The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples

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1 The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples Oliver Hinder, Stanford University X The deferred acceptance algorithm has been the most commonly studied tool for computing stable matchings. An alternate less-studied approach is to use integer programming formulations and linear programming relaxations to compute optimal stable matchings. Papers in this area tend to focus on the simple ordinal preferences of the stable marriage problem. This paper advocates the use of linear programming for computing stable matchings with more general preferences: complements, substitutes and responsiveness, by presenting a series of qualitative and computational results. First, we show how linear programming relaxations can provide strong qualitative insights by deriving a new approximate rural hospital theorem. The standard rural hospital theorem, which states that every stable outcome matches the same doctors and hospitals, is known to fail in the presence of couples. We show that the total number of doctors and hospitals that change from matched to unmatched, and vice versa, between stable matchings is, at most, twice the number of couples. Next, we move from qualitative to computational insights, by outlining sufficient conditions for when our linear program returns a stable matchings. We show solving the stable matching linear program will yield a stable matching (i) for the doctor-optimal objective (or hospital-optimal), when agent preferences obey substitutes and the law of aggregate demand, and (ii) for any objective, when agent preferences over sets of contracts are responsive. Finally, we demonstrate the computational power of our linear program via synthetic experiments for finding stable matchings in markets with couples. Our linear program more frequently finds stable matchings than a deferred acceptance algorithm that accommodates couples. Categories and Subject Descriptors: C.2.2 [Theory and Foundations]: Market design 1. INTRODUCTION Perhaps the most famous matching market is the National Residency Matching Program (NRMP) in U.S. Each year the NRMP matches 16,000 newly graduating doctors to their first medical position at a hospital 1. Deciding how to match doctors and hospitals is challenging, because each side has preferences concerning matches. Many doctors would like to live in a particular city or work at a prestigious hospital. Hospitals may want doctors in certain specializations and prefer the best students. Following a round of interviews the doctors and hospitals construct preference lists. The centralized clearinghouse uses these preference lists to match the doctors to hospitals. The matching is said to be stable if there is no hospital and doctor who both prefer each other over their current match. Stability implies the market is in equilibrium and matchings do not change. Stable matchings are popular because of theoretical and empirical evidence that markets function effectively when they are operated by a centralized mechanism that outputs a stable matching [Kagel and Roth 2000; Roth 2002; Roth and Sotomayor 1992]. Now assuming we wish to design a market to be stable, two questions that naturally arise are: i. Under what conditions can we efficiently find stable matches? Being able to quickly find stable matchings is important if we are to use them in real markets. Moreover, it is not clear that they will exist, for example, in markets with couples. ii. Can we find the optimal stable matching? We might want to pick the best stable matching according to some objective. For example, we might want to maximize the number of doctors that are employed. Under what conditions can we efficiently find stable matchings? We know that stable matchings always exist when the market can be split into two sides (i.e. doctors and hospitals) and agent preferences are substitutable [Hatfield and Milgrom 2005]. In this case, we can find a stable matching quickly using the deferred acceptance algorithm. However, in many markets agent preferences contain complementaries. In labor markets, couples create complementaries because they have preferences over pairs of jobs (e.g. they want to live in the same city). We can construct examples of small labor markets where the introduction of one couple causes no stable matchings to exist [Roth 1984]. However, empirical evidence suggests that these examples are rare: in real markets we seem to always be able to find a stable matching [Kojima et al. 2014; Roth and Peranson 1999]. Accordingly, Ashlagi et al. 1 There are 16,000 U.S. students that participated in the match in We exclude foreign doctors because they are significantly less competitive in the match [Ashlagi et al. 2014]. The author is supported at Stanford by the PACCAR INC fellowship. Author s addresses: Oliver Hinder, Management Science and Engineering, Stanford.

2 X:2 Oliver Hinder [2014] show in random markets if the number of couples grows sub-linearly with the market size, then the probability that a stable matching exists tends to one. We can find these stable matchings using a variation of the deferred acceptance algorithm that accommodates couples. While the deferred acceptance algorithm is the typical approach for finding stable matchings [Gale and Shapley 1962], it is not the only approach. One recent suggestion for finding stable matchings in the presence of complementaries is based on Scarf s algorithm [Biro et al. 2013]. Scarf s algorithm guarantees to find a fractional stable matching, even when no integral stable matching exists. When the matchings found are integer the algorithm guarantees to have found a stable matching. Furthermore, we can round the solutions found to find approximately feasible stable matchings [Nguyen and Vohra 2014]. Unfortunately, finding a Scarf solution is PPAD complete, so it is unlikely we can solve it efficiently [Kintali 2008]. Can we find the optimal stable matching? Given a stable matching exists, we would like to choose the best stable matching. Policymakers often ask if they could shift more doctors from the urban hospitals to the unpopular rural hospitals [Kamada and Kojima 2010; Roth 1984]. The rural hospital theorem famously states that, if agent preferences satisfy substitutes and the law of aggregate demand then, the same positions will be filled and the same doctors will be employed in every stable matching [Hatfield and Milgrom 2005; Roth 1984]. This implies we cannot shift doctors from urban hospitals to rural hospitals and maintain stability. However, this result is well-known to fail in the presence of couples [Aldershof and Carducci 1996]. Nonetheless, we might expect adding a small number of couples into the market would not cause large variations in employment levels. There is empirical evidence to support this intuition: when Roth and Peranson [1999] ran experiments with the deferred acceptance algorithm they found the rural hospital theorem was rarely violated. However, this has not been demonstrated rigorously, because there is no reason to believe the deferred acceptance algorithm (and its modifications) should give a stable matching that minimizes or maximizes employment. Even if who is matched does not change between stable outcomes, there still could be other differences. Some doctors or hospitals may get better positions in different stable matchings. The deferred acceptance algorithm can produce the stable matching that is unanimously optimal for either the doctors or the hospitals [Hatfield and Milgrom 2005]. However, it is unclear that policymakers necessarily want to optimize for just one side of the market. For example, they may want to pick the stable matching with the maximum social welfare or the most fair stable matching [Sethuraman et al. 2006]. Vande Vate [1988] gave a linear program that allows us to compute the optimal stable marriage with a linear objective. Unfortunately, the value of this particular linear program is limited for two reasons. Firstly, stable marriage matches are one-to-one, but in real markets matches are many-to-many: couples are matched with a pair of hospitals and hospitals are matched to many doctors. Secondly, real world objectives are frequently non-linear. For example, when allocating students to schools one might want to optimize diversity [Kominers and Sönmez 2013; Hafalir et al. 2013]. With this objective, in good allocations, schools will have a mixture of students e.g. some students from poor households and some students from wealthy households. Maximizing this mixture is fundamentally a non-linear objective Our contributions This paper aims to progress our understanding of these two fundamental questions in markets using linear programming. We introduce a new approximate rural hospital theorem that applies when there are couples in the market. We also prove that our linear program can find stable matchings (i) for the doctor-optimal objective (or hospital-optimal), when agent preferences obey substitutes and the law of aggregate demand, and (ii) for any objective, when agent preferences over sets of contracts are responsive. Furthermore, we demonstrate in synthetic experiments that linear programming frequently finds stable matchings with couples. We will now dissect the contributions of each section: Section 2 presents definitions necessary for the rest of the paper. Section 3 presents an integer programming formulation of stable matching with arbitrary preferences. Relaxing the integer requirements yields the stable matching linear program. This generalizes the stable marriage linear program [Vande Vate 1988] to arbitrary preferences over sets of contracts. Section 4 provides an approximate rural hospital theorem that applies when couples exist in markets. We show the total number of doctors and hospitals who switch from matched to unmatched and vice versa, between two stable matchings, is at most twice the number of couples who change position. We demonstrate this result is tight and show it indicates that, in random markets, the rural hospital theorem is only slightly violated because couples infrequently change position.

3 The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples X:3 Section 5 gives conditions that guarantee our linear program finds a stable matching. It is known that linear programming finds stable matchings for the stable marriage problem [Vande Vate 1988]. We broaden this result, by showing that solving the stable matching linear program will yield a stable matching (i) for the doctor-optimal objective (or hospital-optimal), when agent preferences obey substitutes and the law of aggregate demand, and (ii) for any objective, when agent preferences over sets of contracts are responsive. The latter result implies we are able to optimize over the sum of arbitrary functions on the set of contracts allocated to agents. Consequently, we can optimize non-linear objectives such as socio-economic diversity, subject to stability constraints. Section 6 demonstrates the power of linear programming and integer programming for finding stable matchings when couples are present. The deferred acceptance algorithm has proven to be an effective heuristic for finding stable matchings with couples. We show our linear relaxation is also effective. We solve random instances using our linear relaxation and report a stable matching when the solution integer. We find that the linear program more frequently finds stable matchings than a deferred acceptance algorithm that accommodates couples [Ashlagi et al. 2014]. We also observe that a commercial integer programming solver quickly proves or disproves the existence of stable matchings. Section 7 concludes the paper Related work The stable marriage linear program was first introduced and proved to be naturally integer by Vande Vate [1988]. Following this, Roth et al. [1993] used linear programming and duality arguments to re-establish many important theorems in stable marriages and provided a new proof that the stable marriage linear program is naturally integer. This paper is particularly relevant to our work, because we leverage some of the proof ideas. Later, Abeledo and Rothblum [1995] showed when this linear program is applied to the stable roommate problem the polytope is always non-empty, even though there may exist no stable matching. In response, Teo and Sethuraman [1998] created a linear programming formulation of the stable roommate problem that was feasible if and only if there exists a stable matching. In 2000, Baïou and Balinski demonstrated that, for the special case of college admissions, one can construct a naturally integer linear program with fewer variables, but an exponential number of constraints; they also provided a polynomial time oracle for finding a violated constraint. In 2006, Sethuraman et al. explored the structure of solutions to this linear program. Their results demonstrated how to use linear programming to solve a special case of the hospital-resident matching with couples. They illustrated that, if couples are restricted to ranking hospitals by geographic region, then linear programming can be used to prove or disprove the existence of a stable matching. Another relevant paper is by Biró et al. [2014] who showed that, on synthetic examples, an integer programming solver can quickly find the maximum cardinality matching for the hospital-resident problem with couples and ties. Integer programming solvers essentially solve a series of linear programming sub-problems. Therefore by providing theoretical and empirical evidence to show linear programming is efficient at solving stable matching problems (with couples) our paper helps explain why integer programming is able to solve large problem instances. One of the purposes of our work is to generalize the stable marriage linear program to a manyto-many contracts framework where agent preferences can satisfy more general conditions. Hatfield and Milgrom [2005] expanded stable marriages (and other models of stable matchings) to one-to-many matchings, in a framework known as matching with contracts. In this model, agents have preferences over set of contracts that they sign with each other. This model was further extended to many-tomany matchings by Hatfield and Kominers [2014]. These papers show that the deferred acceptance algorithm finds the doctor-optimal or hospital-optimal stable matching, when preferences satisfy substitutes. However, the deferred acceptance algorithm is unable to find the optimal stable matching for an arbitrary objective. 2. DEFINITIONS This section introduces definitions necessary for the remainder of the paper. The notation is an extension of matching with contracts framework to many-to-many matchings, which gives a generic way of expressing a large variety of stable matching problems [Hatfield and Milgrom 2005]. A stable matching problem is specified by a finite set of contracts X, a set of agents A and a choice function C a : 2 Xa 2 Xa for each agent a A. Each contract x X represents an agreement between the set x A of agents where x A 2. There may be multiple contracts between the same set of agents, which might, for example, express different terms of employment. In literature, it is generally assumed that agents have pairwise preferences, that is contracts are signed between two agents. More formally,

4 X:4 Oliver Hinder an agent a has pairwise preferences if x A = 2 for every x X a, where X a = {x X : a x A } is the set of contracts that agent a is able to sign. However, as we will show later, removing the assumption of pairwise preferences is an alternative way of modeling the introduction of couples to the market. The choice function C a (S) specifies the subset of contracts agent a A would choose from the set of available contracts S X a. A contract x can only be in the choice set C a (S) of an agent a if a x A. We assume that there exists some underlying utility function u a : 2 Xa R such that such that u a (S) = u a (S ) where C(S) = arg max T S {u a (T )}. This implies that agents have strict orderings over sets of contracts i.e. < S 1 < a S 2 < a... < a S n. Example 2.1 (Choice functions). If an agent a has the preferences {x 1, x 2 } > a {x 3 } > a over contracts then C a ({x 1, x 2, x 3 }) = {x 1, x 2 } and C a ({x 1, x 3 }) = {x 3 }. A matching of contracts M : A 2 Xa is function where M(a) = M a denotes the set of contracts allocated to agent a. We say an agent a signs the contract x if x M a. A matching M must satisfy the following property, which states if agent α x A signs the contract x all agents a x A sign the contract x. We describe a contract x X as used if x M a for all a x A and unused if x M a for all a x A. Clearly if an matching M is a matching each contract is either used or unused. An matching M is individually rational if for any a A that C a (M a ) = M a. Individual rationality implies that agents will not keep contracts they do not want. We use Q a = {S X a : C a (S) = S} to denote all the sets that are individually rational for the agent a. Therefore an alternative way to state individual rationality is that M a Q a. We will describe a matching M as stable if for every unused contract x X at least one agent a x A does not want to sign it: x C a (M a x). We overload M a x to mean M a {x}. We emphasize this implies only contracts can block allocations, not sets of contracts, so this not as strong as other definitions of stability in literature 2 [Hatfield and Milgrom 2005]. We now define three basic restrictions on agent preferences: substitutes, the law of aggregate demand and responsiveness. A choice function C a obeys substitutes if x, x X, B X and x C a (B) implies x C a (B x ). When all agent preferences satisfy substitutes there will always exist a stable matching. A choice function C a obeys the law of aggregate demand if C a (U V ) C a (U) for all U, V X. An agent a has responsive preferences if there exists q a such that (i) C a (x) = x implies S = C a (S) for any S q i and (ii) the contracts can be listed x 1 a... a x n such that if i > j and x j C a (S x i x j ) then x i C a (S x i x j ). If a choice function is responsive then it follows that it satisfies substitutes and the law of aggregate demand. Finally, the most basic preferences are ordinal preferences, where agents only ever want at most one contract, this is described formally as C a (S) 1 for all S Q a. Example 2.2 (Types of preferences). The preferences: {x 1 } > {x 2 } > {x 3 } > are ordinal, {x 1, x 2 } > do not obey substitutes or the law of aggregate demand and {x 2, x 3 } > {x 1, x 2 } > {x 1 } > {x 2 } > {x 3 } > obey substitutes and the law of aggregate demand, but are not responsive. For the next definition we will assume the market is pairwise. A matching market is two-sided if the set of agents A can be partitioned into a set of doctors D and hospitals H such that no two doctors or two hospital share contracts. Formally, this means for any x X and {a 1, a 2 } = x A either a 1 D and a 2 H or a 1 H and a 2 D. Furthermore we will denote the doctor and hospital associated with contract x as x D and x H respectively. The next definition does not assume the market is two-sided. We will now define the hospital-resident problem with couples. In this matching market each single doctor and position at a hospital will be an agent, furthermore each couple will be an agent that signs at most one contract with one or two hospital positions. Consequently, up to three agents may sign one contract. The market is partitioned into single doctors and couples in the set D and the positions in the set H. However, because we allow three agents to sign a contract the market is not two-sided according to our definition. The set of couples is K D, the set of single doctors is D \ K. Couples can sign one contract which falls into two different categories: i. Single application. One contract with one hospital where only one member is employed. ii. Joint application. One contract with two hospitals where each member is employed by a different hospital. It is possible in two different contracts one hospital is matched to the same member of the couple. In this case, hospitals break ties between contracts in favor of the couples preferences. See the Example But, is equivalent to these definitions if agent preferences obey substitutes.

5 The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples X:5 All agents, including the couples, have ordinal preferences. The complementary preferences of the couples have been introduced by allowing contracts to be signed between more than two agents. We wish to find a stable matching. We remark that this is trivially equivalent to standard definitions [Ashlagi et al. 2014; Kojima et al. 2014]. Example 2.3 (A hospital-resident problem with couples). Consider a market with the following agents one couple k, doctor d and two hospitals h 1 and h 2. Suppose the following contracts are possible x 1 = ((k m, h 1 ), (k f, h 2 )), x 2 = ((k m, h 1 )), x 3 = ((d, h 1 )). The contract x 1 represents when k m is matched to h 1 and k f is matched to h 2. It is a three way contract because x 1 A = {k, h 1, h 2 }. All other contracts are pairwise contracts. Now suppose the hospital h 1 prefers k m over d and the couple k prefers x 1 over x 2. Clearly h 1 is indifferent between x 1 and x 2. Therefore we break ties in the hospitals preferences over contracts in favour of the couples. So the preference list of the hospital h 1 is x 1 > h1 x 2 > h1 x 3. We remark that this allows the couples to express complementary preferences over positions while only having ordinal preferences over contracts Generation of random preferences In this paper, we use synthetically generate preferences to indicate how the algorithms may behave on real problem instances. A simple approximation to agent preferences is to assume they are generated uniformly at random in the following way. Each doctor uniformly at random ranks 10 positions. All other positions are unacceptable. Each couples uniformly at random ranks 30 pairs of positions. All other positions are unacceptable. Each position consists of a single hospital which uniformly at random ranks the doctors. There are three variables that we can adjust when generating these examples: the size of the market, the applicants in a couple and the number of available positions. 3. FORMULATING STABLE MATCHING AS AN INTEGER PROGRAM This section presents an integer program with a solution set that exactly corresponds to the set of stable matchings. This provides a generalization of the stable marriage linear program [Vande Vate 1988] to the matching with contracts framework. We will now proceed to present the constraints that form the matching integer program, we will then present the stability constraints which combined with matching integer program forms the stable matching integer program (Figure 1). Finally we discuss four possible objectives (Figure 2). Fig. 1: Stable matching integer program Let z x {0, 1} be whether contract x is used. Let y a,b {0, 1} be whether the set B of contracts is allocated to agent a. Therefore y a,b = 1 is equivalent to B = M a. Set constraint. Each agent is allocated exactly one set of contracts (may be the emptyset). B Q a y a,b = 1 a A (1) Matching constraint. If a contract x X is used then all agents a x A must sign it. z x = y a,b x X, a x A (2) B Q a:x B Non-negativity constraint. z x, y a,b 0 Integer constraints. z x, y a,b Z Stability constraint. If a contract x X is unused then at least one agent a x A does not want to sign it. z x + y a,b 1 x X (5) a x A B Q a:x C a(b x) Constraint (1) and (3) guarantee that y a,b 1 and hence by (2) that z x 1. Combining this with constraint (4) implies that z x {0, 1} and y a,b {0, 1}. (3) (4)

6 X:6 Oliver Hinder We can think of the matching integer program formed from constraints (1), (2), (3) and (4) as a generalization of bipartite matching. We call the linear program naturally integer when all extreme points are integer. Consequently, we can remove the integer constraints to solve the linear program. While bipartite matching is naturally integer, the linear relaxation of the matching integer program can have fractional extreme points. Furthermore, even when the market is two-sided, finding the optimal integer matching can be easily reduced to a maximum set packing problem 3, so it is NP-hard [Motwani and Raghavan 2010, chapter 8]. Therefore, it is somewhat surprising that by adding the stability constraint and insisting on responsive agent preferences the linear program becomes naturally integer (Corollary 5.7). The linear program formed from constraints (1), (2), (3) and (5) is called the stable matching linear program. When the binary constraint (4) is added it becomes the stable matching integer program. It is already known when the market is two-sided and agents have ordinal preferences this linear program becomes equivalent to the stable marriage linear program and therefore is naturally integer [Vande Vate 1988]. Further, we make the observation that any stable matching corresponds to an integer solution to linear program and visa versa. This holds independent of agent preferences. Fig. 2: Four relevant objectives for this paper. Maximum employment objective. Maximize the number of contracts signed. max x X z x (6) Stable marriage objective. Maximize a linear function over the contracts. max x X c x z x (7) Doctor-optimal objective. Maximize the utility of the doctors. max u d (B)y d,b d D B Q d Arbitrary objective. Maximize a linear function over sets of contracts allocated to agents. max f a (B)y a,b a A B Q a The function u a : X a R is the utility agent a gains by being allocated the set of contracts B. The function f a : X a R is any arbitrary function. (8) (9) However, it is possible that our stable matching linear program will not yield an integer solution. Nonetheless, we can identify conditions on preferences and the objective when our linear program is guaranteed to give a stable matching. Assume that the market is two-sided. Since the stable marriage linear program is naturally integer [Vande Vate 1988], the linear program will have an integer optimal solution, for any linear objective over the contracts (see (7)), when preferences are ordinal. Furthermore, as we will show Section 5.2, the linear program will have an integer optimum: (i) for the hospital-optimal objective (or doctor-optimal, see (8)), when agent preferences satisfy substitutes and the law of aggregate demand and (ii) for any objective (see (9)), when agent preferences are responsive. Finally, as we will show in the next section, using the maximum employment objective (see (6)), assuming the market is two-sided and agent preferences satisfy substitutes and the law of aggregate demand, the same agents will be matched in every stable matching. Furthermore, even when there are couples present in the market and agents have ordinal preferences we are able to bound how many agents switch from matched to unmatched and vice versa. To see the stable matching linear program simplified for ordinal preferences (but not necessarily pairwise preferences over contracts) see Appendix B. 3 Let {S 1,..., S n} = S be a maximum set packing problem. This is equivalent to a problem with one hospital h 1 and n doctors: d 1,...d n. Set Q h = S and Q di = {S i }. Set the objective to max n i=1 y d i,s i.

7 The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples X:7 4. AN APPROXIMATE RURAL HOSPITAL THEOREM WITH COUPLES In this section, the main result we will prove is that the total number of doctors and hospitals that change employment between stable matchings is at most the number of applicants in a couple (twice the number of couples). We begin this section by explaining the result, then provide an example to show the result is tight and show this result provides a useful guideline in random markets. The remainder of the section is used to prove the result. Let y and y be solutions to the stable matching linear program that correspond to two different stable matchings. Let m(y, y ) be the number of couples that change position between stable matching y and y. Let n a and n a be the employment level of agent a in solution y and y respectively. For a hospital the employment level corresponds to the number of doctors employed. For a doctor the employment level is one if she or he is employed and zero otherwise. For a couple the employment level is equal to two if both members are employed, one if one member is employed and zero if no members are employed. Recall that A is the set of agents, so includes doctors, couples and hospitals. Our result, which will be proved at the end of this section, is: THEOREM 4.1. The rural hospital theorem with couples. n a n a 2m(y, y ) a A To the best of our knowledge our paper gives the first statement and proof of this result. For fractional solutions to the stable matching linear program we define: { } m(y, y ) = max (z x z x z x), (z x z x z x) number of couples x X k x X k One advantage of proving this result using linear programming is that these bounds apply to fractional solutions. Consequently, we can find a fractional solution to the stable matching linear program and use it to bound properties of the set of stable matchings, even when we have not found a stable matching (since finding a stable matching with couples in NP-complete [Ronn 1990]). We now present an example with one couple and three single doctors. By duplicating this market m times we can show that our theorem is tight. The preferences are listed in Table I. The couple c comprises of two doctors c m and c f ; the three doctors are d 1, d 2 and d 3. These preferences yield two possible stable matchings shown in Table II. We observe that between stable matching 1 and stable matching 2 that d 1 switches to unemployed and d 3 switches to employed. Therefore a D n a n a = 2 = 2m(y, y ) which implies the result is tight. We remark that if d 1 was removed from the market then a A (n a n a ) = 2 and if d 3 was also removed from the market then a H n a n a = 2. This implies we cannot strengthen the result by moving the absolute values outside of the summation or making the summation only over one side of the market. h 1 c m > d 1 h 2 d 2 > c f h 3 c m > d 2 h 4 c f > d 3 c {(c m, h 1 ), (c f, h 2 )} > {(c m, h 3 ), (c f, h 4 )} d 1 h 1 d 2 h 3 > h 2 d 3 h 4 Table I: Example of agent preferences h 1 h 2 h 3 h 4 Stable matching 1 d 1 d 2 c m c f Stable matching 2 c m c f d 2 d 3 Table II: Set of stable matchings that result from the preferences in Table I Figure 3 shows that Theorem 4.1 provides a good guideline in markets with randomly generated preferences (see Subsection 2.1). The black line shows that for each couple that changes position between the stable matchings y and y we increase the violation of the rural hospital theorem

8 X:8 Oliver Hinder ( a A n a n a ) on average by 1.6. As shown by the red line, our theorem states that this value cannot be more than 2. This indicates that in random markets the rural hospital theorem rarely violated, because couples rarely change positions between stable matchings. Interestingly, when couples change position this rarely affects the employment of the couples. Most of the violations of the rural hospital theorem are by the doctors (44%) or hospitals (48%) rather than the couples (8%) a A n a na m(y, y ) Fig. 3: Scatter plot of m(y, y ) against a A n a n a where y and y are maximum and minimum employment stable matchings with 1000 doctors, 10% of applicants in a couple and the same number of hospitals as doctors (found using integer programming solver). The area of the dot is proportional to the number of observations. The red line is the upper bound produced by our theorem (y = 2x), the black line the best fit of the line (y = 1.6x) Proof of the Approximate Rural Hospital Theorem with Couples Typical proofs of the rural hospital theorem utilize the existence of a doctor-optimal and hospitaloptimal matchings [Roth 1984; Hatfield and Milgrom 2005]. Unfortunately, couples break the lattice structure, so it is no longer possible to use this proof technique. To prove the approximate rural hospital theorem with couples we utilize the complementary slackness property of linear programming and elegant arguments based on duality. Now, we begin the proof. First, we will introduce the dual of the stable matching linear program with the maximum employment objective. Given a feasible solution to the primal we are able to construct a corresponding feasible solution to the dual. Using weak complementary slackness (Theorem 4.4) we deduce our rural hospital theorem with couples. We will assume that agent preferences satisfy substitutes and the law of aggregate demand over the contracts. The arbitrary preferences of couples over pairs of positions are introduced by allowing the possibility of three way contracts ( x A = 3). Our technique employed is similar to Roth et al. [1993]. min a A α a x X γ x 0 η a x B κ a,x 1 γ x + x X a:x C a(b x) (10) γ x = f a,b (α, κ, γ) B Q a ; a A (11) κ a,x = g x (κ, γ) x X (12) a x A γ x 0, α a, κ a,x R (13) The dual of the stable matching linear program with the maximum employment objective (see (6)) is given by constraints (10)-(13). Constraints (11) and (12) correspond to primal variables y a,b and z x respectively. Dual variables η a, κ a,x and γ x correspond to constraints (1), (2) and (5) respectively.

9 The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples X:9 Consider the following substitution (η, z, κ) = π(y) = (η a (y), γ x (y), κ a,x (y)). η a (y) = z x a A (14) x X a γ x (y) = z x x X (15) κ a,x (y) = z x + y a,t a x A ; x X (16) T Q a:x C a(t x) LEMMA 4.2. If y is feasible to the primal then (η(y), κ(y), γ(y)) is feasible for the dual. In particular f a,b (η(y), κ(y), γ(y)) T Q a y a,t ( T B ) + 0 and g x (κ(y), γ(y)) ( x A 2)z x We give the proof in Appendix A. We now present three immediate and well-known corollaries assuming that the market is pairwise and therefore there are no couples ( x A = 2). These results are new because they apply to fractional solutions to our linear program. We will use this in Section 5. These results rely on the following fact, called complementary slackness. Definition 4.3. Let (y, π) be the duality gap between primal solution y and dual solution π that is: c T y b T π = (y, π) THEOREM 4.4 (PG BERTSIMAS AND TSITSIKLIS [1997]). Weak complementary slackness. Let y be a solution to the primal and π be a solution to the dual. Define u i = π i (a (i, ) y b i ) and v i = (c j π T a (,j) )y j where a (i, ) is the ith row and a (,j) is the jth column of the primal constraint matrix A. Then 0 i u i + j v j = c T y b T π = (y, π) Where u i, v i 0 We will let z and z correspond to solution y and y respectively to the stable matching linear program (see constraint (2)). COROLLARY 4.5. If the market is pairwise, then the primal objective of y is equal to the dual objective of π(y). Therefore any feasible solution to the primal is optimal. PROOF. Substituting (14), (15) and (16) into the objective (10) yields: η a γ x = ( x A 1)z x = (2 1)z x = z x a A x X x X x X x X COROLLARY 4.6. If the market is pairwise, then z x > 0 for one stable solution y implies (5) is satisfied at equality for all fractional stable matchings. PROOF. Follows from complementary slackness of (5) and γ x. This corollary essentially says if a contract x is used in one stable allocation, then in any stable allocation M with contract x unused, exactly one of the two agents {a 1, a 2 } = x A would like to add the contract x to her set of contracts M a. COROLLARY 4.7. Fractional rural hospital theorem. If the market is pairwise, then each agent signs the same weighted number of contracts in every fractional stable allocation. PROOF. Follows from complementary slackness of y a,b and f a,b (η, κ, γ). We now present the results that enable the inclusion of couples. LEMMA 4.8. Lower bound on duality gap. (y, π(y )) a A (η a η a ) + + x X ( x A 2)z x z x

10 X:10 Oliver Hinder PROOF. y a,b f a,b (η(y ), κ(y ), γ(y )) y a,b y a,t ( T B ) + (17) a A B Q a a A B Q a T Q a + y a,t T y a,b B (18) a A T Q a B Q a = (η a η a ) + (19) a A Line 17 holds by Lemma 4.2. Line 18 brings the summation inside the brackets and applies constraint 1. Line 19 follows by the definition of η a. (y, π(y )) y a,b f a,b (η(y ), κ(y ), γ(y )) + ( x A 2)z x z x (20) a A B Q a x X (η a η a ) + + ( x A 2)z x z x (21) a A x X Line 20 holds by weak complementary slackness. Line 21 holds by the previous inequality. This next lemma is equivilant to the standard rural hospital theorem when the market is pairwise ( x A = 2). LEMMA 4.9. Rural hospital theorem with complements. η a η a ( x A 2) (z x + z x 2z x z x) a A x X PROOF. ( (y, π(y)) + (y, π(y )) = ( x A 1) z x ) ( z x + ( x A 1) z x ) z x x X x X x X x X = x X ( x A 2) (z x + z x) Combining this with lemma 4.8 yields the result. We now present our main result. We now assume that this is a matching market with couples. THEOREM 4.1. The rural hospital theorem with couples. n a n a 2m(y, y ) a A PROOF. We observe that for the couples k K: n k n k = x X k ( x A 2)(z x z x ) + η k η k (22) And of course n a n a = η a η a for a A \ K. Now: n a n a ( x A 2)(z x z x ) + ( x A 2) (z x + z x 2z x z x) (23) a A k K x X k k K x X k = 2 { } max ( x A 2) (z x z x z x), ( x A 2) (z x z x z x) (24) k K x X k x X k 2 { } max (z x z x z x), (z x z x z x) (25) k K x X k x X k = 2m(y, y ) (26)

11 The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples X:11 Line 23 follows from line 22 and lemma 4.9. Line 24 comes from expanding the absolute signs on line 23 to a maximum and re-arranging. Line 25 is because 3 x A. Line 26 follows from the definition of m(y, y ). We observe this implies that the aggregate difference in the number of doctors employed at different stable matchings is at most the number of couples: COROLLARY Couples aggregate employment theorem. (n d n d) m(y, y ) d D While we only proved these result only when doctors and hospitals have ordinal preferences, we hope to extend it to when agents preferences satisfy substitutes and the law of aggregate demand. 5. WHEN ARE SOLUTIONS TO THE LINEAR PROGRAM INTEGER? This section identifies conditions when the stable matching linear program finds a stable outcome. In Subsection 5.1, we show that the if the market is two-sided, agent preferences are substitutable and obey the law of aggregate demand then each fractional solution to the stable matching linear program has a doctor-optimal and hospital-optimal rounding, which is a stable matching. We show this implies, when the linear program has the doctor-optimal (or hospital-optimal) objective, it will produce a stable matching. In Subsection 5.2, we prove that the if agents also have responsive preferences over sets of contracts the linear program is naturally integer. This an important result because it allows us to optimize over the sum of arbitrary functions over the contracts allocated to each agent. This would allow us to pick the stable matching that, for example, maximizes socio-economic diversity Doctor-optimal and hospital-optimal rounding In this section, we will show how we can round a fractional stable solution. In particular, each fractional stable solution has a doctor-optimal and hospital-optimal stable rounding, which is a stable matching. We use assume that the market is two-sided and the preferences satisfy substitutes and the law of aggregate demand. In the case of integer solutions this result is not new and is described as the lattice structure of stable matchings [Hatfield and Milgrom 2005]. The result is a generalization of Roth et al. [1993, lemma 12] to the stable marriage problem to stable matching. We now introduce the concept of the doctor-optimal and hospital-optimal rounding. Definition 5.1 (Doctor-optimal and hospital-optimal rounding). Let M(z) where M d (z) = C d ({x : z x > 0}) for every d D be the doctor-optimal solution rounding of the fractional solution z. Similarly, let M(z) where M h (z) = C h ({x : z x > 0}) for every h H be the hospital-optimal solution rounding of the fractional solution z. LEMMA 5.2. C h (B M h (z)) = B for all B such that y h,b > 0. Moreover C h (M h (z)) = M h (z) hence M(z) is individually rational. PROOF. We will start by proving C h (B M h ) = B. Let x M h, d = x D and h = x H this implies x C d ({u : z u > 0}) and hence by subsitutes B Q d :x C d (B x) y d,b = 0. The stability constraint (5) and set constraint (1) imply that: y h,b 1 = B Q h :C h (B x)=b 0 B Q h :C h (B x) B y h,b B Q h y h,b It follows that C h (B x) = B for all y h,b > 0 hence by substitutes C h (B M h ) = B. We will now prove C h (M h ) = M h. Let x M h. Now by construction of M there exists some B such that y h,b > 0 and x B. Now x B = C h (B M h ) by substitutes x C h (M h ). Hence M h = C h (M h ). The proofs for M(z) are identical by symmetry. THEOREM 5.3. M(z) is a stable matching.

12 X:12 Oliver Hinder PROOF. By lemma 5.2 we know M satisfies individual rationality. It remains to show that any contract x X which doctor d = x D wants will be rejected by the hospital h = x H. Let M d C d (M d x) by definition of M d and substitutes it follows that B Q d :C d (B x)=b y d,b = 0 then by the stability constraint (5) we can deduce that: y h,b = 1 B Q h :x C h (B x) Hence x C h (B x) for all y h,b > 0. If v M h then v B = C h (B M h ) = C h (B M h x) for some y h,b > 0 where the first equality holds by lemma 5.2 and the second by substitutes. By substitutes v C h (M h x), hence M h C h (M h x). We will now use the law of aggregate demand to show M h = C h (M h x) and the result will follow. Using the law of aggregate demand observe that n d M d by definition of M and that n h M h by lemma 5.2. However by definition of M we know that d D n d d D M d = h H M h h H n h. Combining this with d D n d = h H n h yields n d = M d. So M h x B = M h for all y a,b > 0 as required. The proofs for M are identical by symmetry. This result immediately implies the following corollary. COROLLARY 5.4. The stable matching linear program with the doctor-optimal (or hospital-optimal) objective has a unique integer optimum when agent preferences satisfy substitutes and the law of aggregate demand. Also, it is irrelevant what the utilities of the doctors are, since there is a unique doctor-optimal solution for any doctor utility functions with the same preference ordering. We remark it is already known the deferred acceptance algorithm has this property when preferences are substitutable [Hatfield and Kominers 2014] Response preferences make the linear program naturally integer In this subsection, we add the assumption that agents have responsive preferences. Under this assumption, we will show the stable matching linear program is naturally integer. Furthermore, for each agent a there is at most X a variables y a,b where y a,b > 0 is possible in a feasible solution. By identifying these variables we are able to solve a smaller tractable linear program. Consequently, we can optimize over a A B Q a f a (B)y a,b, for any arbitrary f a : Q a R in polynomial time. To the best of our knowledge this is the first polynomial time algorithm for this problem. It worth noting that it is already well-known that we can optimize over objectives of the form x X c xz x when preferences are responsive [Vande Vate 1988] (since responsive preferences can trivially be represented using ordinal preferences). LEMMA 5.5. For any stable matching y and agent a A the variables y a,ma and y a,ma are strictly positive. PROOF. We will prove this for the hospitals. The proof for the doctors are identical. Consider the least favorite x M h, now by definition of M h there exists some B with y a,b > 0 and x B. Let x M h by lemma 5.2 x B = C h (M h B) by reflexivity this implies x C h (M h B) = B. Hence M h B. By Corollary 4.7 and Theorem 5.3 it follows that M h = B. Suppose that there are only two sets B and B with y h,b > 0 and y h,b > 0. From the paragraph above we know that either M h = B or M h = B, for simplicity say M h = B. Now M h = C h (M h B ) = B by lemma 5.2. Now observe that by substitutes C h (B B B ) = C h (C h (B B ) B ). Hence by induction y h,mh > 0. COROLLARY 5.6. For any agent a A we can order the possible sets it can be matched to in a stable matching B 1,..., B n such that if x B i, x B i+1 and x a x then x B i. PROOF. Consider two stable matchings M and M, and any agent a A. By our previous lemma, without loss of generality, C a (M a M a) = M a. Now let x M a, x M a and x a x. Now x M a = C a (M a M a) so by responsiveness x C a (M a M a) = M a. Therefore there are no more than X a sets which a particular agent a can be matched to in different stable matchings. Since preferences are responsive we can enumerate these trivially via the stable

13 The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples X:13 marriage linear program of Vande Vate [1988]. This implies our stable matching linear program only needs O( X ) variables. Next, we show this linear program is naturally integer. THEOREM 5.7. When all agents have responsive preferences the set of extreme points of the stable matching linear program is identical to the set of integer stable matchings. PROOF. The fact that all integer stable matchings are extreme points of the linear program is trivial. The difficulty is proving that any extreme point is an integer stable matching. Let y and z be the integer vector corresponding to M. Let: y δ a,b = y a,b δy a,b 1 δ B Q a ; a A z δ x = z x δz x 1 δ x X For some 1 > δ > 0. We aim to show for some small δ, y δ is feasible. This holds trivially for constraint (1) and constraint (2). By lemma 5.5 y a,b = 1 implies y a,b > 0, so if δ < y a,b for all y a,b > 0 then (3) satisfied. It remains to show constraint (5) is satisfied. Our only issue is if (5) is satisfied at equality for y then the constraint must be satisfied at equality for y. So assume (5) is satisfied at equality for some contract x and solution y. If z x > 0 then the result holds via corollary 4.6. If B Q h :x C h (B x) y h,b + z = 0 then the result holds trivially. To show this assume z x = 0 and B Q h :x C h (B x) y h,a + z = 1. This implies x C h (M h x) and therefore x C h (B M h x) = C h (B x) by lemma 5.2. It follows that: z x + y h,b = 1 B Q h :x C h (B x) Since we have assumed the stability constraint (5) holds at equality: y d,b = 0 B Q d :x C d (B x) And by substitutes x C h ({t : z t > 0}) = M, hence: B Q d :x C d (B x) y d,b = 0 Thus constraint (5) is satisfied at equality for y and contract x. It is a little surprising we can optimize over non-linear objectives since without the stability constraints, even when the market is two-sided, this problem can be shown to be NP-hard (see Section 3). This result could be useful in stable matching such as school choice when we want to optimize socioeconomic diversity [Kominers and Sönmez 2013]. Socio-economic diversity is fundamentally a nonlinear objective on the set of students allocated to a school, because we want schools to be balanced in each school we want not too many students from poor households and not too many students from wealthy households. A simple example might be f h (B) = w(b)p(h)/qh 2 where w(b) is the number of students from wealthy households and p(h) is the number of students from poor households. 6. THE STABLE MATCHING LINEAR PROGRAM WITH COUPLES In this section, we demonstrate that linear programming is an effective heuristic for finding stable matchings with couples, by performing synthetic experiments. In particular, we show that the linear program with the doctor-optimal objective frequently has an integer optimal solution and therefore is a good heuristic for finding a stable matching. We also observe that using an integer programming solver with the doctor-optimal objective rapidly finds a stable matching. We compare these approaches to sorted deferred acceptance [Ashlagi et al. 2014] and show our methods more frequently find stable matches. We already know that the modifications of the deferred acceptance algorithm seem to frequently find stable matchings in real markets and markets generated by simple random distributions [Roth and Peranson 1999; Kojima et al. 2014; Ashlagi et al. 2014]. Of course, since the couples problem is NP-hard, we cannot expect to find an algorithm that guarantees to find a stable matching and a quick runtime. However, an integer program guarantees the former and a linear program guarantees the

14 X:14 Oliver Hinder latter. The question remains therefore whether, in practice, the integer program can be solved quickly and the linear program frequently finds a stable matching. For the following numerical experiments, we answer both of these questions affirmatively. To test how effective a particular method is at finding a stable solution, we need some distribution for generating preferences. The method for randomly generating preferences is outlined in Subsection 2.1. There are three variables that we adjust in our experiments: the size of the market, the applicants in a couple and the number of available positions. The results are presented in the following figures. The three methods we use to find a stable matching are: i. A linear program with the doctor-optimal objective 4. If the solution is naturally integer, we report a stable matching, if the solution is fractional then we are unable to find a stable matching using this method. ii. A integer program with the doctor-optimal objective. We solve the integer program using Gurobi [Gurobi Optimization 2014] which is branch and bound solver that uses our linear program to deduce the existence or non-existence of a stable matching. This always proves or disproves the existence of a stable matching, but might take a very long time to do so. iii. Sorted deferred acceptance (SoDA) a deferred acceptance algorithm that accommodates couples. This will be our benchmark. While the algorithm will find stable matchings in large random markets with a small number of couples, for arbitrary preferences it does not guarantee to find a stable matching (nor a polynomial runtime). Figure 4 and 5 show the runtime of the various methods when the proportion of couples in the market varies and the size of the market varies. In Figure 4, we see that all the methods quickly terminate, even as number of doctors in the market increases 5. Figure 5 shows that as the number of couples grows, both SoDA and the integer program methods tend to be quick until the number of applicants in a couple reaches some threshold, at which point the runtime increases sharply. We can solve linear programs efficiently, so as we see in Figure 4, the runtime increases gradually as the number of couples grows. average time (seconds) Integer Program Linear Program SoDA number of doctors Fig. 4: Time different methods take to finish as the number of doctors varies. 10% of applicants are in a couple and there are the same number of positions as doctors 6. Figure 6, 7 and 8 show how frequently our different methods find stable solutions as we change the size of the market, the proportion of applicants in couples and the number of available positions 4 Specifically we set u a(s) as the position of S in a s preference list. Formally, u a(s) = {S Q a : S < a S} i.e. take the preferences S 1 < a S 2 < a...s n and make u a(s i ) = i. 5 Figure 4 shows that when the proportion of couples is fixed at 10% the runtime of the linear program is more than SoDA or the integer program. The fact that the integer program is, in this instance, quicker to solve than the linear program is unexpected, since to solve the integer program we solve a series of linear programs. This unexpected event occurs because Gurobi s integer program presolve reduces the original integer program to a significantly smaller integer program [Johnson et al. 2000]. Furthermore, it rarely takes more than a few branches and cuts (if any) to solve the integer program.