FACULTY WORKING PAPER NO. 1080 Stable Matching with Multiple Partners and Lattice Structure Charles E. Blair rn\E OBRfiRX Pi College of Commerce and Business Administration Bureau of Economic and Business Research University of Illinois, Urbana-Champaign BEBR FACULTY WORKING PAPER NO. 1080 College of Commerce and Business Administration University of Illinois at Urbana-Champaign October, 1984 Stable Matching with Multiple Partners and Lattice Structure Charles E. Blair, Associate Professor Department of Business Administration Stable Matching with Multiple Partners and Lattice Structure by Charles Blair Abstract We continue recent work, on the matching problem for firms and workers, and show that, for a suitable ordering, the set of stable matchings is a lattice. Acknowledgement: 1 wish to thank Alvin Roth for many helpful discussions . Digitized by the Internet Archive in 2011 with funding from University of Illinois Urbana-Champaign http://www.archive.org/details/stablematchingwi1080blai 1. Introduction : In [2] Gale and Shapley considered the problem of forming a set of men and women into couples. A matching was called stable if we could not find a man in one couple and a women in another who would be happier together than with their current partners. [2 J showed that stable matchings always exist, regardless of the preferences of the players. Conway [4] showed that for a fixed set of players the set of all stable matches is a distributive lattice. Kelso and Crawford [3] generalized this model to one involving firms and workers, where a firm hires sets of workers. In effect, the model allows polygamy by one set of players. [3] showed that a stable match exists provided the preferences the firms have over sets of workers hired satisfy a "gross substitutes condition." Roth [5, 6] further generalized this model to allow each worker to work for a set of firms, thus allowing polygamy and polyandry. Again, the existence of a stable matching was shown, provided a gross substi- tutability assumption on the preferences was satisfied. Some examples were constructed showing that the lattice structure present in the monogamous case did not generalize in a straightforward manner. This paper continues the study of multi-partner matching in [5, 6]. Section 2 describes the model. In section 3, we re-examine the notion of stability and show that it does not coincide with the idea of the core. Additional assumptions are suggested which imply relationships between the core and the set of stable outcomes. Section 4 proves the existence of stable outcomes. In section 5 we show that the set of stable outcomes does possess a lattice structure, for a suitably chosen partial ordering. Section 6 gives some open problems and section 7 gives some examples. -2- 2 . Description of the Model and Notation We will have a set of firms F and workers W. These are often lumped together as the set of players P ■ F U W. A firm i hires a worker j at a salary s. We will use S to stand for the set of all possible salaries. In this paper, P and S will be assumed finite, although it is hoped that the framework given here will extend to other cases. We will use "preferred" in the sense of "strictly preferred," indif- ference not allowed. For sets A,B the notation ACB allows the pos- sibility A=B. A-B will be used for set-theoretic difference (there is no ambiguity because the elements do not have an addition operation). For brevity, we will use AUx and A-x in place of the more careful AU{x} and A-{x}. PxS A matching will be specified by a function f:P+2 such that for all i,jeP; s,teS (HI) If (i,s)ef(j), then (j,s)ef(i) (H2) (i,s)ef(j) and (i,t)ef(j) implies s=t (H3) If ieF f(i)CWxS (HA) If ieW f(i)CFxS Intuitively (i,s)ef(j) means (if ieF) firm i hires worker j at salary s. Note that f is completely specified by its values f(i), ieF (or ieW). Each player has a preference among various possible combinations of partners and salaries. For ieP, ACPxS C.(A) is the set of partners and -3- salaries in A that i would like best. We will assume that, for each i,jeP; A,BcPxS; seS: (CI) C.(A)CA (C2) If C.(A)CBCA, then C i (B) = C.(A) (C3) If (j,s)eC.(A), then C jL (A)-(j ,s)CC i (A-( j ,s) ) It is understood that C. chooses from among feasible combinations. Thus, if i cannot hire j at salary s, then (j,s)^C.(A) for any A. Similarly, (j,s)eC.(A) and (j,t)eC.(A) implies s=t. We shall be making repeated use of the following consequences of (C1)-(C3). Proposition 2.1 : C.(C.(A)) - C.(A) Proof : By (CI) C. (A)CC. (A)CA. Then use (C2). Q.E.D. Proposition 2.2: C. (AUB)/UCC. (A) . c x x Proof : Induction on size of (Al/B)-A. If size is zero C.(AUB) = C.(A). If xeAUB-A, then by induction hypothesis C. (AUB-x)rtACC. (A) . By (C3), C.(AUB)flACC.(AUB-x)rtA if x£k. Q.E.D. Proposition 2.3 : C.(AtfB) = C. (C. (A)UB)) . 1 1 i 2. Proof : From Prop. 2.2, C. (AUB)flACC. (A) and C. (AUB>lBCC. (B)CB. Thus C.(AVB)CC.(A)UBCAim and we may apply (C2). Q.E.D. For i,jeP; seS; and f a matching we will say "i wants (j,s) in f" if (j ,s)eC. (f (i)U(j ,s) ) . When there is no ambiguity we may say "i wants (j,s)." -4- 3 . Stable and Superstable Matchings We will define a matching f to be stable if for all i,jeP and seS (51) C.(f(i)) - f(i) (52) If (j,s)eC.(f(i)i;(j,s)) then C.(f(j)U(i,s))Cf(j). (S2) says that if i wants (j,s) then either j does not want (i,s) or i and j are already matched in f at salary s. Thus (S1)-(S2) give the usual notion of stability. However, certain pathologies may occur: Example 3 .1: 1 2 3 a,b b,c a,c b,c a,c a,b a b c b a a c c b 1.2 2,3 1,3 2.3 1,3 1,2 12 3 2 11 3 3 2 In this example and all subsequent ones, numbers will be firms, small letters will be workers. Unless otherwise stated, S will consist of one member. In this example, the first choice for firm 1 would be to hire workers a and b. His worst choice (other than hiring nobody) would be hiring c. The only stable matching gives 1 to a, 2 to b, 3 to c. However, giving 2 and 3 to a, 1 and 3 to b, and 1 and 2 to c would make everyone happier. Similar difficulties can be created by salaries. This definition of stability is equivalent to that in [5, 6]. See especially [6, Lemma 2], -5- Example 3.2 : 12 a b a ,b a ,b 1 ,2 1 .2 ay y a ay y a a„,b„ a„,b„ 1„»2„ 1..2. 8' 3 8' 8 $' S 8' 8 The salaries are a,8,y» The matching in which everybody works for everybody at salary 8 is stable, but giving everybody their first choice (a works for 1 at salary a, etc.) is also stable and makes everyone happier. The definition of stability needs to be strengthened in this con- text if it is to coincide with the concept of the core. For i,jeP we will say i and j are matched in f if (j,s)sf(i) for some seS. Two raatchings f,g are similar iff for all i,jeP i and j are matched in f iff i and j are matched in g. A stable matching f is superstable if g similar to f and every ieP does not prefer f(i) to g(i) implies f=g. In other words, f is Pareto optimal among the raatchings similar to f. Proposition 3.3 : If h is stable there is a superstable f similar to h. Proof : If h is not superstable, there is h' similar to h where every ieP does not prefer h(i) to h'(i) and h'(j) is preferred to h(j) 2 for at least one j. The stability of h implies the stability of h' . The argument can now be repeated with h' replacing h. The finiteness of P,S imply we eventually obtain a superstable matching. Q.E.D. 2 We assume that players are not indifferent between any two sets of partners, as in [6, p. 49], -6- To show that superstable matchings are in the core we need to make additional assumptions about the preferences. (Al) If h(i) is preferred to f(i) and (j,s)eh(i) is preferred to (j,t) for (j,t)ef(i), then (j,s)eC i (f (i)(/(j,s)). (A2) If i is matched with j in g implies i matched with j in f, and i prefers f(i) to g(i), and keP is not matched with i in f [or g] , then i prefers f(i)U(k,s) to g(i)l/(k,s). 3 (Al) is similar in spirit to the substitutability property [6, p. 51], which in turn is equivalent to (C1)-(C3). It can be shown that (Al) implies (C1)-(C3), but example 3.1 shows the converse does not hold. (A2) is similar to the assumption (GS) in [3, p. i486]. The rationale for (A2) is that if an employer prefers one combination of salaries to another for a group of employees, he will still prefer that combination if the group is part of a larger group. If the preferences satisfy (A1)-(A2) then superstable matchings are in the core. Theorem 3.4 ; If (A1)-(A2) hold and f is superstable then there does not exist a matching h and nonempty TCP such that every ieT pre- fers h(i) to f(i) and h(i)Cf(i) for every i^T. Proof ; Suppose there were such h,T. We first show that if i,jeP are matched in h they must be matched in f. If i,j were not matched in f then they must be members of T. By (Al), there is (j,s)eh(i) with (j,s)eC i (f(i)U(j,s)) and (i ,s)eC . (f ( j )U(i,s) ) . But this would contradict the stability of f. 3 (Al) was suggested by Roth (private communication) -7- Define g(i) = h(i )U{ ( j ,s) | ( j ,s)ef (i) and i and j are not matched in h} . If i^T, g(i)=f(i). If ieT, i prefers h(i) to f(i). Since f is stable, f(i) is preferred to { ( j ,s) | ( j ,s)ef (i) and i and j are matched in h} . Repeated use of (A2) shows that g(i) is preferred to f(i) for all ieT. By the preceding paragraph, g is similar to f. But this contradicts the superstability of f. Q.E.D, Thus, if (Al) and (A2) are assumed, we can always obtain a member of the core from a stable matching by making salary adjustments. Two special cases should be mentioned. If S consists of only one member, only (Al) need be assumed and every stable matching will be superstable. The same holds if each worker will only work for one firm. In the rest of the paper, however, we will not assume (A1)-(A2) and will confine our attention to the set of stable matchings. One reason for doing so is that this seems the most natural way to extend the results from the monogamous case. Also, only the functions C. need be considered, and not the preference orderings. -8- 4 . Existence of a Stable Matching Lemma 4.1 : Suppose the matching given by f(i) = C.(A.), ieF, A.CWxS satisfies the condition l (M) If (j,s)e(WxS)-A jL , then C . (f ( j )U(i,s) )Cf ( j ) [ (M) may be interpreted as saying that if (j,s) is not in A. then j does not want i,s).] A Then, for any LeW, the new matching f'(i) = C (A') where A^ = A. - (L,s) if (i,s)ef(L) - (^(fCD) = A. otherwise l satisfies (M) with f replaced by f, A. by A!. Proof : If j*L, then (j,s)^A! implies (j,s)^A.. By (M), C.(f(j)U(i,s))Cf(j). By (C3), (j,s)eC m (A m ) implies (j,s)eC m g and g >_ h then f >_ h. Proof : C.(f(i)Ug(i)l/h(i)) = [Prop. 2.3] = C. (f (i)Ug(i) ) - f(i). Q.E.D. It is also easy to show f >_ g and g >. f implies f = g, so we have a partial ordering. Next we show that for stable matchings interchanging the roles of F and W replaces >_ by <_. Proposition 5.2 : If f >_ g and g is stable, then for all isW C.(f(i)Ug(i)) = g(i). Proof : For isW and (j,s)ef(i) f >_ g and Prop. 2.2 imply (i,s)£C.(g(j)U(i,s)). Since g is stable (S2) implies C. (g(i)U(j ,s))qg(i) for all (j,s)ef(i). Since (j ,s)eC i (f (i)Ug(i) ) implies (j ,s)eC i (g(iX/(j ,s)) we have C.(f (i)Ug(i) )Cg(i)Cf (i)Ug(i) . Hence [(C2)] C.(f (i)Ug(i) ) = C.(g(i)) = [(SI)] = g(i). Q.E.D. For stable matchings when the preferences satisfy (A1)-(A2) the ordering takes a simpler form. Proposition 5.3 : If (A1)-(A2) hold then f _> g [f , g stable] iff for all ieF (GSL1) i does not prefer g(i) to f(i). (GSL2) If (j,s)ef(i) and (j,t)eg(i) then i does not prefer (j,t) to (j,s). Proof : The "only if" is easy. To discuss "if", we first show that every jsW does not prefer f(j) to g(j). If this were not the case then for some j there would be (i,s)ef(j) - g(j) such that (i,s)eC . (g( j )U(i,s) ) . -11- (Al) implies (j ,s)eC i (g(i)U(j ,s) ) , hence C i (g(i)Uj ,s) )^g(i) , which con- tradicts the stability of g. Next, let (j,s)eg(i) - f(i). Since j prefers g(j) to f(j) (i,s)eC.(f(j)U(i,s)). By stability, (j ,s)^C i (f (i)U(j ,s)) . Thus C.(f(i)Ug(i)) = f(i). Q.E.D. If S consists of one member, only (GSL1) is relevant. Next we show that there is a maximal stable f. Theorem 5.4 ; Let f be the stable matching constructed in Theorem 4.2. If g is stable, f >_ g. Proof : Recall that a sequence of matchings f,f',... is constructed, In each case f(i) = C.(A.) ieF. We show that g(i)CA., from which C.(f(iX/g(i)) = [Prop. 2.3] C (A.Ug(i)) = C.(A.) = f(i). At the begin- ning A. = WxS. If g(i)CA.. and A! is obtained as in Lemma 4.1, we must & l ° l l show that g(i)CA!. This is immediate if A! - A. . If A! = A. - (L,s) l 1111 we complete the proof by showing (L,s)^g(i). Since f >_ g by induction hypothesis, Prop. 5.2 implies C. (f (L)Ug(L) ) = g(L). If (i,s)eg(L), then Prop. 2.2 implies (i,s)eC T (f (L)(Ai ,s)) . But the definition of A! Li 1 requires that if A! = A. - (L,s) then (i,s)ef(L) - C,(f(L)). This 11 Li contradiction establishes that A! = A. - (L,s) implies (i,s)^g(L) hence [(HI)] (L,s)^g(i). Q.E.D. Next we show that the stable matchings form a lattice under >_. For technical reasons we will be concerned with the set of firms that want a specified worker at various salaries. For f a matching, jeW define I(j,f) = {(i,s)|(j,s)eC.(f(i)(j(j,s))} -12- Proposition 5.5 : If C.(f(i)) = f(i) for all ieF then f(k)G(k,f) for all keW. Proposition 5,6 : If f(i) = C.(A.) and g(i) = C.(a!) where A.CA.' c i* j/ on / i N i' ii for all ieF then I(k,g)C(k,f ) for all keW. Proof : If (i,s)el(j,g), (j ,s)eC.(g(i)U(j ,s) ) = C. (A^J(j ,s)) , hence (j ,s)eG 1 (A jL U(j,s)) = (^(f (i)U(j ,s)). Q.E.D. Lemma 5.7 : Let h(i) = C.(A.), ieF. Fix jeW. Let h f (i) = C(a!) where *[ ■ A.l/(j,s) if (i,s)eC (I(j,h)) = A. otherwise. l Then if f > h and f is stable, f >_ h'. Moreover, if h(k)CC k (I(k,h)) , then h , (k)CC.(I(k,h , )) 1 for all keW. Proof : First note that f >_ h implies f(j>Cl(j,h), since C (f(i)Uh(i)) = f(i) implies (j ,s)eC (h(i)U(j ,s) ) for (i,s)ef(j). Thus if (i,s)eC.(I(j,h)), then (i,s)eC . (f ( j )U(i,s)) , by Prop. 2.2. Since f is stable, this implies C. (f (i)U(j ,s) ) = f(i). Thus, if a! - A.l/(j,s) we have C.(f (i)Uh' (i)) = C ± (f (iX/A^Cj ,s)) = C.(f(i)l/A.) - C.(f(i)Uh(i)) = f(i). If A^ = A., C.(f(i)Uh»(i)) = C.(f(i)Uh(i)) = f(i) is immediate, so we have established f _> h' . By proposition 5.6, I(k,h' )CI(k,h) . Thus, h(k)CC k (I(k,h)) implies h(k)Oh'(k)C(Prop. 5.5) h(k)ni(k,h' XX^CKk^' ) ) . If (i,s)eh' (k)-h(k) we must have k=j and (i,s)eC.(I(j,h)), hence (i,s)eC.(I(j,h')). Q.E.D, -13- Lemma 5.8 : Let f, g be stable raatchings and h(i) = C. (f (i)Cfe(i)) , ieF. If e is stable, e >_ f and e >_ g, then e l^ h. Moreover, h(k)CC k (I(k,h)), for all keW. Proof : C^eCDOhCi)) - C.(e(i)Uf (i)Ug(i)) - C.(e(i)Ug(i)) = e(i). By proposition 5.5, h(k)CI(k,h). Since f is stable we have C (f(k)U(i,s)) = f(k) for any (i,s)el(k,f ) , hence C k (I(k,f)) = f(k). By proposition 5.6, I(k,h)CI(k,f ) . Hence f (k/)h(k)Cf (k)m(k,h)CC k (I(k,h)) Similarly we obtain g(k)Ah(k)CC k (I(k,h)) , hence h(k)CCL (I(k,h)) . Q.E.D. Theorem 5.9 : The set of stable raatchings under > is a lattice. Proof : It is sufficient to show how to construct the l.u.b. of stable f,g. The g.l.b. follows from duality, interchanging F and W (note Proposition 5.2). Let h(i) = C. (f (i)(/g(i)) for ieF. If h is stable, then h is the l.u.b., by Lemma 5.8. Also h(k)CC, (I(k,h)) implies (Prop. 2.2) C,(h(k)) = h(k). Thus, if h is not stable (S2) must be violated, i.e., for some ieF, jeW i and j are not matched and t want to be matched at some salary. For this j, let h , A. be obtained from h, A. - f(i)Ug(i) as in Lemma 5.7. As before, if h' is stable, it is the l.u.b. If h' is not stable, then (S2) must be violated and the argument can be repeated. The finiteness of S and P implies that the process terminates with a stable matching. In section 7 we give an example to show that, unlike the monogamous case, the lattice may fail to be distributive. -14- 6 . Some Open Questions 1. These results show that many features of the original Gale- Shapley model persist in the more general model. In fact, the similarity of proofs suggests that the later models are really virtually identical to the original one. Can a reduction be given which converts a f irms-and-workers model to a monogamous marriage model so that stable marriages in one correspond to stable matchings in the other? The non-distributivity phenomenon makes this unlikely. Less ambitious would be a reduction of the general case to the case in which S consists of one member. 2. The collection of stable matchings in the monogamous case has been shown to be a finite distributive lattice and [1] shows that no further refinement is possible. The most natural representation of a f.d.l. is as a collection of sets closed under unions and inter- sections. Find a natural correspondence between such a collection and the collection of stable matchings. 3. Can every finite lattice occur as a set of stable matchings? 4. As previously noted, different preference orderings can yield the same C. . If C. satisfy (C1)-(C3), is there a corresponding prefer- ence ordering satisfying (Al)? 5. Is the set of superstable matchings a lattice? If not, are there natural additional assumptions which make it one? 6. What about cases where P and S are infinite? There seem to be two ways to still get existence of matchings. One is to impose well- ordering requirements, the other is to topologize and use compactness and continuity. -15- 7 . Two Examples Example 7.1 : The natural extension of the ordering used by Conway in the raonogomous case would be to define f ]^ g iff i does not prefer g(i) to f(i) for all ieF. This ordering would not give a lattice. a,b e d h J h k a,d d e a h J a b,d b 8 9 10 11 12 13 L k m,p q P n,q k L b,n,p m p q m,q n abdehjkL 4 10 23 5689 7 1 1,3 2 4,6 5 7,9 8 1 1 m n p q 13 10 11 12 10 13 10,12 11,13 Define matchings f in which 1 gets b, 2d, 3e, 4a, 5h, 6 j , 7k, 8L, 9k, 10m, lip, 12q, 13n, and g in which Ibd, 2e, 3d, 4h, 5 j , 6h, 7a, 8k, 9L, 10m, lip, 12q, 13n. f and g are both stable. Two minimal stable upper bounds on f, g are Bound 1: lab 2e 3d 4h 5j 6h 7k 8L 9k 10m lip 12q 13n. Bound 2: lad 2e 3d 4h 5j 6h 7k 8L 9k lObnp llq I2p 13mq. However, in the special case when (Al) holds and S has one member this partial ordering coincides with that in section 5, and hence gives a lattice. (See Proposition 5.3) -16- Example 7.2 ; The lattice constructed in section 5 may fail to be distributive even if (Al) holds and S has one member. a b c d e f g bed h i j h i j ... ae af ag g 2,3, 4 1 1 1 2 3 4 5 6 7 2,3 2 3 4 5 6 7 2 3 4 3,4 2,4 1 2 3 4 5 6 7 1,2 1,3 1,4 m l = b, c,d a,e a,f j h i g 1 2 m 2 = b,c,d a,e i a,g h f j 3 m 3 = b, c,d h a,f a,g e i j The matchings m , nu , m_ (in til 1 hires b, c and d; 2 hires a and e; etc.) are all stable. The l.u.b. of m and m_ has every firm get its first choice. The g.l.b of this matching and m is nu . But the g.l.b. of m and m_ and the g.l.b. of nu and m_ both have every worker get his first choice (a works for 2, 3, and 4, etc.). Thus distri- butivity fails. -17- Ref erences 1. C. Blair, "Every Finite Distributive Lattice is a Set of Stable Hatchings," Journal of Combinatorial Theory . 2. D. Gale and L. Shapley, "College Admissions and the Stability of Marriage," American Math. Monthly 69 (1962), pp. 9-15. 3. A. Kelso and V. Crawford, "Job Matching, Coalition Formation, and Gross Substitutes," Econometrica 50 (1982), pp. 1483-1504. 4. D. Knuth, Marriage Stables (pp. 87-92), Montreal University Press 1976. 5. A. Roth, "Conflict and Coincidence of Interest in Job Matching," to appear, Mathematics of Operations Research . 6. A. Roth, "Stability and Polarization of Interests in Job Matching," Econometrica 52 , pp. 47-57. D/244