Stable Matching in Large Economies

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1 Stable Matching in Large Economies Yeon-Koo Che, Jinwoo Kim, Fuhito Kojima June 16, 2015 Abstract We study stability o two-side many to one two-sided many-to-one matching in which irms preerences or workers may exhibit complementarities. Although such preerences are known to jeopardize stability in a inite market, we show that a stable matching exists in a large market with a continuum o workers, provided that each irm s choice changes continuously as the set o available workers changes. Building on this result, we show that an approximately stable matching exists in any large inite economy. We extend our ramework to ensure a stable matching with desirable incentive and airness properties in the presence o indierences in irms preerences. JEL Classiication Numbers: C70, D47, D61, D63. Keywords: two-sided matching, stability, complementarity, strategy-prooness, large economy We are grateul to Nikhil Agarwal, Nick Arnosti, Eduardo Azevedo, Péter Biró, Aaron Bodoh-Creed, Pradeep Dubey, Piotr Dworczak, Tadashi Hashimoto, John William Hatield, Johannes Hörner, Yuichiro Kamada, Michihiro Kandori, Scott Kominers, Ehud Lehrer, Jacob Leshno, Bobak Pakzad-Hurson, Jinjae Park, Parag Pathak, Marcin Peski, Larry Samuelson, Ilya Segal, Rajiv Sethi, Bob Wilson, and seminar participants at Hanyang, Korea, Kyoto, Paris, Seoul National, Stanord, and Tokyo universities as well as the ASSA Meeting 2015, NBER Market Design Workshop, INFORMS 2014, International Conerence on Game Theory at Stony Brook 2014, Monash Market Design Workshop, SAET 2014, and Midwest Economic Theory Conerence 2014 or helpul comments. Taehoon Kim, Janet Lu and Xingye Wu provided excellent research assistance. Che: Department o Economics, Columbia University ( yeonkooche@gmail.com); Kim: Department o Economics, Seoul National University ( jikim72@gmail.com); Kojima: Department o Economics, Stanord University ( uhitokojima1979@gmail.com). We acknowledge inancial support rom the National Research Foundation through its Global Research Network Grant (NRF-2013S1A2A ). Kojima grateully acknowledges inancial support rom the Sloan Foundation. 1

2 1 Introduction Since the celebrated work by Gale and Shapley (1962), matching theory has taken center stage in the ield o market design and in economic theory more broadly. In particular, the successul application o matching theory in medical matching and school choice has undamentally changed how these markets are organized. An essential requirement in the design o matching markets is stability namely, there should be no incentives or participants to block (i.e., side-contract around) the suggested matching. Stability is crucial or the long-term sustainability o a market; unstable matching would be undermined by the parties side-contracting around it either during or ater a market. 1 When one side o the market is under centralized control, as with school choice, blocking by a pair o agents on both sides is less o a concern. Nonetheless, even in this case, stability is desirable rom the perspective o airness because it eliminates justiied envy, i.e., envy that cannot be explained away by the preerences o the agents on the other side. In the school choice application, i schools preerences are determined by test scores or other priorities that a student eels entitled to, eliminating justiied envy appears to be important. Unortunately, a stable matching exists only under restrictive conditions. It is well known that the existence o a stable matching is not generally guaranteed unless the preerences o participants or example, irms are substitutable. 2 In other words, complementarity can lead to the nonexistence o a stable matching. This is a serious limitation with respect to the applicability o centralized matching mechanisms because complementarities o preerences are a pervasive eature o many matching markets. Firms oten seek to hire workers with complementary skills. For instance, in proessional sports leagues, teams demand athletes that complement one another in terms o the skills they possess as well as the positions they play. Some public schools in New York City seek diversity in their student bodies with respect to their skill levels. US colleges tend to assemble classes that are complementary and diverse in terms o their aptitudes, lie backgrounds, and demographics. 1 Table 1 in Roth (2002) shows that unstable matching algorithms tend to die out while stable algorithms survive the test o time. 2 Substitutability here means that a irm s demand or a worker never grows when more workers are available. More precisely, i a irm does not wish to hire a worker rom a set o workers, then it never preers to hire that worker rom a larger (in the sense o set inclusion) set o workers. The existence o a stable matching under substitutable preerences is established by Kelso and Craword (1982), Roth (1985), and Hatield and Milgrom (2005), while substitutability was shown to be the maximal domain or existence by Sönmez and Ünver (2010), Hatield and Kojima (2008), and Hatield and Kominers (2014). 2

3 To better organize such markets, we must achieve a clearer understanding o complementarities; without such an understanding, the applicability o centralized matching will remain severely limited. In particular, this limitation is important or many decentralized markets that might otherwise beneit rom centralization, such as the markets or college and graduate admissions. Decentralized matching leaves much to be desired in terms o eiciencies and airness (see Che and Koh (2015)). However, the exact beneits rom centralizing these markets and the method or centralization remain unclear because o the instability that may arise rom participants complementary preerences. This paper takes a step orward in accommodating complementarities and other orms o general preerences. In light o the general impossibility, this requires us to weaken the notion o stability in some way. Our approach is to consider a large market. Speciically, we consider a market that consists o a large number o workers/students on one side and a inite number o irms/colleges with large capacities on the other, 3 and we ask whether stability can be achieved in an asymptotic sense, i.e., whether participants incentives or blocking disappear as the number o workers/students and as the capacities o irms/colleges grow large. Our notion o stability preserves the motivation behind the original notion o stability: as long as the incentive or blocking is suiciently weak, the instability and airness concerns will not be serious enough to jeopardize the mechanism. We irst consider a continuum model with a inite number o irms and a continuum o workers. Each worker seeks to match with one irm at most. Firms have preerences over groups o workers, and, importantly, their preerences may exhibit complementarities. Finally, a matching is a distribution o workers across irms. Our model generalizes Azevedo and Leshno (2014), who assume that irms have responsive preerences (a special case o substitutable preerences). Our main result is that a stable matching exists i irms preerences exhibit continuity, i.e., i the set o workers chosen by each irm varies continuously as the set o workers available to that irm changes. This result is quite general because continuity is satisied by a rich class o preerences including those exhibiting complementarities. 4 The existence o a stable matching ollows rom two results: (i) a stable matching can be characterized as a ixed point o a suitably deined mapping over a unctional space, and (ii) such a ixed point exists, given the continuity assumption. The construction o the ixed point mapping in the present paper diers rom the ones in the existing matching literature, 3 Our large market model is meant to approximate some labor market and school choice problems. For example, a typical school choice problem involves a handul o schools each admitting hundreds o students. 4 For instance, it allows or Leontie-type preerences with respect to alternative types o workers, in which irms desire to hire all types o workers in equal size (or density). 3

4 including Adachi (2000), Hatield and Milgrom (2005) and Echenique and Oviedo (2006), among others. In particular, the existence o a ixed point is established by means o the Kakutani-Fan-Glicksberg ixed point theorem a generalization o Kakutani s ixed point theorem to unctional spaces which appears to be new to the matching literature. We next use our continuum model to approximate a large inite economies. More speciically, we demonstrate that or any large inite economy that is suiciently close to our continuum economy (in terms o the distribution o worker types and irms preerences), an approximately stable matching exists in that the incentives or blocking are arbitrarily small. Although the basic model assumes that irm preerences are strict, our ramework can be extended to allow or indierence in irm preerences. Accommodating such indierence is particularly important in the school choice context in which schools preerences are generated by coarse priorities that land many students in the same priority class. Indierent irm preerences raise a urther issue about airness, as the standard notion o stability does not ensure that workers who are perceived by irms as equivalent are treated equitably. In the school choice context, in particular, it may thus be especially important to strengthen the notion o stability to eliminate discrimination among such workers (in addition to eliminating justiied envy). How such workers are treated also raises an issue regarding incentives because inequitable treatment among workers who are perceived by irms as equivalent may lead to untruthul reporting by workers. To accommodate irm indierence, we represent a irm s preerence as a choice correspondence (as opposed to a unction). We then extend both the ixed point characterization (via a correspondence deined on a unctional space) and the existence result. We inally show that a matching mechanism exists that satisies both the stronger notion o stability ( strong stability, as deined by Kesten and Ünver (2014)) and strategy-prooness or workers. Further, our general model accommodating irms preerence indierences has natural applications to ractional/time share matching models. These models examine how schools/irms and students/workers can share time or match probabilistically in a stable manner in a inite economy (see Sotomayor (1999), Alkan and Gale (2003), and Kesten and Ünver (2014), among others). Our continuum model lends itsel to the study o such a probabilistic/time share environment; we can simply interpret types in dierent subsets within the type space as probabilistic/time units available to alternative (inite) workers. Our novel contribution is to allow or more general preerences or irms, including complementarities as well as indierences. As discussed above, accommodating indierence is important in school choice design, and complementarities are also relevant because some 4

5 schools (such as those in New York City) seek diversity in their student bodies. 5 Although our existence result can accommodate general preerences including some complementarities, we do not claim to handle all orms o complementarities. We show later in Example 2 that irm preerences exhibiting lumpy demands (which may result rom ixed cost in production or increasing returns to scale) violate our continuity assumption, and a stable matching does not necessarily exist in such a situation. In view o this, the key contribution o our analysis is not to claim that all problems go away but rather to clariy what types o complementarities become not so problematic and what types still remain a problem in large economies. Relationship with the Previous Literature The present paper is connected with several strands in the previous literature. Most importantly, it is related to the growing literature on matching and market design. Since the seminal contributions o Gale and Shapley (1962) and Roth (1984), stability has been recognized as the most compelling solution concept in matching markets. 6 demonstrated by Sönmez and As argued and Ünver (2010), Hatield and Milgrom (2005), Hatield and Kojima (2008), and Hatield and Kominers (2014) in various situations, the substitutability condition is necessary and suicient to guarantee the existence o a stable matching with a inite number o agents. Our paper contributes to this line o research by showing that substitutability is not necessary or the existence o a stable matching when there is a continuum o agents on one side o the market, and that an approximately stable matching exists in large inite markets. Our study was inspired by recent research on matching with a continuum o agents by Abdulkadiroğlu, Che and Yasuda (2015) and Azevedo and Leshno (2014). 7 As in the present study, these authors assume that there are a inite number o irms and a continuum 5 As discussed below in the Conclusion, we investigate additional issues. First, we study a setting with substitutable preerences, provide a condition or uniqueness o a stable matching, and apply it to a setup that generalizes Azevedo and Leshno (2014) by allowing or airmative action constraints. Second, we generalize our basic model to a model that involves matching with contracts. 6 See Roth (1991) and Kagel and Roth (2000) or empirical and experimental evidence on the importance o stability in labor markets and Abdulkadiroğlu and Sönmez (2003) or the interpretation o stability as a airness concept in school choice. 7 Various recent studies on large matching markets are also related but ormally dierent, such as Roth and Peranson (1999), Immorlica and Mahdian (2005), Kojima and Pathak (2009), Kojima and Manea (2010), Manea (2009), Che and Kojima (2010), Lee (2014), Liu and Pycia (2013), Che and Tercieux (2015b), Che and Tercieux (2015a), Ashlagi, Kanoria and Leshno (2014), Kojima, Pathak and Roth (2013), and Hatield, Kojima and Narita (2015). 5

6 o workers. In particular, Azevedo and Leshno (2014) show the existence and uniqueness o a stable matching in that setting. However, as opposed to the present study, these authors assume that irms have responsive preerences which is a special case o substitutability. One o our contributions is that we show that the restrictions on preerences (such as responsiveness or even substitutability) that are almost universally assumed in the literature are unnecessary to guarantee the existence o a stable matching in the continuum markets. 8 An independent study by Azevedo and Hatield (2014) also analyzes matching with a continuum o agents. 9 Consistent with our study, these authors ind that a stable matching exists even when not all agents have substitutable preerences. However, the two studies have several notable dierences. First, Azevedo and Hatield (2014) consider a large number (a continuum) o irms each employing a inite number o workers; thus, they consider a continuum o agents on both sides o the market. By contrast, we consider a inite number o irms each employing a large number (a continuum) o workers. These two models thus provide complementary approaches or studying large markets. For example, in the context o school choice, many school districts consist o a small number o schools that each admit hundreds o students, which its well with our approach. However, in a large school district such as New York City, the number o schools admitting students is also large, and the Azevedo and Hatield (2014) model may oer a reasonable approximation. Second, Azevedo and Hatield (2014) assume that there is a inite number o both irm and worker types, which enables them to use Brouwer s ixed point theorem to demonstrate the existence o a stable matching. By contrast, we put no restriction on the number o workers types and thus allow or both inite and ininite numbers o types, and this generality in type spaces requires a topological ixed point theorem rom a unctional analysis. To the best o our knowledge, this type o mathematics has never been applied to two-sided matching, and we view the introduction o these tools into the matching literature as one o our methodological contributions. Our model also has the advantage o subsuming the previous work by Azevedo and Leshno (2014) as well as many o the other studies mentioned above that assume a continuum o worker types. Finally, although these authors also consider many-to-many matching, our applications to time share and probabilistic matching models also allow or many-to-many matching. 8 Section S.3 o the Supplementary Notes generalizes the condition or uniqueness o stable matching. under substitutable but not necessarily responsive preerences. This result nests one o the uniqueness results by Azevedo and Leshno (2014) as a special case. 9 Although not as closely related, our study is also analogous to Azevedo, Weyl and White (2013), who demonstrate the existence o competitive equilibrium in an exchange economy with a continuum o agents and indivisible objects. 6

7 Our methodological contribution is also related to another recent advance in matching theory based on the monotone method. In the context o one-to-one matching, Adachi (2000) deines a certain operator whose ixed points are equivalent to stable matchings. His work has been generalized in many directions by Fleiner (2003), Echenique and Oviedo (2004, 2006), Hatield and Milgrom (2005), Ostrovsky (2008), and Hatield and Kominers (2014), among others, and we also deine an operator whose ixed points are equivalent to stable matchings. However, these previous studies also impose restrictions on preerences (e.g., responsiveness or substitutability) so that the operator is monotone and enables the application o Tarski s ixed point theorem to demonstrate the existence o stable matchings. By contrast, we distinguish our study rom the previous literature by not imposing responsiveness or substitutability restrictions on preerences; instead, we rely on the continuum o workers along with continuity in irms preerences to guarantee the continuity o the operator (in an appropriately chosen topology). This approach allows us to use a generalization o the Kakutani ixed point theorem, a more amiliar tool in traditional economic theory that is used in existence proos o general equilibrium and Nash equilibrium in mixed strategies. The current paper is also related to the literature on matching with couples. Like a irm in our model, a couple can be seen as a single agent with complementary preerences over contracts. Both Roth (1984) and unpublished work by Sotomayor show that a stable matching does not necessarily exist in a context that includes the presence o a couple. Klaus and Klijn (2005) provide a condition to guarantee the existence o stable matchings in such a context; moreover, recent work by Kojima, Pathak and Roth (2013) presents conditions under which the probability that a stable matching exists even in the presence o couples converges to one as the market becomes ininitely large, and similar conditions have been urther analyzed by Ashlagi, Braverman and Hassidim (2014). Pycia (2012) and Echenique and Yenmez (2007) study many-to-one matching with complementarities as well as with peer eects. Although our paper is dierent rom these studies in various respects, it complements them by ormalizing the sense in which inding a stable matching becomes easier in a large market even in the presence o complementarities. The remainder o this paper is organized as ollows. Section 2 oers an example that illustrates the main contribution o our paper. Section 3 describes a matching model in the continuum economy. Section 4 establishes the existence o a stable matching under general, continuous preerences. In Section 5, we use this existence result to show that an approximately stable matching exists in any large inite economy. In Section 6, we extend our analysis to the case in which irms may have multi-valued choice mappings (i.e., choice correspondences) and apply the analysis to study airness and incentive compatibility, in 7

8 addition to time share/probabilistic matching models. Section 7 concludes. 2 Illustrative Example Beore proceeding, we illustrate the main contribution o our paper with an example. We irst illustrate how complementary preerences may lead to the non-existence o a stable matching when there is a inite number o agents. To this end, suppose that there are two irms, 1 and 2, and two workers, θ and θ. The agents have the ollowing preerences: θ : 1 2 ; 1 : {θ, θ } ø; θ : 2 1 ; 2 : {θ} {θ } ø. In other words, worker θ preers 1 to 2, and worker θ preers 2 to 1 ; irm 1 preers employing both workers to employing neither, which the irm in turn preers to employing only one o the workers; and irm 2 preers worker θ to θ, which it in turn preers to employing neither. Firm 1 has a complementary preerence, which creates instability. To illustrate this, recall that stability requires that there be no blocking coalition. Due to 1 s complementary preerence, it must employ either both workers or neither in any stable matching. The ormer case is unstable because worker θ preers irm 2 to irm 1, and 2 preers θ to being unmatched, so θ and 2 can orm a blocking coalition. The latter case is also unstable because 2 will only hire θ in that case, which leaves θ unemployed; this outcome will be blocked by 1 orming a coalition with θ and θ that will beneit all members o the coalition. Can stability be restored i the market becomes large? I the market remains inite, the answer is no. To illustrate this proposition, consider a scaled-up version o the above model: there are q workers o type θ and q workers o type θ, and they have the same preerences as previously described. Firm 2 preers type-θ workers to type-θ workers and wishes to hire in that order but at most a total o q workers. Firm 1 has a complementary preerence or hiring identical numbers o type-θ and type-θ workers (with no capacity limit). Formally, i x and x are the numbers o available workers o types θ and θ, respectively, then irm 1 would choose min{x, x } workers o each type. When q is odd (including the original economy, where q = 1), a stable matching does not exist. 10 To illustrate this, irst note that i irm 1 hires more than q/2 workers o each type, then irm 2 has a vacant position to orm a blocking coalition with a type-θ worker, 10 We sketch the argument here; Section S.1 o the Supplementary Notes provides the argument in uller orm. When q is even, a matching in which each irm hires q 2 o each type o workers is stable. 8

9 who preers 2 to 1. I 1 hires ewer than q/2 workers o each type, then some workers will remain unmatched (because 2 hires at most q workers). I a type-θ worker is unmatched, then 2 will orm a blocking coalition with that worker. I a type-θ worker is unmatched, then irm 1 will orm a blocking coalition by hiring that worker and a θ worker (possibly matched with 2 ). Consequently, exact stability is not guaranteed, even in a large market. Nevertheless, one may hope to achieve approximate stability. This is indeed the case with the above example; the magnitude o instability diminishes as the economy grows large. To illustrate this, let q be odd and consider a matching in which 1 hires q+1 2 workers o each type, whereas 2 hires q 1 2 workers o each type. This matching is unstable because 2 has one vacant position it wants to ill, and there is a type-θ worker who is matched to 1 but preers 2. However, this is the only possible block o this matching, and it involves only one worker. As the economy grows large, i the additional worker becomes insigniicant or irm 2 relative to its size, which is what the continuity o a irm s preerence captures, then the payo consequence o orming such a block must also become insigniicant, which suggests that the instability problem becomes insigniicant as well. This can be seen most clearly in the limits o the above economy. Suppose there is a unit mass o workers, hal o whom are type θ and the other hal o whom are type θ. Their preerences are the same as described above. Suppose irm 1 wishes to maximize min{x, x }, where x and x are the measures o type-θ and type-θ workers, respectively. Firm 2 can hire at most 1 o the workers, and it preers to ill as much o this quota as 2 possible with type-θ workers and ill the remaining quota with type-θ workers. In this economy, there is a (unique) stable matching in which each irm hires exactly one-hal o the workers o each type. To illustrate this, note that any blocking coalition involving irm 1 requires taking away a positive and identical measure o type-θ and type-θ workers rom irm 2, which is impossible because type-θ workers will object to it. Additionally, any blocking coalition involving irm 2 requires that a positive measure o type-θ workers be taken away rom irm 1 and replaced by the same measure o type-θ workers in its workorce, which is impossible because type-θ workers will object to it. Our analysis below will demonstrate that the continuity o irms preerences, which will be deined more clearly, is responsible or guaranteeing the existence o a stable matching in the continuum economy and approximate stability in the large inite economies in this example. 9

10 3 Model o a Continuum Economy Agents and their measures. There is a inite set F = { 1,..., n } o irms and a unit mass o workers. Let ø be the null irm, representing the possibility o workers not being matched with any irm, and deine F := F {ø}. The workers are identiied with types θ Θ, where Θ is a compact metric space. Let Σ denote a Borel σ-algebra o space Θ. Let X be the set o all nonnegative measures such that or any X X, X(Θ) 1. Assume that the entire population o workers is distributed according to a nonnegative (Borel) measure G X on (Θ, Σ). In other words, or any E Σ, G(E) is the measure o workers belonging to E. For normalization, assume that G(Θ) = 1. To illustrate, the limit economy o the example rom the previous section is a continuum economy with F = { 1, 2 }, Θ = {θ, θ }, and G({θ}) = G({θ }) = 1/2. 11 In the sequel, we shall use this as our leading example or purposes o illustrating the various concepts we develop. Any subset o the population or subpopulation is represented by a nonnegative measure X on (Θ, Σ), such that X(E) G(E) or all E Σ. Let X X denote the set o all subpopulations. We urther say that a nonnegative measure X X is a subpopulation o X X, denoted as X X, i X(E) X(E) or all E Σ. We use XX to denote the set o all subpopulations o X. Given the partial order, or any X, Y X, we deine X Y (join) and X Y (meet) to be the supremum and inimum o X and Y, respectively. 12 X Y and X Y are welldeined, i.e., they are also measures belonging to X, which ollows rom the next lemma, whose proo is in Section S.2.1 o the Supplementary Notes. Lemma 1. The partially ordered set (X, ) is a complete lattice. The join and meet o X and Y in X can be illustrated with examples. Let X := (x, x ) and Y := (y, y ) be two measures our leading example, where x and x are the measures o types θ and θ, respectively, under X, and likewise y and y under Y. Then, their join and meet are respectively measures X Y = (max{x, y}, max{x, y }) and X Y = (min{x, y}, min{x, y }). Next, consider a continuum economy with types Θ = [0, 1] and suppose the measure G admits a bounded density g or all θ [0, 1]. In this case, it easily ollows that or 11 Henceorth, given any measure X, X(θ) will denote a measure o the singleton set {θ} to simpliy notation. 12 For instance, X Y is the smallest measure o which both X and Y are subpopulations. It can be shown that, or all E Σ, (X Y )(E) = sup X(E D) + Y (E D c ), D Σ which is a special case o Lemma 1. 10

11 X, Y G, their densities x and y are well deined. 13 Then, their join Z := (X Y ) and meet Z := (X Y ) admit densities z and z deined by z(θ) = max{x(θ), y(θ)} and z (θ) = min{x(θ), y(θ)} or all θ, respectively. As usual, or any two measures X, Y X, X + Y and X Y denote their sum and dierence, respectively. Consider the space o all (signed) measures (o bounded variation) on (Θ, Σ). We endow this space with a weak- topology and its subspace X with the relative topology. Given a w sequence o measures (X k ) and a measure X on (Θ, Σ), we write X k X to indicate that (X k ) converges to X as k under weak- topology and simply say that (X k ) weakly converges to X. 14 Agents preerences. We now describe agents preerences. Each worker is assumed to have a strict preerence over F. Let a bijection P : {1,..., F } F denote a worker s preerence, where P (j) denotes the identity o the worker s j-th best alternative, and let P denote the (inite) set o all possible worker preerences. We write P to indicate that is strictly preerred to, according to P. For each P P, let Θ P Θ denote the set o all worker types whose preerence is given by P, and assume that Θ P is measurable and G( Θ P ) = 0, where Θ P denotes the boundary o Θ P. 15 Because all worker types have strict preerences, Θ can be partitioned into the sets in P Θ := {Θ P : P P}. We next describe irms preerences. We do so indirectly by deining a irm s choice unction, C : X X, where C (X) is a subpopulation o X or any X X and satisies the ollowing revealed preerence property: or any X, X X with X X, i C (X) X, then C (X ) = C (X). 16 Note that we assume that the irm s demand is 13 X([0, θ ]) X([0, θ]) G([0, θ ]) G([0, θ]) N θ θ, where N := sup s g(s). Thus, X([0, θ)) is Lipschitz continuous, and its density is well deined. 14 We use the term weak convergence because it is common in statistics and mathematics, although weak- convergence is a more appropriate term rom the perspective o unctional analysis. As is well w known, X k X i Θ hdx k hdx or all bounded continuous unctions h. See Theorem 8 in Θ Appendix A or some implications o this convergence. 15 This is a technical assumption that acilitates our analysis. The assumption is satisied i, or each P P, Θ P is an open set such that G( P P Θ P ) = G(Θ): all agents, except or a measure-zero set, have strict preerences, a standard assumption in matching theory literature. The assumption that G( Θ P ) = 0 is also satisied i Θ is discrete. To see it, note that E := E E c, where E and E c are the closures o E and E c, respectively. Then, we have E = E and E c = E c, so E E c = E E c =. Hence, the assumption is satisied. 16 This property must hold i the choice is made by a irm optimizing with a well-deined preerence relation. See, or instance, Hatield and Milgrom (2005), Fleiner (2003), and Alkan and Gale (2003) or some implicit and explicit uses o the revealed preerence property in the matching theory literature. 11

12 unique given any set o available workers. In Section 6.1, we consider a generalization o the model in which the irm s choice is not unique. Let R : X X be a rejection unction deined by R (X) := X C (X). By convention, we let C ø (X) = X, X X, meaning that R ø (X)(E) = 0 or all X X and E Σ. In our leading example, the choice unctions o irms 1 and 2 are given respectively by C 1 (x 1, x 1) = (min{x 1, x 1}, min{x 1, x 1}) and C 2 (x 2, x 2) = (x 2, min{ 1 x 2 2, x 2}), when x i [0, 1] o type-θ workers and 2 x i [0, 1] o 2 type-θ workers are available to irm i, i = 1, 2. In sum, a continuum matching model is summarized as a tuple (G, F, P Θ, C F ). Remark 1. Our model takes irms choice unctions as a primitive, which oers us some lexibility in describing their preerences, in particular the preerences over alternatives that are not chosen. This approach is also adopted by other studies in matching theory, which include Alkan and Gale (2003) and Aygün and Sönmez (2013), among others. An alternative, albeit more restrictive, approach would be to assume that each irm is endowed with a complete, continuous preerence relation over X. Maximization with such a preerence will result in an upper hemicontinuous choice correspondence deined over X. 17 Assuming a unique optimal choice will then give us a choice unction (which is also continuous); however, our results generalize to the case in which each irm s choice is not unique, as will be shown in Section 6.1. Matchings, and their eiciency and stability requirements. A matching is M = (M ) F, such that M X or all F and F M = G. Firms choice unctions can be used to deine a binary relation describing irms preerences over matchings. For any two matchings, M and M, we say that M M (or irm preers M to M ) i M = C (M M ). 18 We also say M M i M M and M M. The resulting preerence relation amounts to taking a minimal stance on the irms preerences, limiting attention to those revealed via their choices. Given this preerence relation, we say that M F M i M M or each F. To discuss workers welare, ix any matching M and any irm. Let D (M) := M (Θ P ) and D (M) := P P F : P P P F : P M (Θ P ) (1) Recently, Aygün and Sönmez (2013) have clariied the role o this property in the context o matching with contracts. 17 This result also relies on the act that the set o alternatives X is compact, a act we establish in the proo o Theorem This is known as the Blair order in the literature. See Blair (1984). 12

13 denote the measure o workers assigned to irm or better (according to their preerences) and the measure o workers assigned to irm or worse (again, according to their preerences), respectively, where M (Θ P ) denotes a measure that takes the value M (Θ P E) or each E Σ. Starting rom M as a deault matching, the latter measures the number o workers who are available to irm or possible rematching. Meanwhile, the ormer measure is useul or characterizing the workers overall welare. For any two matchings M and M, we say that M Θ M i or each F, D (M) D (M ). In other words, or each irm, i the measure o workers assigned to or better is larger in one matching than in the other, then we can say that the workers overall welare is higher in the ormer matching. Equipped with these notions, we can deine Pareto eiciency and stability. Deinition 1. A matching M is Pareto eicient i no matching M M exists such that M F M and M Θ M. Deinition 2. A matching M = (M ) F is stable i 1. (Individual Rationality) For all P P, we have M (Θ P ) = 0 or any satisying ø P, and or each F, M = C (M ); and 2. (No Blocking Coalition) There are no F and M X such that M M and M D (M). Condition 1 o this deinition requires that each matched worker preers being matched to being unmatched and that no irm wishes to unilaterally drop any o its matched workers. Condition 2 requires that there is no irm and no set o workers who are not matched together but preer to be. When Condition 2 is violated by and M, we say that and block M. M Remark 2 (Equivalence to group stability). We say that a matching M is group stable i Condition 1 o Deinition 2 holds and, 2. There are no F F and M F X F such that M M and M D (M) or all F. This deinition strengthens our stability concept because it requires that matching be immune to blocks by coalitions that potentially involve multiple irms. Such stability concepts with coalitional blocks are analyzed by Sotomayor (1999), Echenique and Oviedo (2006), and Hatield and Kominers (2014), among others. Clearly, any group stable matching is stable, because i Condition 2 is violated by a irm and M, then Condition 2 is 13

14 violated by a singleton set F = {} and M {}. The converse also holds. To see why, note that i Condition 2 is violated by F F and M F, then Condition 2 is violated by any and M such that F because M M and M D (M), by assumption. 19 As in the standard inite market, stability implies Pareto eiciency: Proposition 1. I a matching is stable, then it is Pareto eicient. Proo. See Section S.2.2 o the Supplementary Notes. 4 The Existence o a Stable Matching in the Continuum Economy Finding a stable matching is tantamount to identiying the opportunities available to each participant that are compatible with those opportunities available to all other participants, given their preerences. The mutually compatible opportunity sets are endogenous and o a ixed point character. For each irm, its opportunities in a stable matching (the workers available to the irm) are precisely those workers who have no better choice than. To identiy such workers, the opportunities available to those workers (i.e., the irms that would like to hire them) must be identiied. In turn, to identiy these irms requires identiying the workers available to them, thus coming ull circle. This logic suggests that a stable matching is associated with a ixed point o a mapping or more intuitively, a stationary point o a process that repeatedly revises the set o available workers to the irms based on the preerences o the workers and the irms. Formally, we deine a map T : X n+1 X n+1, where T (X) = (T (X)) F or each X X n+1. For each F, the map T : X n+1 X is deined by T (X)(E) := P :=P (1) G(Θ P E) + P : P (1) R P (X P )(Θ P E), (2) 19 By requiring M D (M) or all F in Condition 2, our group stability concept implicitly assumes that workers who are considering participation in a blocking coalition with F use the current matching (M ) as a reerence point. This means that workers are available to irm as long as they preer to their current matching. However, given that a more preerred irm F may be making oers to workers in D (M) as well, the set o workers available to may be smaller. Such a consideration would result in a weaker notion o group stability. Any such concept, however, will be equivalent to our notion o stability because in this remark we establish that even the most restrictive notion o group stability the concept using D (M) in Condition 2 is equivalent to stability, while stability is weaker than any group stability concept described above. 14

15 where P F, which is called the immediate predecessor o, is a irm that is ranked immediately above irm according to P. 20 The map can be interpreted as a tâtonnement process in which an auctioneer quotes budgets o workers that irms can choose rom. As in a classical Walrasian auction, the budget quotes are revised based on the preerences o the market participants, reducing the budget or irm (i.e., making a smaller work orce available) when more workers are demanded by the irms that rank ahead o it and increasing the budget otherwise. Once the process converges, one reaches a ixed point, having ound the workers who are truly available to irms those who are compatible with the preerences o other market participants. Alternatively, the mapping can be seen as a process by which irms rationally adjust their belies about available workers based on the preerences o the other market participants. The ixed point o the mapping then captures the workers that irms can iteratively rationalize as being available to them. To illustrate, ix a irm. Consider irst the worker types θ Θ P or which is at the top o their preerence P (i.e., = P (1)). Firm can rationally believe that all such workers are available to it, which explains the irst term o (2). Consider next the worker types θ Θ P or which ranks second-best according to P. Firm can rationally believe that, o this group, only those who would be rejected by their top-choice irm are available to it, which explains the second term o (2). Now, consider the worker types or whom ranks third-best. Firm analogously reasons that only those workers who would be rejected by their irst- and second-choice irms would be available to it. However, the workers who would be rejected by their top choice are available to the second-best irm, according to the earlier rationalization. This in turn rationalizes s belie that the workers available to are precisely those who are available to but rejected by the second-best irm. In general, or all worker types, the same iterative process o belie rationalization establishes the validity o (2). Remark 3. Our map can be rewritten to mimic Gale and Shapley s deerred acceptance algorithm, in which irms and workers take turns rejecting the dominated proposals in each round. Speciically, we can write T = Ψ Φ, where, or each proile X = (X ) F X n+1 o workers, the map Φ (X) := G R (X ) returns those workers who are not rejected by each irm, and or Y = (Y ) X n+1, the map Ψ (Y ) := G Y P (Θ P ) P P 20 An immediate predecessor o is ormally deined such that P P and i P or F, then P P. 15

16 returns those workers available to each irm the workers who remain ater removing those who would be accepted by irms that they rank higher than. The map written in this way resembles those developed in the context o inite matching markets (e.g., see Adachi (2000), Hatield and Milgrom (2005), and Echenique and Oviedo (2006)), but the construction here diers because it deals with a richer space o worker types. As will also be clear, this new construction is necessary or a new method o proo characterizing the ixed point. Theorem 1. M is a stable matching i and only i M = C (X ), F, where X = T (X). Proo. This result ollows as a corollary o Theorem 7. For details, see Appendix A. Example 1. To illustrate how a stable matching can be ound rom the T mapping, consider our leading example. We can denote candidate measures o available workers by a tuple: (X 1, X 2 ) = (x 1, x 1; x 2, x 2) [0, 1 2 ]4, where X 1 = (x 1, x 1) are the measures o workers o types θ and θ available to 1 and X 2 = (x 2, x 2) are the measures o workers o types θ and θ available to 2. Since 1 is the top choice or θ and 2 is the top choice or θ, according to our T, all o these workers are available to the respective irms. Thus, without loss we can set x 1 = G(θ) = 1 2 and x 2 = G(θ ) = 1 2 and consider ( 1 2, x 1; x 2, 1 2 ) as our candidate measures. To compute T ( 1, 2 x 1; x 2, 1 ), we irst consider the choice o each 2 irm given its respective measures o available workers. Note that C 1 ( 1, 2 x 1) = (x 1, x 1), so R 1 ( 1 2, x 1) = ( 1 2 x 1, 0). Similarly, C 2 (x 2, 1 2 ) = (x 2, 1 2 x 2), so R 2 (x 2, 1 2 ) = (0, x 2). Now, applying our ormula rom (2), we obtain T ( 1, 2 x 1; x 2, 1) = ( 1, x 2 2 2; 1 2 x 1, 1 ). Thus, 2 ( 1, 2 x 1; x 2, 1) is a ixed point o T i and only i ( 1, 2 2 x 1; x 2, 1) = ( 1, x 2 2 2; 1 2 x 1, 1), or 2 x 1 = x 2 = 1. The optimal choice rom the ixed point then gives a matching 4 ( ) 1 2 M = 1 θ θ 1θ +, θ where the notation here (an analogous notation is used throughout) indicates that each o the irms 1 and 2 is matched to a mass 1 4 o worker types θ and θ. This matching M is stable. We now introduce a condition on irm preerences that ensures the existence o a stable matching. Deinition 3. Firm s preerence is continuous i, or any sequence (X k ) k N and X in w X such that X k X, it holds that C (X k ) w C (X). 16

17 As suggested by the name, the continuity o a irm s preerences means that a irm s choice changes continuously with the distribution o available workers. Under this assumption, we obtain a general existence result as ollows: Theorem 2. I each irm s preerence is continuous, then a stable matching exists. Proo. This result ollows as a corollary o Theorem 4. For details, see Appendix A. To prove that T admits a ixed point, we irst demonstrate that the continuity o irm preerences implies that the mapping T is also continuous. We also veriy that X is a compact and convex set. Continuity o T and compactness and convexity o X allow us to apply the Kakutani-Fan-Glicksberg ixed point theorem to guarantee that T has a ixed point. Then, the existence o a stable matching ollows rom Theorem 1, which shows the equivalence between the set o stable matchings and the set o ixed points o T. Many complementary preerences are compatible with continuous preerences and consequently with the existence o a stable matching. Recall Example 1, in which irm 1 has a Leontie-type preerence: it wishes to hire an equal number o workers o types θ and θ (speciically, the irm wants to hire type-θ workers only i type-θ workers are also available, and vice versa). As Example 1 shows, a stable matching exists despite extreme complementarity: since the irm s preerences are clearly continuous in that example, the existence o a stable matching is implied by Theorem 2. A stable matching may not exist even in the continuum economy unless all irms have continuous preerences, as the ollowing example illustrates. Example 2 (Role o continuity). Consider the ollowing economy, which is modiied rom Example 1. There are two irms 1 and 2, and two worker types θ and θ, each with measure 1/2. Firm 1 wishes to hire exactly measure 1/2 o each type and preers to be unmatched otherwise. Firm 2 s preerence is responsive subject to the capacity o measure 1/2: it preers type-θ to type-θ workers, and preers the latter to leaving a position vacant. Given this, C 1 violates continuity, while C 2 does not. As beore, we assume θ : 1 2 ; θ : 2 1. No stable matching exists in this environment. To illustrate, consider the ollowing two cases: 1. Suppose 1 hires measure 1/2 o each type o workers (i.e., all workers). In such a matching, none o the capacity o 2 is illed. Thus, such a matching is blocked by 2 17

18 and type-θ workers (note that every type-θ worker is currently matched with 1, so they are willing to participate in the block). 2. Suppose 1 hires no worker. Then, the only candidate or a stable matching is one in which 2 hires measure 1/2 o the type-θ workers (otherwise 2 and unmatched workers o type θ would block the matching). Then, because 1 is the top choice o all type-θ workers and type-θ workers preer 1 to ø, the matching is blocked by a coalition o 1/2 o the type-θ workers, 1/2 o the type-θ workers, and 1. The continuity assumption is important to the existence o a stable matching; this example shows that nonexistence can occur even i only one irm has a discontinuous choice unction. This example also suggests that non-existence can reemerge when some lumpiness is reintroduced into the continuum economy (i.e., one irm can only hire a minimum mass o workers). However, this kind o lumpiness may not be the most natural in a continuum economy, which is unlike a inite economy, where lumpiness is a natural consequence o the indivisibility o each worker. 5 Approximate Stability in Finite Economies As we have seen in the illustrative example in Section 2, no matter how large the economy is, as long as it is inite, a stable matching does not necessarily exist. This motivates us to look or an approximately stable matching in a large inite economy. In this section, we build on the existence o a stable matching in the continuum economy to demonstrate that approximate stability can be achieved i the economy is inite but suiciently large. To analyze economies o inite sizes, we consider a sequence o economies (Γ q ) q N indexed by the total number o workers q N. In each economy Γ q, there is a ixed set o n irms, 1,..., n, that does not vary with q. As beore, each worker has a type in Θ. The worker distribution is normalized with the economy s size. Formally, let the (normalized) population G q o workers in Γ q be deined so that G q (E) represents the number o workers with types in E divided by q. A subpopulation X q is easible in economy Γ q i X q G q, and it is a measure whose value or any E is a multiple o 1/q. Let X q denote the set o all easible subpopulations. Note that G q, and thus every X q X q, belongs to X, although it does not have to be an element o X, i.e., a subpopulation o G. Let us say that a sequence o economies (Γ q ) q N converges to a continuum economy Γ i the measure G q o worker types converges weakly to the measure G o the continuum economy, that is, G q w G. To ormalize approximate stability, we irst represent each irm s preerence by a cardinal utility unction u : X R deined over normalized distributions o workers 18

19 it matches with. And, this utility unction represents a irm s preerence or each inite economy Γ q as well as or the continuum economy. 21 We assume that u (X) rom matching with a subpopulation X X is continuous in X in weak-* topology. This utility unction rationalizes irm s choice in both continuum and inite economies in the sense that irm chooses a subpopulation that is easible and maximizes u in the respective economies. 22 We assume that the choice unction, C, in the continuum economy is a unique maximizer o u, i.e., or any X X, C (X) = arg max u (X ) X X is a singleton, which implies that C is continuous. 23 In each inite economy Γ q, or any subpopulation o available workers X q X q, each irm chooses a subpopulation o X q in X q that maximizes its utility, u. The continuity o the utility unction then implies that each irm s optimal choice in the inite economy (which we do not assume to be unique) converges to the optimal choice in the limit economy as q. 24 An example is a sequence o replica economies in which each irm has a responsive preerence (same or all economies in the sequence) but aces a capacity o rk in the r-old replica economy (where the total number o workers q increases proportionally to r). A matching in inite economy Γ q is M q = (M q ) F such that M q X q or all F and F M q = Gq. The measure o available workers or each irm at matching M q (X q ) n+1 is D (M q ), where D ( ) is deined as in (1). 25 Note that because M q or each F is a multiple o 1/q, so is D (M q ), which means that D (M q ) is easible in Γ q. We now deine ɛ-stability in inite economy Γ q. Deinition 4. A matching M q (X q ) n+1 in economy Γ q is ɛ-stable i (i) it is individually rational and (ii) u ( M q ) < u (M q ) + ɛ or any F and M q X q with M q D (M q ). Condition (i) o this deinition is identical to the corresponding condition or exact stability, soɛ-stability relaxes stability only with respect to condition (ii). Speciically, ɛ- stability does not require that there be no blocking coalition, 26 but rather that the utility 21 The assumption that the same utility unction applies to both inite and limit economies is made or convenience. The results in this section hold i, or instance, the utilities in inite economies converge uniormly to the utility in the continuum economy. 22 To guarantee the existence o such a utility unction, we may assume, as in Remark 1, that each irm is endowed with a complete, continuous preerence relation. Then, because the set o alternatives X is a compact metric space, such a preerence can be represented by a continuous utility unction according to the Debreu representation theorem (Debreu, 1954). 23 This is an implication o Berge s maximum theorem. 24 This statement is ormally proven in Lemma 7 o Appendix B. 25 To be precise, D (M q ) is given as in (1) with G and M being replaced by G q and M q, respectively. 26 A blocking coalition is unavoidable i a stable matching does not exist, as demonstrated in Section 2. 19