Matching with Quantity

Transcription

1 Matching with Quantity David Delacrétaz September 12, 2016 Preliminary and Incomplete: Please do not cite or circulate without permission. Abstract We consider matching problems without transfers and with exogenous priorities where some agents demand two units of the same object while others only demand one unit. Applications of this model include the matching of children to day-care centers, students to exchange programs and refugees to localities. We show that, in this environment, the set of stable matchings may not possess same properties as in the canonical school choice model and may even be empty. We propose an algorithm to find an undominated stable matching whenever one exists. I would like to thank Georgy Artemov, Ivan Balbuzanov, Sven Feldman, Bettina Klaus, Scott Duke Kominers, Edwin Ip, Simon Loertscher, Claudio Mezzetti, Ellen Muir, Alex Nichifor, Andras Niedermayer, Peter Norman, Marek Pycia, Alex Teytelboym, Neil Thakral, Utku Unver, Cédric Wasser, Tom Wilkening, Jun Xiao and the audiences at the Australasian Economic Theory Workshop 2015 and 2016 for their valuable comments. I gratefully acknowledge the support of the Faculty of Business and Economics, the Department of Economics and the Centre For Market Design at the University of Melbourne. All errors are my own. Department of Economics, Level 4, FBE Building, University of Melbourne, 111 Barry St, Victoria 3010, Australia. ddelacretaz@unimelb.edu.au. 1

2 Preface Thesis Title: Essays in Microeconomic Theory and Market Design Supervisors: Prof. Simon Loertscher, Prof. Claudio Mezzetti, Dr. Jun Xiao My thesis consists of three papers, each of which makes a separate contribution to the field of market design: (I) Two-Sided Allocation Problems, Matching with Transfers, and the Impossibility of Ex Post Efficienty (with Simon Loerscher, Leslie Marx and Tom Wilkening, submitted) (II) Matching with Quantity (III) Refugee Resettlement (with Scott Kominers and Alex Teytelboym) Paper I starts with a simple observation: a buyer and a seller who trade are complements in the sense that the sum of their marginal contributions exceeds the gains from trade. Trade does not occur if either one of them leaves, therefore the sum of their marginal contributions equals twice the gains from trade. In any mechanism that incentivizes agents to truthfully reveal their type, agents must receive their marginal contribution. The complementarity between the buyer and the seller then implies that any such mechanism runs a deficit: what the buyer pays does not cover what the seller receives. This sheds a new light on a seminal result by Myerson and Sattersthwaite (1981). We extend this insight to more complex problems with many buyers and many sellers who can consume and produce packages of objects and provide a sufficient condition on buyer utility and seller cost functions for the result to hold. Paper II (this paper) considers an extension to the well-known school choice model (Abdulkadiroglu and Sönmez, 2003), where some agents want two units of the same object. The paper shows that this problem may have an empty set of stable matchings or many undominated stable matching instead of an optimal one. It proposes an algorithm to reduce the size of the problem by eliminating agent-object pairs that cannot be part of a stable matching. This allows finding an undominated stable matching every time one exists. Possible applciations include but are not limited to: day care matching as some children attend part-time and others full-time, matching with couples as each couple requires two places, refugee matching as families have different sizes and different needs and university exchange as some students visit for a semester and others for the whole year. Paper III is devoted to an application of the theory developed in Paper II: the resettlement of refugees within a specific country. Refugees accepted in a country are usually relocated in different areas that can provide them with a house as well as various services (e.g. school or hospital places). We design a matching program that allocates families to local authorities while taking preferences on both sides into account. The algorithm developed is adapted from the one in Paper II and finds a refugee-undominated stable matching whenever one exists. If the set of stable matchings is empty, quotas are adapted to obtain a slightly altered problem for which a stable matching exists. 2

3 1 Introduction Finding a place for their child in a day care center is essential for parents who want to reconcile career and family. A large amount of evidence suggests that early childhood programs are important in the development of children, for example Chetty et al. (2011) found that these programs have a long-term impact on future earnings. Yet, finding a place can prove a very difficult task for parents. In Australia, a report from the Productivity Commission (2014) documented the dissatisfaction of parents who have difficulties finding a place that fits their needs despite a large increase in the number of places offered and government funding. Following in the footsteps of Gale and Shapley (1962), economists have applied matching theory to various markets such as school choice (Abdulkadiroglu and Sönmez 2003) and kidney exchange (Roth, Sönmez and Ünver 2005, 2007). These designs have considerably improved the way these markets operate and the welfare they create by reducing transactions costs and yielding a more efficient and fairer allocation of resources. The matching of children to day care centers, however, has remained decentralized around the world. The Australian Productivity Commission (2014) reported many parents complaints about the current decentralized system. In particular, parents tend to accept the first place that becomes available, even if it does not completely fit their needs, because they cannot afford to risk not placing their child at all. An important difference between the matching of children with day care centers and students with schools is that children often attend day care part-time while school attendance is always one a full time basis. A single place at a day care center can therefore be split between two or more children attending on different days. This seemingly small twist on the well-known school choice model considerably complicates the search for a good algorithm, which may offer an explanation as to why the matching of children with day care centers has remained decentralized. 1 Matching problems where agents demand different quantities can arise in various contexts. Student exchange agreements between universities, such as those of the Erasmus program, often contain this feature. These bilateral agreements allow students from each university to study at the other for a year or a semester. The home university is responsible for selecting which students it will send to its partner. Places are often competitive so students may apply to go to more than one partner university, in order of preference. The selection can be thought of as a matching problem: students have preferences over partner universities and have different priorities for each of them depending on the quality of their application. Capacities are determined by the exchange agreement, which can specify either a maximum number of students that can be selected or a maximum number of semesters that can be used. If all agreements are of the former type, then the model is similar to school choice and the deferred-acceptance algorithm will select the student-optimal matching among 1 Another potential difficulty with day care matching is that children may enter or exit at any time, making the problem dynamic. The largest intake however takes place once a year when the older children start school or kindergarten and optimizing this static problem is likely to greatly improve the way the market operates. 3

4 those that eliminate justified envy. If at least some of the agreements are of the latter type, then students going on exchange for a year at one of these partner universities will use two units of capacity while those going for a semester will only use one. Exchange has become an increasingly important part of university degrees and, in a globalized world, demand for it is likely to keep growing in the future. Designing mechanisms for universities to efficiently cope with this demand and match students and partner institutions in the best way possible requires to study models where agents demand different quantities. Refugee resettlement (Delacrétaz, Kominers and Teytelboym, 2016) has a similar structure. People accepted as refugees in a given country are generally resettled across various localities that can provide them with accommodation and services (e.g. school places or hospital beds). A multidimensional quantity component arises in this problem as families are of different sizes and have different service requirements. Given the number of international crises, the need for refugee resettlement is unlikely to decrease in the next few years and it will become increasingly important for host countries to organize this in an efficient manner. This paper develops a model that accounts for that difference. Agents (e.g. parents, students, families) have ordinal preferences over objects (e.g. child care centers, host universities, localities) that are available in multiple units. For each object, agents are ranked according to exogenous priorities. The only difference with the canonical school choice model is that some of these agents demand two units of the same object. They represent families wanting to place their child full-time in a day care center or students wanting to go on exchange for a year. The model and the insights gathered throughout its analysis can be generalized in various ways, for example parents may want their child to visit a day care center on specific days of the week. Besides avoiding the cost of additional notation, sticking to this simple model has the benefit of departing from the canonical school choice model in the smallest possible way while still introducing complementarity in agent preferences. 2 Models with complementarities do not have the same structure as those usually studied in the literature and understanding how and why they differ is instrumental in order to design central clearinghouses for markets containing that specificity. It is our hope that the approach developed in this paper provides a benchmark that can be extended in various ways to fit different applications. A central concept in matching theory is stability. A matching is stable if there does not exist any agent who prefers an object to the one he receives and at least the number of units he demands are either unassigned or in the hands of agents with a lower priority. Stability constitutes an essential fairness criterion by ensuring that a matching respects the priorities in place. 3 It is well-known 2 An agent who demands two units does not derive any utility from receiving only one and therefore sees the two units as complements to each other. 3 Stability was first introduced by Gale and Shapley (1962) in their marriage market. A matching is stable if no pair of a man and a woman prefer each other to their current respective partners. In a two-sided matching problem such as Gale and Shapley s (1962), stability is essential to ensure agents remain with the partner they are assigned. In a one-sided matching problem such as school choice, stability constitutes a fairness criterion as it ensures a student can only miss out on a seat at a school he would have preferred if all seats at that school are assigned to students with 4

5 (Gale and Shapley 1962, Adbulkadiroglu and Sonmez 2003) that the set of stable matchings in the school choice model is nonempty and forms a lattice, which guarantees the existence of an optimal stable matching. 4 We show that this is no longer true in our model. The set of stable matchings may be empty and, if nonempty, it may contain multiple (agent-)undominated stable matchings instead of an (agent-)optimal stable matching. 5 the deferred-acceptance algorithm may fail. When an (agent-)optimal stable matching exist, We develop an algorithm that adapts the deferredacceptance algorithm in order to iteratively identify agents and objects that are not matched together in any stable matching. whenever the set of stable matchings is nonempty. We use these insights to find an (agent-)undominated stable matching The algorithm presented in this based on a simple principle that lies, though not in an obvious way, at the heart of the deferred-acceptance algorithm. In a model where each agent demands at most one unit, if an agent is rejected by an object in the deferred-acceptance algorithm, then he is not matched with that object in any stable matching. The deferred-acceptance algorithm can be interpreted as identifying such agent-object pairs until a stable matching is found. If some agents demand two units of the same good, the deferred-acceptance algorithm may lead to an agent being rejected by an object even though the two could be matched together in a stable matching. This happens because an agent demanding two units may be rejected even though one unit is available. That unit could then be assigned to an agent who demands only one unit but has been previously rejected because of his lower priority. The Top-Down Bottom-Up (TDBU) algorithm adapts makes use of these insights. The top-down part of the algorithm adapts the object-proposing deferred acceptance algorithm. In any round and for each object, agents receive a guarantee if their priority is high enough so that in every stable matching they are matched with that object or one they prefer. In the subsequent rounds, agents no longer contest objects they like less than one they have been guaranteed. In the terminology of the deferred acceptance algorithm, this corresponds to agents rejecting objects. The bottom-up part of the algorithm adapts the agent-proposing deferred acceptance algorithm. For each object, agents are rejected if their priority is so low that they cannot be matched with that object in any stable matching. These agents no longer contest the object in the following rounds. Importantly, rejections only occur when it can be determined that the agent and object are not matched together in any stable matching. This sometimes involves leaving an agent on hold in case a unit becomes subsequently available. The TDBU algorithm simplifies the matching problem by reducing the number of objects that each agent is contesting. When the algorithm terminates, a matching can a higher priority. In this paper we extend the definition of stability to a model where some agents want two units of a same object. 4 All other stable matchings make all students weakly worse-off. There also exists a pessimal stable matching, often referred to as the school optimal stable matching. 5 The agent-optimal stable matching is unique and such that all agents are weakly better-off than in all other stable matchings. An agent-undominated stable matching is such that, in every other stable matching, at least one agent is strictly worse-off. 5

6 be constructed by matching every agent with their favorite object among the ones they are still contesting. That matching does not have any blocking pair and dominates all stable matchings, however it may not be feasible because of agents being left on hold. In order to find a stable matching, one needs to identify more agents and objects who necessarily create blocking pairs if they are matched together. We do this by selecting specific agents and objects and look at what happens if the former stops contesting the latter. Running the TDBU algorithm again allows reducing further the number of agent-object pairs that need to be considered. We develop a search algorithm that finds candidates for an (agent-) undominated stable matching in this way. It ends by either identifying an (agent-)undominated stable matching or returning that the set of stable matchings is empty. This paper relates to several strands of the matching literature. Kennes, Monte and Tumennasan (2014) study the assignment of children to day care centers in a dynamic context, taking into account the fact that children may move from one center to another once they have secured a place. The authors do not consider the part-time feature of childcare and effectively build a dynamic extension of the school choice model framed in the context of day care. While these dynamic issues constitute interesting problems and may allow improving the system further, the sucess encountered by the reforms of school choice systems across the world suggests that taking care of the static problem can already greatly improve the way the market operates. Static mechanisms may also be easier to implement in practice. Sönmez and Switzer (2013) design a matching market for the American Army that has a similar flavor to the model studied in this paper in the sense that it specifies the duration for which a pair will be matched together: cadets are matched to army branches for a certain number of years. The two models differ in the fact that the term is designed to give cadets an incentive to commit for a longer period and has no impact on the capacity of each branch. In this paper, agents will affect the capacity of the object with which they are matched differently depending on the number of units they receive. Budish and Cantillon (2012) and Kojima (2013) study the matching of college students to classes. These models incorporate multi-unit demand since each student wants to attend many (typically four) classes. A given student and a given class may however only be matched once so a student will never take up two units of capacity of the same class. Crucially, student preferences are assumed to be responsive. This ensures the existence of a unique student-optimal stable matching, which can always be found using the deferred-acceptance algorithm. Martinez et al. (2004) characterize the set of stable matchings in a two-sided many-to-many matching problem. Their assumption about preferences is that they are substitutable. Substitutability is a slightly more general assumption than responsiveness that ensures the set of stable matching is nonempty and forms a lattice whose extrema can be found using the deferred acceptance algorithm. This paper introduces complementarity in some of the agents preferences, as a result the properties described above may disappear. 6

7 The closest model to this paper is the couple problem of the National Resident Matching Program (NRMP). The aim of the program is to match medical graduates to hospitals for their residency program. Graduates can apply as singles or as couples, in which case they rank pairs of hospitals in order of preferences. While hospitals submit a ranking of applicants and are therefore restricted to submitting responsive preferences, the possibility for medical graduates to apply as couples may introduce complementarities in their preferences: a graduate may be interested in a hospital if his or his partner gets a job in the same city or area but not otherwise. Our model can be framed into a couple problem, an agent demanding two units can be thought of as a couple where both partners want a place at the same hospital and will look for another hospital if this is not possible. The complementarity introduced in our simple model is enough to obtain the key feature of the couple s problem: there may not exist any stable matching and even if some exist the deferredacceptance may fail to select any of them. A satisfying engineering solution has been developed for this specific market (Roth and Peranson 1997, Roth 2003). Singles are matched first using the deferred-acceptance algorithm. Couples are then introduced one at a time, following a random order. Whenever a couple enters, blocking pairs (medical graduates and hospitals who would rather be matched together than with their current partner) are matched together until none remains. This algorithm may fail to find a stable matching even when one exists (Kojima 2007) and does not provide any guidance for the case where none exists. This has however not been a problem in practice as, to this day, it has found a stable matching every year. This may continue for a long time, Kojima, Pathak and Roth (2013) showed that a stable matching exists with probability one in large markets. The NRMP is indeed a very large market and, although the algorithm lacks clear theoretical properties, it has worked very well and is likely to continue doing so in the future for this specific market. The approach of this paper is however different. From a practical perspective, it is unclear how the matching found by the NRMP algorithm compares to other stable matchings in tems of doctor or hospital welfare. Our approach allows finding a stable matching that is undominated in terms of agent welfare. From a theoretical perspective, the TDBU algorithm places bounds on the set of stable matchings and can provide a useful point of departure to study that set s structure. It is our hope that the insights gathered throughout will be useful to better understand and design theory-based solutions for more general models such as the couple problem. Echenique and Yenmez (2007) study a (two-sided) college admission model where students have preferences over colleges as well as over the set of colleagues with whom they will attend. This models allow for complementarity since a student may be interested in a college as long as other students go there as well but not otherwise. As such, the set of stable matchings does not have its usual lattice structure and may even be empty. The authors propose an algorithm based on each stable matching being a fixed point of an increasing function. It either finds all stable matchings or indicates that none exists. Kojima (2007) adapts this approach to couple problems and by extension to our model. While this algorithm provides a solution for the type of matching problems studied in this paper, we believe our approach has two important advantages. From a theoretical point of 7

8 view, as mentioned above, the TDBU algorithm identifies bounds on the set of stable matchings. From a practical point of view, the computational feasibility of the fixed point algorithm is unclear. Echenique and Yenmez (2007) admit they cannot guarantee that it will not need to check every possible matching although they argue it is unlikely. The Top-Down Bottom-Up (TDBU) algorithm greatly reduces the size of the matching problem in a few simple steps. The search part may be computationally more intensive 6 but does not require looking through every possible matching since agents do not contest all objects. Another computational advantage is that our algorithm only looks for one stable matching and guarantees that it is undominated. Kojima s (2007) algorithm requires finding all stable matchings before one can be identified as undominated. The remainder of the paper is organized as follows. Section 2 lays out the setup. Section 3 provides illustrative examples about the structure of the set of stable matchings and the problems associated with the deferred-acceptance algorithm in this model. Section 4 presents the Top-Down Botton-Up algirhthm and shows how it can be used, in some cases, to find the optimal stable matching. Section 5 is devoted to additional techniques that allow further reducing the complexity of the matching problem. The main algorithm, which allows finding an undominated stable matching whenever one exists, is introduced in 6 and Section 7 concludes. 2 Setup A matching problem consists of a set A of agents and a set O of objects. The set of agents is partitioned into two subsets S and D. Agents in S are the single-unit agents and demand one unit. Agents in D are the double-unit agents and demand two units of the same object. 7 define w a {1, 2} such that w a := 1 if a S and w a := 2 if a D as the demand of agent a. The demand vector, w, is the A -dimensional vector containing the demand if all agents. Each object o is available in q o 1 units. We refer to q o as the quota of object o and defined the quota vector, q, as the O -dimensional vector containing all quotas. We assume that there exists a null object whose quota is large enough to accommodate all agents, that is q = a A w a = 2 D + S. Agents have strict, transitive and ordinal preference relations over all objects. The preference relation of agent a is denoted a and o a o signifies that agent a prefers o to o. The preference profile is a A -tuple of preference relations := ( s1,..., s S, d1,..., d D ), which contains the 6 McDermid and Manlove (2010) show that finding whether or not a stable matching exists in this model is an NP-complete problem. 7 In the school choice model, the agents are students, the objects are schools and each unit represents a seat at a specific school. All students are single-unit agents. In the context of day care, the agents are children (or their parents), the objects are day care center and each unit corresponds to a part-time place. Those children who want a part-time place are single-unit agents and those who want a full-time place are double-units agent. In the context of an exchange program, the agents are students, the objects are partner universities and each unit represents a semester. Students who want to go away for a semester are single-unit agents and those who want to go away for a year are double-unit agents. We 8

9 preference relations of all agents. An object is acceptable to an agent if the latter prefers it to the null object. For each object, agents are ranked in order of priority. π o represents the priority relation for object o and consists of a strict and transitive ranking of all agents. a π o a signifies that agent a has a higher priority than agent a for object o. For simplicity, we assume that the null object does not have a priority relation, which is without loss of generality since its capacity is large enough to accommodate all agents. The priority profile is defined as a O 1-tuple of priority relations π := (π o1,..., o O 1 ) containing the preference relations of all but the null object. 8 A contract is an agent-object pair (a, o). We denote by X A O the set of active contracts and say that agent a contests object o if (a, o) X. A matching problem, denoted M, π, w, q, X, is fully defined by its preferences and priority profiles, its demand and quota vectors and its set of contracts. We call M a full matching problem if X = A O and a reduced matching problem if X A O. All matching problems are initially full but we create reduced matching problems by eliminating contracts from the set of active ones. Given a matching problem M, we denote by A M o the set of agents who contest object o and by Oa M the set of objects that are contested by agent a. A M is partitioned into S M := A M S and D M := A M D. The top-choice of agent a in matching problem M =, π, w, q, X, denoted o M a, is the object a prefers among those he contests. Formally, o = o a if (a, o) X and o a o for any o o such that (a, o ) X. Conversly, A M o denotes the subset of agents who have o as their top-choice. That is, for any a A, a A M o if o = o M a. A M o is partitioned into its single-units agents S M o := A M o S and its double-unit agents D M o := A M o D. All subsets of agents defined below are partitioned in the same way between single- and double-unit agents. For any (a, o) X, we denote by ÂM (a,o) the subset of agents who contest object o and have a higher priority than a. Formally,  M (a,o) = {a A (a, o ) X, a π o a}. We analogously define ǍM (a,o) to be the subset of agents contesting object o who have a lower priority than a. The subset of agents with a higher priority than a at object o (regardless of whether or not they contest o) is denoted  (a,o), that is  (a,o) := {a A a π o a}. Observe that for any full matching problem M =, π, w, q, A O,  (a,o) = ÂM (a,o). Ǎ (a,o) is defined analogously as the subset of agents with a lower priority than a for object o. Observe that { (a,o), Ǎ (a,o) } is a partition of A \ {a}. We similarly define ÔM (a,o) and ǑM (a,o) to be the set of objects contested by a in matching problem M that he likes more, respectively less, than o. Ô (a,o) and Ǒ (a,o) are defined analogously for all objects, regardless of whether a contests them. We drop all dependencies on M and where there is no risk of confusion. A matching µ is a mapping defined on the set A O such that for all a A and o O: (i) µ(a) O. 8 In order to keep the model as simple as possible, it is assumed that even the agent with the lowest priority can get an object so long as its units are not claimed by other agents. This is natural in a one-sided model where only the welfare of agent matters. The assumption is without loss of generality since an eligibility threshold under which an agent cannot be matched to an object can be introduced in the model by ranking the objects after the null one in the agent s preferences. 9

10 (ii) µ(o) A. (iii) µ(a) = o if and only if a µ(o). Condition (i) states that each agent is matched with exactly one object, which may be the null object. Conditions (ii) and (iii) ensure that each object is matched exactly with those agents that are matched with it. A matching is feasible if all objects are available in enough units to satisfy the demand of agents with whom they are matched. That is, µ is feasible if a µ(o) w a q o for all o O. We say that a contract (a, o) A O belongs to matching µ if µ(a) = o. We denote by X(µ) the subset of contracts that belong to matching µ, formally X(µ) := {(a, o) A O µ(a) = o}. We say that a matching µ dominates another matching µ if it makes all agents weakly better-off and at least one agent strictly better-off. That is, µ dominates µ if µ µ and either µ(a) = µ (a) or µ(a) a µ (a) for all a A. 3 Stability and Deferred-Acceptance The model described above would be identical to the canonical school choice model (Abdulkadiroglu and Sönmez, 2003) if it were not for the fact that some agents want two units. In the remainder of this section, we illustrate the impact that this seemingly small difference has on the set of stable matchings. We begin by precisely defining stability in this model. A single-unit agent s and an object o form a blocking pair in matching µ if at least one unit of o is either unassigned or assigned to an agent with a lower priority. Formally, s and o form a blocking pair in µ if o s µ(o) and a µ(o) Â(s,o) w a q o 1. A double-unit agent d and an object o form a blocking pair in matching µ if at least two unit of o are either unassigned or assigned to an agent with a lower priority. Formally, d and o form a blocking pair in µ if o d µ(o) and a µ(o) Â(d,o) w a q o 2. This naturally leads to defining stability. Definition 1 (Stability). A matching is stable if it is feasible and there is no blocking pair in it. The novel part of this definition, which is otherwise standard in the matching literature, relates to blocking pairs involving double-unit agents. Double-unit agents are only interested in an object if they can get two units of it. As a result, there can be stable matchings where a double-unit agent misses out on an object he prefers to the one he is assigned and one unit of an object is either unassigned or assigned to a (single-unit) agent with a lower priority. Such a matching is still fair because, even though he has a higher priority for it, the agent is not interested in that unit. 3.1 Structure of the Set of Stable Matchings Stable matchings can be partially ranked in terms of agent welfare. An undominated stable matching (USM) is a stable matching that is not dominated by any other stable matching. If 10

11 there exists a unique undominated stable matching, then it dominates all other stable matchings and we refer to it as the optimal stable matching (OSM). If all agents demand one unit, as in the school choice model, it is well known that an optimal stable matching exists (Gale and Shapley, 1962, Abdulkadiroglu and Sönmez, 2003). This is no longer true in this model, as we illustrate next by presenting a matching problem where the set of stable matchings is empty (Example 1) and a matching problem where multiple undominated stable matchings exist (Example 2). Example 1 (No Stable Matching). There are two single-unit agents, s 1 and s 2, one double-unit agents, d 1 and two non-null objects o 1 and o 2. The preferences, priorities and quotas are detailed below. s1 : o 1, o 2, d1 : o 1,, o 2 π o1 : s 2, d 1, s 1 q o1 = 2 s2 : o 2, o 1, π o2 : s 1, s 2, d 1 q o2 = 1 We now show that this matching problem does not have any stable matching. If µ(d 1 ) = o 2 then µ is not feasible. If µ(d 1 ) = n then d 1 and o 1 form a blocking pair unless µ(s 2 ) = o 1 and s 1 and o 1 form a blocking pair unless µ(s 1 ) = o 1. In this case the unique unit of o 2 is unassigned, hence s 2 and o 1 form a blocking pair. If µ(d 1 ) = o 1 then s 2 and o 1 form a blocking pair unless µ(s 2 ) = o 2. As all units of o 1 and o 2 are assigned, s 1 is matched with the null object and forms a blocking pair with o 2. It follows that the set of stable matchings is empty. Example 2 (Multiple Undominated Stable Matchings). There are two single-unit agents s 1 and s 2, two double-unit agents d 1 and d 2 and two non-null objects o 1 and o 2. The preferences, priorities and quotas are details below s1 : o 1, o 2, d1 : o 1,, o 2 π o1 : s 2, d 1, s 1, d 2 q o1 = 2 s2 : o 2, o 1, d2 : o 2,, o 1 π o2 : s 1, d 2, s 2, d 1 q o2 = 2 We now show that this matching problem has two undominated stable matchings: µ and µ such that X(µ) = {(s 1, o 1 ), (s 2, o 1 ), (d 1, ), (d 2, o 2 )} and X(µ ) = {(s 1, o 2 ), (s 2, o 2 ), (d 1, o 1 ), (d 2, )}. In µ, s 1 and d 2 are not involved in any blocking pair since they receive their first preference. s 2 is not involved in any blocking pair either since both units of o 2 are assigned to d 2, who has a higher priority. The same is true for d 1 since one unit of o 1 is assigned to s 2, who has a higher priority. µ is therefore a stable matching. Observe that µ is stable despite the fact that a unit of o 1 is assigned to s 1, who has a lower priority than d 2. d 2 and o 1 do not form a blocking pair because d 2 prefers two units of to one unit of o 2. To show that µ is undominated, suppose to the contrary that there exists a stable matching µ that dominates µ. Since s 1 and d 2 receive their first preference under µ, it must be that µ(s 1 ) = µ(s 1 ) = o 1 11

12 and µ(d 2 ) = µ(d 2 ) = o 2. Then d 1 and o 1 form a blocking pair unless µ(s 2 ) = o 1. As all units of o 1 and o 2 are assigned, feasibility dictates that µ(d 2 ) =. It follows that µ = µ, hence µ is an undominated stable matching. µ can be shown to also be an undominated stable matching as well through an analogous reasoning. Consequently, this matching problem has two undominated stable matchings, µ favors s 1 and d 2 while µ favors This matching problem has two undominated stable matchings, µ, which favors s 1 and d 2 and µ, which favors s 2 and d 1. In fact, it can be shown that µ and µ are the only two stable matchings in the problem. 3.2 Deferred-Acceptance We have seen the the above two examples that an optimal stable matching is not guaranteed to exist, through either the emptiness of the set of stable matchings or the multiplicity of undominated stable matchings. An obvious implication is that, contrary to the school choice model, the deferredacceptance algorithm is not guaranteed to find the optimal stable matching. A natural question is then whether the deferred-acceptance algorithm finds the optimal stable matching whenever it exists. We answer by the negative in the following example. Example 3 (DA fails to find OSM). There are three single-unit agents s 1, s 2 and s 3, two doubleunit agent d 1 and d 2 and three non-null objects o 1, o 2 and o 3. The preferences, priorities and quotas are presented below. s1 : o 1, o 3, o 2, d1 : o 1, o 3, o 2, π o1 : s 3, d 1, d 2, s 1, s 2 q o1 = 2 s2 : o 3, o 2, o 1, d2 : o 3,, o 2, o 1 π o2 : s 1, s 2, d 2, s 3, d 1 q o2 = 1 s3 : o 2, o 1, o 3, π o3 : s 3, d 1, d 2, s 1, s 2 q o3 = 2 An optimal stable matching exists in this problem: µ such that X(µ ) = {(s 1, o 1 ), (s 2, o 2 ), (s 3, o 1 ), (d 1, o 3 ),(d 2, )}. We show this in two parts by proving first that µ is stable and second that it dominates any other stable matching. s 1 receives his first preference and are not involved in any blocking pair. All other agents receives their second preference but their first one is assigned to an agent with a higher priority. For s 2 and d 2, both units of o 3 are assigned to d 1. For s 3, the only unit of o 2 is assigned to s 2 and for d 1 only one unit of o 1 is available since the other is assigned to s 3. µ is stable. Suppose now there exists a matching µ that is not dominated by µ. Then at least one agent is better-off under µ than under µ, which implies at least one of the following: µ(s 2 ) = o 3, µ(s 3 ) = o 2, µ(d 1 ) = o 1 or µ(d 2 ) = o 3. If µ (s 2 ) = o 3, then both d 2 forms a blocking pair with o 3 unless µ (s 3 ) = o 3, then s 3 forms a blocking pair with o 1 and µ si not stable. If µ(s 3 ) = o 2, then s 2 forms a blocking pair with o 2 unless he is matched with o 3, which contradicts stability. If µ (d 1 ) = o 1, then 12

13 s 3 forms a blocking pair with o 1 unless he is matched with o 2, which contradicts stability. Finally, if µ (d 2 ) = o 3, d 1 forms a blocking pair with o 3 unless he is matched with o 1, which contradicts stability. We now show that Gale and Shapley s (1962) deferred-acceptance algorithm does not find µ. The algorithm works as follows: The agent-proposing deferred-acceptance algorithm (Gale and Shapley, 1962) is defined as follows: Round 1: All agents propose to the object of their first preference. All objects tentatively accept proposals in order of priority up to their quota and reject all others. If all proposals are tentatively accepted, the algorithm ends and all agents receive their first preference. Otherwise, the algorithm continues in Round 2. [...] Round k: All agents whose proposal was tentatively accepted in Round k 1 propose to the same object again. All agent whose proposal was rejected in Round k 1 propose to their next preferred object. All objects tentatively accept proposals in order of priority up to their quota and reject all others. If all proposals are tentatively accepted, the algorithm ends and all agents receive the object of their proposal. otherwise, the algorithm continues in Round k + 1. Table 1 displays the steps of the deferred-acceptance algorithm when applied to this matching problem. A matching µ DA is obtained after five rounds when all proposals are accepted: X(µ DA ) = {(s 1, o 2 ), (s 2, o 1 ), (s 3, o 1 ), (d 1, o 3 )}. Clearly, µ µ DA. In fact, µ DA is not stable since s 1 prefers o 1 to o 2 and has a higher priority than s 2. Round 1 Round 2 Round 3 Round 4 Round 5 Round 6 s 1 o 1 s 1 o 3 s 1 o 3 s 1 o 3 s 1 o 2 s 1 o 2 s 2 o 3 s 2 o 2 s 2 o 2 s 2 o 2 s 2 o 2 s 2 o 1 s 3 o 2 s 3 o 2 s 3 o 1 s 3 o 1 s 3 o 1 s 3 o 1 d 1 o 1 d 1 o 1 d 1 o 1 d 1 o 3 d 1 o 3 d 1 o 3 d 2 o 3 d 2 o 3 d 2 d 2 d 2 d 2 Table 1: Deferred-Acceptance Algorithm applied to the Matching Problem from Example 3. The deferred-acceptance algorithm fails to find a stable matching even though an optimal one exists. In Round 1, s 1 is rejected by o 1 because d 1 is tentatively assigned both units. In Round 2, d 1 is rejected by o 1 because s 3 has a higher priority. As s 3 only uses one unit, the second one is unassigned so s 1 and o 1 form a blocking pair. The final matching is not stable unless another agent with a higher priority than s 1 is assigned that unit in a subsequent round. This does not happen in this example, the second unit of o 1 is finally given to s 2, whose priority for o 1 is lower than s 1 s. A solution would have been to allow s 1 to remain on hold for o 1 instead of rejecting him straight away. He would 13

14 then receive the second unit of o 1 in Round 2 and the OSM would be found. This is the essence of the Top-Down Bottom-Up algorithm, which we introduce next. 4 Top-Down Bottom-Up This section is devoted to the Top-Down Bottom-Up algorithm, which we use to eliminate contracts that cannot be part of any stable matching and thus reduce the number of contracts that need to be considered. We provide intuition for our approach in Section 4.1, introduce the algorithm in Section 4.2, provide an example in Section 4.3 and discuss the implications of our findings in Section Approach The general idea of the Top-Down Bottom-Up algorithm is to identify contracts that are not part of any stable matching and eliminates them. This principle lies at the heart of the deferred acceptance algorithm, in that sense the TDBU algorithm can be thought of as a generalization of the DA algorithm that is suitable in an environment where agents may want different quantities of an object. To see this, consider the school choice environment, where all agents demand one unit. In any round of the DA algorithm, an object o available in q o units rejects an agent a if it has received at least q o proposals from agents with a priority higher than a s. If a and o are matched together, then at least one agent with a priority higher than a s, say a, is not matched with o. a and o form a blocking pair unless a is matched with an object he prefers. That object must have rejected him in a previous round so a cannot be matched to that object without taking the place of someone with a higher priority. By induction, a blocking pair exists. The bottom-up part of the algorithm adapts the agent-proposing deferred-acceptance algorithm. An agent is rejected by an object if the number of units he demands exceeds the number of units of the object that are not tentatively assigned to an agent with a higher priority. As units are tentatively assigned to agents who consider the object to be their top-choice, this can be thought of as agents proposing and objects tentatively accepting proposals up to their capacity. A special provision allows single-unit agents, in some cases, to not be rejected despite all units of the object being tentatively assigned to agents with a higher priority. This is designed to avoid situations such as the one presented in Example 3 and ensure that only contracts that are not part of any stable matching are eliminated. The top-down part of the algorithm adapts the object proposing deferred-acceptance algorithm. An agent receives a guarantee from an object if the total number of units demanded by him and agents with a higher priority does not exceed the object s quota. A guarantee means that, in any stable matching, the agent is either matched with that object or with one he prefers. In the language of the deferred-acceptance algorithm, guarantees correspond to objects proposing to agents. If an agent receives a proposal, he is guaranteed to be match with at worst that object in any stable 14

15 matching. The consequence of a guarantee in the algorithm is that all contracts involving the agent and an object he likes less can be eliminated. Throughout the algorithm, contracts are eliminated both from the top-down and the bottom-up until all a round occurs where no new contract can be eliminated. As we show in Proposition 1, the matching constructed by giving all agents their top-choice is the optimal stable matching if and only if it is feasible. We deal with the case where it is non-feasible in Section 6. We now illustrate the top-down bottom-up with an example before introducing it formally. 4.2 Description In any given round of the TDBU algorithm, a matching problem M with a set of contracts X enters. The guarantee number and rejection number of each contract is calculated as follows. The guarantee number of contract (a, o), G M (a,o), is the sum of all units demanded by agents with a higher priority for o plus the units demanded by a himself. The rejection number of contract (a, o) in matching problem M, R(a,o) M, is defined in an almost analogous way but only those agents who have o as their top-choice are taken into account. Formally, G M (a,o) := w a + w a and R(a,o) M := w a + w a, (1) a ÂM (a,o) a ÂM (a,o) AM o Agent a receives a guarantee for object o if their contract s guarantee number is below o s quota (G M (a,o) q o). In this case, agent a is assigned o or an object he likes more in any stable matching. To see this, notice that a is matched with an object he likes less, he and object o form a blocking pair because there are not enough agents with a priority for o higher than a s to prevent a from being matched with o. There exists a special case where a single-unit agent can receive a guarantee despite having a guarantee number that exceeds the quota. Suppose that, for some object o, the guarantee number of a double-unit agent d is q o + 1. Suppose also that o is the top-choice of all single-unit agents with a higher priority than d. Then the single-unit agent with the highest priority among those who have a lower priority than d, say s, receives a guarantee. To see this, suppose there exists a stable matching where s is matched with an object he likes less than o. By stability, all agents who have a priority higher than d (hence a guarantee) and consider o to be their top-choice must be matched with o. Among them are, by assumption, all single-unit agents with a higher priority than s. Since R (d,o) = q o 1, the difference between q o and the number of these single-unit agents is odd. It follows that an odd number of units remains to be assigned once all agents with a guarantee and for whom o is the top-choice are matched with o is odd. Of the remaining agents, only double-unit agents have a higher priority than s, therefore at least one unit is either unassigned or assigned to a single-unit agent with a lower priority than s. s and o consequently form a blocking pair. 15

16 Formally, any s S such that G M (s,o) > q o receives a guarantee for o if (i) There exists d D such that G (d,o) = q o + 1; and (ii) For all s ŜM (s,o), G (s,o) < q o and o M s = o. (2) A double-unit agent d is rejected from object o if their contract s rejection number is larger than o s quota (R(d,o) M q o). Such a contract cannot belong to a stable matching, if it did the priority of agents with a higher priority than d for whom o is the top-choice would be violated. Rejection decisions involving single-unit agents are more complex because they may take an object s last unit that double-unit agents do not want unless they can get a second one. Consequently, single-unit agents may remain on hold in two cases.. First, suppose that a double-unit agent d 1 has o as his top-choice and rejection number q o + 1. d 1 is rejected in this round and does not contest o in the next one, hence the rejection number of any agent with a lower priority will fall by two. If a single-unit agent s 1 has the next priority, his current rejection number is q o + 2 but in the next round it may fall to q. This continues to be true if there are more double-unit agents with o as their top choice and a priority between d 1 and s 1. For example, if there exists d 2 with top choice o and a priority lower than d 1 but higher than s 1, his rejection number is q o + 3 and s 1 s rejection number is q o + 4. As both d 1 and d 2 are rejected, s 1 s rejection number falls to q o in the next round so he should not be rejected in this one. Second, suppose that d 1 still has o as his top choice but rejection number q o while s 1 has rejection number q o + 1. It is possible that, in a subsequent round, o becomes the top-choice of another single-unit agent, say s 2, with a priority higher than d 1. As a result, d 1 s rejection number moves to q o + 1 and he is rejected, allowing s 1 s rejection number to fall to q o. We are now in a position to formally define the rejection rule for single-unit agents. A single-unit agent s is on hold for an object o if either R(s,o) M > q o or there exists d D such that d π o s and o d o d. We denote by H M the set of contracts involving a single-unit agents and on object for which he is on hold. For any (s, o) X, s is rejected by o if (s, o) H M and at least one of the following is satisfied: (i) There exists s such that (s, o) H M A M o and s π o s; or (ii) R (s,o) > q o and there exists s A o such that s π o s and R(s M,o) = q o; or (3) (iii) R (s,o) q o and for any s A M o such that R (s,o) < q o 1, o M s = o. We say that a single-unit agent remains on hold for a given object if it is on hold for that object but not rejected as neither of the above conditions is satisfied. Part (i) of Condition (3) considers the case where multiple single-unit agents are on hold. If a double-unit agent is rejected, only one unit can be freed, hence at most one single-unit agent can remain on hold for any given object. Part (ii) considers the case where the last unit is tentatively assigned to a single-unit agent. No unit will be freed in this case, hence no agent can remain on hold. 16

17 Part (iii) considers the case where there is no single-unit agent with contesting o without it being his top-choice. If a single-unit agent s rejection number is beyond the quota, its number can only go down if the object becomes the top-choice of another single-unit agent. If this cannot happen, no agent can remain on hold for that object. Once all contracts have been considered for guarantee and rejection, a subset of contracts is eliminated as follows. For each contract where the agent was given a guarantee, contracts involving that agent and objects he likes less are eliminated. All contracts where the agent was rejected are eliminated. If at least one contract was eliminated, the algorithm continues in the next round where the remaining contracts enter. Otherwise the algorithm terminates. We now summarize and formally introduce the algorithm. Top-Down Bottom-Up (TDBU) Algorithm Round 0: Given a matching problem M, π, w, q, X, let M 1 := M and X 1 := X. The algorithm continues in Round 1. For each contract (a, o) X k : Round k 1: Calculate its guarantee number G M k (a,o) and rejection numbers RM k (a,o). a receives a guarantee for o if G M k (a,o) q o or if a S and Condition (2) is satisfied, a is rejected by o if R M k (a,o) > q o and either a D or Condition (3) is satisfied. Each contract (a, o) is eliminated if a was rejected by o or if a obtained a guarantee for some o such that o a o. Let E k be the set of all eliminated contract. If E k, let X k+1 := X k \ E k and M k+1 :=, π, w, q, X k+1. The algorithm continues in Round k + 1. If E k =, the algorithm terminates and yields TDBU(M) := M k. The Top-Down Bottom-Up algorithm reduces the complexity of the matching problem by eliminating contracts that are incompatible with stability. Given the smaller set of contracts obtrained, it may be possible to accommodate all top-choices, in which case the optimal stable matching has been found. Unfortunately, this is not always possible, in which case finding a stable matching requires eliminating more contracts. Letting ˆM := TDBU(M) and defining µ ˆM to be the matching that assigns each agent their top choice, that is µ ˆM(a) := o ˆM a in the following proposition: for all a A, we summarize these results Proposition 1. µ ˆM does not contain any blocking pair and dominates any stable matching. It is the optimal stable matching if and only if it is feasible. 17