynek szerinte az egyensúlyában szereplő fixpontjait kellene stabil halmaznak nevezni. A karakterizáció segítségével megmutatjuk, hogy a hozzárendelési játékoknál ez a kettő ugyanaz. A dolgozat végén a hozzárendelési játékok egy általánosításával, a többoldalú hozzárendelési játékokkal foglalkozunk. Megmutatjuk, hogy a mag akkor és csak akkor stabil, ha a generáló (poli)mátrix főátlódomináns. Ez csak a legkisebb nemtriviális esetben volt ismert, és még a speciális eset bizonyítása is sokkal bonyolultabb, mint a mienk

Two-Sided Matching Markets: Models, Structures, and Algorithms

Two-sided matching markets are a cornerstone of modern economics. They model a wide range of applications such as ride-sharing, online dating, job positioning, school admissions, and many more. In many of those markets, monetary exchange does not play a role. For instance, the New York City public high school system is free of charge. Thus, the decision on how eighth-graders are assigned to public high schools must be made using concepts of fairness rather than price. There has been therefore a huge amount of literature, mostly in the economics community, defining various concepts of fairness in different settings and showing the existence of matchings that satisfy these fairness conditions. Those concepts have enjoyed wide-spread success, inside and outside academia. However, finding such matchings is as important as showing their existence. Moreover, it is crucial to have fast (i.e., polynomial-time) algorithms as the size of the markets grows. In many cases, modern algorithmic tools must be employed to tackle the intractability issues arising from the big data era. The aim of my research is to provide mathematically rigorous and provably fast algorithms to find solutions that extend and improve over a well-studied concept of fairness in two-sided markets known as stability. This concept was initially employed by the National Resident Matching Program in assigning medical doctors to hospitals, and is now widely used, for instance, by cities in the US for assigning students to public high schools and by certain refugee agencies to relocate asylum seekers. In the classical model, a stable matching can be found efficiently using the renowned deferred acceptance algorithm by Gale and Shapley. However, stability by itself does not take care of important concerns that arose recently, some of which were featured in national newspapers. Some examples are: how can we make sure students get admitted to the best school they deserve, and how can we enforce diversity in a cohort of students? By building on known and new tools from Mathematical Programming, Combinatorial Optimization, and Order Theory, my goal is to provide fast algorithms to answer questions like those above, and test them on real-world data. In Chapter 1, I introduce the stable matching problem and related concepts, as well as its applications in different markets. In Chapter 2, we investigate two extensions introduced in the framework of school choice that aim at finding an assignment that is more favorable to students -- legal assignments and the Efficiency Adjusted Deferred Acceptance Mechanism (EADAM) -- through the lens of classical theory of stable matchings. We prove that the set of legal assignments is exactly the set of stable assignments in another instance. Our result implies that essentially all optimization problems over the set of legal assignments can be solved within the same time bound needed for solving it over the set of stable assignments. We also give an algorithm that obtains the assignment output of EADAM. Our algorithm has the same running time as that of the deferred acceptance algorithm, hence largely improving in both theory and practice over known algorithms. In Chapter 3, we introduce a property of distributive lattices, which we term as affine representability, and show its role in efficiently solving linear optimization problems over the elements of a distributive lattice, as well as describing the convex hull of the characteristic vectors of the lattice elements. We apply this concept to the stable matching model with path-independent quota-filling choice functions, thus giving efficient algorithms and a compact polyhedral description for this model. Such choice functions can be used to model many complex real-world decision rules that are not captured by the classical model, such as those with diversity concerns. To the best of our knowledge, this model generalizes all those for which similar results were known, and our paper is the first that proposes efficient algorithms for stable matchings with choice functions, beyond classical extensions of the Deferred Acceptance algorithm. In Chapter 4, we study the discovery program (DISC), which is an affirmative action policy used by the New York City Department of Education (NYC DOE) for specialized high schools; and explore two other affirmative action policies that can be used to minimally modify and improve the discovery program: the minority reserve (MR) and the joint-seat allocation (JSA) mechanism. Although the discovery program is beneficial in increasing the number of admissions for disadvantaged students, our empirical analysis of the student-school matches from the 12 recent academic years (2005-06 to 2016-17) shows that about 950 in-group blocking pairs were created each year amongst disadvantaged group of students, impacting about 650 disadvantaged students every year. Moreover, we find that this program usually benefits lower-performing disadvantaged students more than top-performing disadvantaged students (in terms of the ranking of their assigned schools), thus unintentionally creating an incentive to under-perform. On the contrary, we show, theoretically by employing choice functions, that (i) both MR and JSA result in no in-group blocking pairs, and (ii) JSA is weakly group strategy-proof, ensures that at least one disadvantaged is not worse off, and when reservation quotas are carefully chosen then no disadvantaged student is worse-off. We show that each of these properties is not satisfied by DISC. In the general setting, we show that there is no clear winner in terms of the matchings provided by DISC, JSA, and MR, from the perspective of disadvantaged students. We however characterize a condition for markets, that we term high competitiveness, where JSA dominates MR for disadvantaged students. This condition is verified, in particular, in certain markets when there is a higher demand for seats than supply, and the performances of disadvantaged students are significantly lower than that of advantaged students. Data from NYC DOE satisfy the high competitiveness condition, and for this dataset our empirical results corroborate our theoretical predictions, showing the superiority of JSA. We believe that the discovery program, and more generally affirmative action mechanisms, can be changed for the better by implementing the JSA mechanism, leading to incentives for the top-performing disadvantaged students while providing many benefits of the affirmative action program

Pareto Dominance of Deferred Acceptance through Early Decision

An early decision market is governed by rules that allow each student to apply to (at most) one college and require the student to attend this college if admitted. This market is ubiquitous in college admissions in the United States. We model this market as an extensive-form game of perfect information and study a refinement of subgame perfect equilibrium (SPE) that induces undominated Nash equilibria in every subgame (SPUE). Our main result shows that this game can be used to define a decentralized matching mechanism that weakly Pareto dominates student-proposing deferred acceptance

Local stability in kidney exchange programs

When each patient of a kidney exchange program has a preference ranking over its set of compatible donors, questions naturally arise surrounding the stability of the proposed exchanges. We extend recent work on stable exchanges by introducing and underlining the relevance of a new concept of locally stable, or L-stable, exchanges. We show that locally stable exchanges in a compatibility digraph are exactly the so-called local kernels (L-kernels) of an associated blocking digraph (whereas the stable exchanges are the kernels of the blocking digraph), and we prove that finding a nonempty L-kernel in an arbitrary digraph is NP-complete. Based on these insights, we propose several integer programming formulations for computing an L-stable exchange of maximum size. We conduct numerical experiments to assess the quality of our formulations and to compare the size of maximum L-stable exchanges with the size of maximum stable exchanges. It turns out that nonempty L-stable exchanges frequently exist in digraphs which do not have any stable exchange. All the above results and observations carry over when the concept of (locally) stable exchanges is extended to the concept of (locally) strongly stable exchanges

Complementary cooperation, minimal winning coalitions, and power indices

We introduce a new simple game, which is referred to as the complementary weighted multiple majority game (C-WMMG for short). C-WMMG models a basic cooperation rule, the complementary cooperation rule, and can be taken as a sister model of the famous weighted majority game (WMG for short). In this paper, we concentrate on the two dimensional C-WMMG. An interesting property of this case is that there are at most $n+1$ minimal winning coalitions (MWC for short), and they can be enumerated in time $O(n\log n)$, where $n$ is the number of players. This property guarantees that the two dimensional C-WMMG is more handleable than WMG. In particular, we prove that the main power indices, i.e. the Shapley-Shubik index, the Penrose-Banzhaf index, the Holler-Packel index, and the Deegan-Packel index, are all polynomially computable. To make a comparison with WMG, we know that it may have exponentially many MWCs, and none of the four power indices is polynomially computable (unless P=NP). Still for the two dimensional case, we show that local monotonicity holds for all of the four power indices. In WMG, this property is possessed by the Shapley-Shubik index and the Penrose-Banzhaf index, but not by the Holler-Packel index or the Deegan-Packel index. Since our model fits very well the cooperation and competition in team sports, we hope that it can be potentially applied in measuring the values of players in team sports, say help people give more objective ranking of NBA players and select MVPs, and consequently bring new insights into contest theory and the more general field of sports economics. It may also provide some interesting enlightenments into the design of non-additive voting mechanisms. Last but not least, the threshold version of C-WMMG is a generalization of WMG, and natural variants of it are closely related with the famous airport game and the stable marriage/roommates problem.Comment: 60 page