On the Correspondence o Contracts to Salaries in (Many-to-Many) Matching Scott Duke Kominers 1, Becker Friedman Institute or Research in Economics, University o Chicago Abstract In this note, I extend the work o Echenique (orthcoming) to show that a model o many-to-many matching with contracts may be embedded into a model o many-tomany matching with wage bargaining whenever (1) all agents preerences are substitutable and (2) the matching with contracts model is unitary, in the sense that every contractual relationship between a given irm worker pair is speciied in a single contract. Conversely, I show that unitarity is essentially necessary or the embedding result. Keywords: Many-to-Many Matching; Stability; Substitutes; Contract Design; Unitarity. JEL: C78 Since its introduction, the matching with contracts model has been extensively generalized (Ostrovsky (2008); Hatield and Kominers (2010, orthcoming)), and has been applied in surprising contexts such cadet branch matching (Sönmez and Switzer (2011); Sönmez (2011)) and the Japanese medical residency match (Kamada and Kojima (2011, orthcoming)). 2 Echenique (orthcoming) has recently shown that under the substitutability condition crucial or many o the results o Hatield and Milgrom (2005), the matching with contracts model directly embeds into the earlier, and seemingly less general, Kelso and Craword (1982) model o many-to-one matching with salaries and gross substitutes preerences. The key insight o Echenique (orthcoming) is that in many-to-one matching with contracts models, contract negotiations between a irm and worker are in a sense separable rom other contract negotiations whenever that irm and worker have substitutable preerences. In this note, I observe that this insight and hence the embedding Corresponding author. Email addresses: skominers@uchicago.edu (Scott Duke Kominers), skominers@gmail.com (Scott Duke Kominers) URL: http://www.scottkom.com/ (Scott Duke Kominers) 1 Department o Economics, University o Chicago, 1126 East 59th Street, Chicago, IL 60637. 2 Hatield and Milgrom (2005) introduced the matching with contracts model as a generalization o the Kelso and Craword (1982) model o many-to-one matching with salaries; the possibility o such a generalization was irst noted in remarks o Craword and Knoer (1981) and Kelso and Craword (1982). Preprint submitted to Games and Economic Behavior December 3, 2011
result o Echenique (orthcoming) applies more generally. Speciically, I show that bargaining over contracts is isomorphic to a irm worker salary bargain whenever agents preerences are substitutable and a irm and worker are allowed to sign at most one contract with each other. This latter condition, which I call unitarity, may be a natural assumption or (many-to-many) irm worker matching markets in which each worker can serve in at most one role at each irm. 3 Unitarity is automatic in the many-to-one matching settings considered by Hatield and Milgrom (2005) and Echenique (orthcoming), as in those settings workers have unit demand or contracts; it is also assumed in the many-to-many model o Klaus and Walzl (2009). 4 I demonstrate (by example) that unitarity is essentially necessary or the Echenique (orthcoming) embedding result. Nonunitary many-to-many matching models such as that o Hatield and Kominers (2010) need not correspond to wage bargaining models, even i all agents have substitutable preerences over contracts. 1. Preliminaries 1.1. Basic Models Extending the ramework o Echenique (orthcoming), I introduce two models o generalized matching among a (inite) set F o irms and a (inite) set W o workers. 1.1.1. Matching with Contracts A model o (many-to-many) matching with contracts is speciied by a set X F W T o contracts, where T is (inite) a set o contractual terms, along with a (one-to-one) utility unction u i : 2 X R or each i F W. For a contract x X, I denote by x F and x W the irm and worker associated to x, respectively. For Y X, I denote by Y i {x Y : i {x F, x W }} the set o contracts in Y associated to i F W. Although the utility unction o each agent i F W is deined (or convenience) over the entire space 2 X, I require that it depend only on the contracts actually available to i, that is, that u i (Y ) = u i (Y i ) or each Y X. 5 The utility unction o i F W determines a preerence relation P i over sets o contracts Y X i and an associated choice unction C i (Y ) max Pi {Z : Z Y i } deined over sets o contracts Y X. 6 The preerences o i F W are substitutable 3 The market designer must exercise care, however, in deining the roles that a worker may serve. As Hatield and Kominers (2010) demonstrate, the most natural contracting model or a matching market may involve splitting jobs into multiple distinct positions (e.g., a night shit and a day shit). 4 Hence, my result extends the Echenique (orthcoming) approach to the setting o Klaus and Walzl (2009). 5 Note that unlike the many-to-one matching with contracts models considered by Hatield and Milgrom (2005) and Echenique (orthcoming), my model does not require that the utility unctions o workers w W exhibit unit-demand, i.e. that they be choice-equivalent to utility unctions on the restricted space {{x} : X i { }. 6 Here, I use the notation max Pi to indicate that the maximization is taken with respect to the preerences o agent i. 2
i or any x, z X and Y X, x / C i (Y {x}) = x / C i (Y {x, z}). 7 A set o contracts Y X is called an allocation. An allocation Y X is said to be individually rational i C i (Y ) = Y i or all i F W, and is said to be unblocked i there does not exist a nonempty set o contracts Z Y such that Z i C i (Y Z) or all i F W. An allocation is stable i it is both individually rational and unblocked. 8 1.1.2. Matching with Salaries A model o (many-to-many) matching with salaries is speciied by a set S R + o possible salaries, a (one-to-one) utility unction v : 2 {} W S R or each F, and a (one-to-one) utility unction v w : 2 F {w} S R or each w W. 9 Without loss o generality, I restrict S to be inite, and identiy it with the set o positive integers {1,..., S} or some S suitably large that no worker is ever hired at salary S, i.e. such that v (B {(, w, S)}) < v (B) or all B ({} (W \ {w}) S). For each F, the utility unction v induces a demand unction D : S F W 2 {} W S deined by D (s) argmax (v (B)). 10 B {} {(w,s w )} The demand unction D w : S F W 2 F {w} S o each worker w W is deined analogously. The demand unction D o F satisies the gross substitutes condition i, or any two salary matrices s and s with s s, (, w, s w ) D (s) = (, w, s w) D (s ) or any w W or which s w = s w. Analogously, the demand unction D w o w W satisies the gross substitutes condition i, or any two salary matrices s and s, or which s w s w, (, w, s w ) D w (s) = (, w, s w) D w (s ) or any F or which s w = s w. 7 Intuitively, this condition means that there are no two contracts x, z X which are sometimes complements in the sense that the availability o z makes x more attractive. 8 A number o alternative stability concepts are available or many-to-many matching settings (Blair (1988); Echenique and Oviedo (2006); Klaus and Walzl (2009)). In general, the choice o solution concept is somewhat immaterial or our exercise: embedding results analogous to Theorem 2 hold so long as the stability concept under consideration or the cases o matching with salaries corresponds to that considered or the case o matching with contracts. Nevertheless, I ix a choice o stability concept or concreteness, using that o Hatield and Kominers (2010, orthcoming), which is neither weaker nor stronger than the setwise stability concept o Echenique and Oviedo (2006). 9 I use the convention that utility unctions or models o matching with contracts are denoted u, while those or models o matching with salaries are denoted v. 10 Note that the demand o only depends upon the salaries s w o workers w at. 3
A (salary) matching is a set B F W S or which (, w, s w ), (, w, s w) B = s w = s w, i.e. a set o irm worker pairs which speciies a unique salary or each pair. A model o matching with salaries may be viewed as a specialized model o matching with contracts having contract set X = F W S and the choice constraint that (, w, s w ), (, w, s w) C i (B) = s w = s w (or all i F W and B F W S): 11 Deining the unction { {(i, w, s iw ) : w W } i F κ i (s) {(, i, s i ) : F } i W, gives D i (s) = C i (κ i (s)). With this observation, salary matchings naturally inherit the notions o individual rationality, unblockedness, and stability described above. 12 1.2. Key Conditions or the Embedding Result I introduce two conditions key to the embedding argument: unitarity and Pareto separability. To the best o my knowledge, the ormer o these two conditions has not been emphasized previously, although it has been discussed abstractly in the literature (Hatield and Kominers (2010)). The latter condition was introduced by Hatield and Kojima (2010). Deinition 1. A model o matching with contracts is unitary i or all Y X, F, x, x C (Y ) = x W x W, w W, x, x C w (Y ) = x F x F. Intuitively, unitarity corresponds to the requirement that a irm and worker sign at most one contract with each other. All many-to-one matching models are unitary, as in such settings all workers w have unit demand: C w (Y ) 1 or all Y X. 13 In modeling many-to-many matching with contracts, unitarity is an added assumption; Klaus and Walzl (2009) impose it, while Hatield and Kominers (2010) do not. 11 This identiication is immaterial to my substantive results; I make it only to simpliy the task o deining stability or many-to-many matching models with salaries. 12 Thus, like in the deinition o stability or matching models with contracts presented above, our stability concept or salary matchings allows agents in a deviating coalition to deviate while maintaining previous relationships with nondeviating agents. As remarked in Footnote 8: Alternative stability solution concepts are available, but the choice o stability concept is not crucial or the existence o an analog o the Echenique (orthcoming) embedding result. 13 The unit demand condition immediately implies the second condition o unitarity. Additionally, it implies that there is no irm worker pair (, w) F W such that X w C (Y ) > 1, so the irst condition o unitarity holds, as well. 4
Hatield and Kojima (2010) show (in their Theorem 3) that in unitary models o matching with contracts, substitutability implies the ollowing Pareto separability condition similar to that o Roth (1984). Deinition 2. The preerences o agent i F W are Pareto separable i or any x, x X i with x x, x F = x F, and x W = x W, Y X such that x C i (Y {x, x }) (1) = Y X such that x C i (Y {x, x }). An equivalent ormulation o Pareto separability is a orm o marginal-utility monotonicity o contracts: the preerences o i F W are Pareto separable i and only i or any x, x X i with x x, x F = x F, and x W = x W, Y X such that u i (Y {x}) > u i (Y {x }) (2) = Y X such that u i (Y {x}) < u i (Y {x }). It is clear that (1) holds i and only i (2) does. This latter ormulation o Pareto separability, (2), leads to an embedding generalizing that o Echenique (orthcoming) whenever agents preerences are subsitutable. 2. Generalizing the Echenique Embedding For a model (X, (u i )) o matching with contracts and a model (S, (v i )) o matching with salaries, an embedding o (X, (u i )) into (S, (v i )) is a one-to-one unction g : X F W S with g(x) {x F } {x W } S or each x X. For Y X and an embedding g o (X, (u i )) into (S, (v i )), I say that g(y ) deines a (stable) matching in (S, (v i )) i (, w, s w ), (, w, s w ) g(y ) implies that s w = s w (and g(y ) is stable in (S, (v i ))). Theorem 1. Suppose that (X, (u i )) is a model o matching with contracts in which the preerences o each agent i F W are substitutable and Pareto separable. Then, there is a model o matching with salaries (S, (v i )) and an embedding g o (X, (u i )) into (S, (v i )) such that 1. or each i F W, the demand unction D i deined by v i satisies the gross substitutes condition, and 2. Y X is a stable allocation in (X, (u i )) i and only i g(y ) deines a stable matching in (S, (v i )). The proo o Theorem 1 closely ollows the argument o Echenique (orthcoming); I present it in Appendix A. As substitutability implies Pareto separability in unitary matching with contracts models (Theorem 3 o Hatield and Kojima (2010)), the ollowing result is an immediate corollary o Theorem 1. 5
Theorem 2. Suppose that (X, (u i )) is a unitary model o matching with contracts in which the preerences o each agent i F W are substitutable. Then, the conclusions o Theorem 1 hold. Theorem 2 generalizes the embedding theorem o Echenique (orthcoming) to unitary many-to-many matching models such as that o Klaus and Walzl (2009). One consequence o Theorem 2 is the existence o stable allocations in unitary many-tomany matching with contracts models with substitutable preerences; this ollows rom existing results on the existence o stable outcomes in many-to-many salary matching models with substitutable preerences. 14 3. The Importance o Unitarity Pareto separability is key to the proo o Theorem 1; it is apparently the only preerence condition needed (in addition to substitutability) or the embedding argument to work. Conversely, Pareto separability appears to be essential or the embedding result. As the Pareto separability o substitutable preerences generally breaks down in nonunitary matching models, it is not clear what generalization o Theorem 2, i any, can be ound or such models. Indeed, the embedding result ails even in simple nonunitary many-to-many matching with contracts settings. For example, 15 suppose that F = {}, W = {w 1, w 2 }, X = {x 1,m, x 1,n, x 2,m, x 2,n }, and that the preerences o take the orm P : {x 1,m, x 1,n } {x 1,m, x 2,n } {x 2,m, x 1,n } {x 2,m, x 2,n } {x 1,m } {x 1,n } {x 2,m } {x 2,n }. In this example, the preerences o are substitutable but are not Pareto separable 16 and (whenever w 1 and w 2 ind all their contracts acceptable) even this simple nonunitary example cannot be embedded into a model o matching with salaries in such a ashion that the relevant substitutability and stability notions correspond. Acknowledgements I thank Vince Craword, John William Hatield, Christian Hellwig, Sonia Jae, Alvin E. Roth, the reeree, and especially Federico Echenique or helpul comments, and grateully acknowledge the support o a National Science Foundation Graduate 14 Hatield et al. (2011) show that stable outcomes exist (in the presence o substitutable preerences) in a model which includes many-to-many salary matching as a special case. 15 The example I present has a natural interpretation: There are two workers, w 1 and w 2, the irst o whom is more talented than the second. The irm has two jobs to ill, a morning position (m) a night position (n), and can hire at most one worker or each job. The irm preers to have the best possible worker in the morning shit. It would like to ill both jobs i possible, but ailing that would preer to hire w 1 i at all possible. 16 Taking i =, Y = {x 1,m }, Y = {x 1,n }, x = x 2,n, and x = x 2,m gives a violation o condition (1). This example also shows that in nonunitary models not even sets o contracts can be marginalutility ranked in a ashion analogous to condition (2). 6
Research Fellowship, a Yahoo! Key Scientiic Challenges Program Fellowship, and a Terence M. Considine Fellowship in Law and Economics unded by the John M. Olin Center at Harvard Law School. Blair, C., 1988. The lattice structure o the set o stable matchings with multiple partners. Mathematics o Operations Research 13, 619 628. Craword, V. P., Knoer, E. M., 1981. Job matching with heterogeneous irms and workers. Econometrica 49 (2), 437 450. Echenique, F., orthcoming. Contracts vs. salaries in matching. American Economic Review. Echenique, F., Oviedo, J., 2006. A theory o stability in many-to-many matching markets. Theoretical Economics 1, 233 273. Hatield, J. W., Kojima, F., 2010. Substitutes and stability or matching with contracts. Journal o Economic Theory 145 (5), 1704 1723. Hatield, J. W., Kominers, S. D., 2010. Contract design and stability in matching markets, Mimeo, Harvard Business School. Hatield, J. W., Kominers, S. D., orthcoming. Matching in Networks with Bilateral Contracts. American Economic Journal: Microeconomics. Hatield, J. W., Kominers, S. D., Nichior, A., Ostrovsky, M., Westkamp, A., 2011. Stability and competitive equilibrium in trading networks, mimeo, Stanord Graduate School o Business. Hatield, J. W., Milgrom, P., 2005. Matching with contracts. American Economic Review 95, 913 935. Kamada, Y., Kojima, F., 2011. Improving eiciency in matching markets with regional caps: The case o the Japan Residency Matching Program, mimeo, Stanord University. Kamada, Y., Kojima, F., orthcoming. Stability and strategy-prooness or matching with constraints: A problem in the Japanese medical matching and its solution. American Economic Review Papers & Proceedings. Kelso, A. S., Craword, V. P., 1982. Job matching, coalition ormation, and gross substitutes. Econometrica 50, 1483 1504. Klaus, B., Walzl, M., 2009. Stable many-to-many matchings with contracts. Journal o Mathematical Economics 45 (7-8), 422 434. Ostrovsky, M., 2008. Stability in supply chain networks. American Economic Review 98, 897 923. Roth, A. E., 1984. Stability and polarization o interests in job matching. Econometrica 52, 47 57. 7
Sönmez, T., 2011. Bidding or army career specialties: Improving the ROTC branching mechanism, mimeo, Boston College. Sönmez, T., Switzer, T. B., 2011. Matching with (branch-o-choice) contracts at United States Military Academy, mimeo, Boston College. 8
Appendix A. Proo o Theorem 1 For (, w) F W, I denote by ˇX w the set o undominated contracts in X X w, i.e. the set deined by ˇX w {x X X w : x X X w such that u ({x }) > u ({x}) and u w ({x }) > u w ({x})}. I order the contracts x 1 w,..., x ˇX w w ˇX w such that l > l = u w ({x l w}) > u w ({x l w}). (A.1) Note that (A.1) implies that l > l = u ({x l w}) < u ({x l w}), (A.2) since otherwise x l w is dominated. I let S = 1+max F,w W ˇX w, set S = {1,..., S}, and let g : X F W S be the map with takes the contract x l w ˇX w to the triple (, w, l) F W S, 17 and takes all dominated contracts x to (x F, x W, S). I set v i (g(c i (Y ))) = u i (C i (Y )) or each Y X, 18 and set each v i to equal 1 + min Y X u i (Y ) everywhere else in its domain. By construction, (S, (v i )) is a model o matching with salaries, and g is an embedding o (X, (u i )) into (S, (v i )) such that Y X is stable in (X, (u i )) i and only i g(y ) deines a stable matching in (S, (v i )). To complete the proo, I now show that or each i F W, the demand unction D i deined by v i satisies the gross substitutes condition. I show that D satisies the gross substitutes condition or each F ; the proo that each D w (w W ) satisies the gross substitutes condition is completely analogous. For a salary vector s S W, I let X s {x s w w : w W } be the set o contracts in X ( w W X w ) associated with the salaries s w, using the convention that i ˇX w < s w then x s w w is some arbitrary dominated contract in X X w. 19 Note that D (s) = g(c (X s )). Now, let { X s + x s w w : S } s w s w. As the preerences o are Pareto separable, this immediately implies that C (X s + ) C (X s ).20 It ollows that C (X s + ) = C (X s ), as Xs + X s. 17 Note that this map g is completely analogous to the embedding o Echenique (orthcoming). As in that embedding, or g to be one-to-one I must, strictly speaking, expand S and choose distinct assignments (x F, x W, S + l) or each dominated x X; I suppress this concern or notational convenience. 18 This is well-deined because the matching with contracts model is unitary. 19 Since dominated contracts are never chosen, it is without loss o generality to assume that (X X w) \ ˇX w is always nonempty. 20 To see this, suppose otherwise that there is some x s+ w C (X s + ) \ C (X s ). Then, there must 9
Now, let s s be salary vectors or which s w = s w and (, w, s w) D (s). I must show that (, w, s w ) D (s ). Suppose otherwise or the sake o contradiction, and note that x s w w / C ( X s ) = C ( X s + ). (A.3) Since X s + X s + and the preerences o are substitutable, (A.3) implies that x s w w / C (X s + ) = C (X s ). But since s w = s w, this means that x s w w / C (X s ), contradicting the act that (, w, s w ) D (s) = g(c (X s )). be some x s w C (X s ) with w = xs W + w u ({x s+ w = xs w W ; Pareto separability (in the orm (2)) then shows that }) > u ({x s w }). But then, by (A.2), s + w < s w, contradicting the act that x s+ w X s +. 10