P ROOFS OF T HEOREMS 9.7, 9.8, AND 9.11 1 Proof of Theorem 9.7

We first establish some lemmas.

LEMMA14.1. Let P be substitutable. Letµ and µ⁰be pre-matchings. If µ ≤^Bµ⁰then, for all w ∈ W and f ∈ F , U (f , µ) ⊆ U (f , µ⁰) and V (w,µ) ⊇ V (w,µ⁰).

PROOF. We prove thatV(w,µ) ⊇ V (w,µ⁰). The proof that U(f ,µ) ⊆ U(f ,µ⁰) is analo-gous.

First, ifV(w,µ⁰) = {;}, then there is nothing to prove, as {;} = V (w,µ⁰) ⊆ V (w,µ).

Suppose thatV(w,µ⁰) 6= {;}, and let f ∈ V (w,µ⁰). Then, w ∈ Ch(µ⁰(f ) ∪ w,P(f )).

Butµ ≤^Bµ⁰, so the definition of≤^Bimplies that, for allf ∈ F , either µ⁰(f ) = µ(f ) so w ∈ Ch (µ(f ) ∪ w, P(f )), or µ⁰(f ) = Ch(µ⁰(f ) ∪ µ(f ),P(f )). Then w ∈ Ch(µ⁰(f ) ∪ w ,P(f )) implies that

w∈ Ch (µ⁰(f ) ∪ w,P(f ))

= Ch(Ch(µ⁰(f ) ∪ µ(f ),P(f )) ∪ w,P(f ))

= Ch(µ⁰(f ) ∪ µ(f ) ∪ {w },P(f )).

The second equality above is from Proposition 2.3 in Blair(1988). (Blair proves that ifP is substitutable, then Ch(A ∪ B,P(f )) = Ch(Ch(A,P(f )) ∪ B,P(f )) for all A and B.) Substitutability ofP implies that w ∈ Ch (µ(f ) ∪ w, P(f )). Then f ∈ V (w, µ) and thus

V(w,µ) ⊇ V (w,µ⁰). ƒ

LEMMA14.2. Let P be strongly substitutable. Let µ and µ⁰be pre–matchings. If µ ≤ µ⁰ then, for all w ∈ W and f ∈ F , U (f , µ) ⊆ U (f , µ⁰) and V (w,µ) ⊇ V (w,µ⁰).

PROOF. We prove thatV(w,µ) ⊇ V (w,µ⁰). The proof that U(f ,µ) ⊆ U(f ,µ⁰) is analo-gous.

First, ifV(w,µ⁰) = {;}, then there is nothing to prove. Suppose that V (w,µ⁰) 6= {;}, and let f ∈ V (w, µ⁰). Then, w ∈ Ch(µ⁰(f ) ∪ w,P(f )). Strong substitutability implies then w∈ Ch (µ(f ) ∪ w, P(f )), as µ⁰(f ) R(f ) µ(f ) because µ ≤ µ⁰. ƒ LetV⁰= {ν ∈ V : ν(s ) R(s ) ; for all s ∈ F ∪ W }. We need to work on the set V⁰instead ofV because ν0andν1are the smallest and largest, respectively, elements ofV⁰. Note thatT(V ) ⊆ V⁰, so there is no loss in working withV⁰.

LEMMA14.3. If P is substitutable, then T|V⁰is increasing whenV⁰is endowed with≤^B. If P is strongly substitutable, then T|V⁰is increasing whenV⁰is endowed with≤.

PROOF. We show thatT|V⁰is increasing whenV⁰ is endowed with order≤^B. That is, wheneverµ ≤^Bµ⁰we have(T µ) ≤^B(T µ⁰). The proof for ≤ follows along the same lines, usingLemma 14.2instead of14.1.

Letµ ≤^B µ⁰and fixf ∈ F and w ∈ W . Lemma 14.1says thatU(f ,µ) ⊆ U(f ,µ⁰). We first show that

Ch(U(f ,µ⁰),P(f )) = Ch([Ch(U(f ,µ⁰),P(f )) ∪ Ch(U(f ,µ),P(f ))],P(f )). (39) To see this, letS⊆ Ch (U (f , µ⁰),P(f )) ∪ Ch(U(f ,µ),P(f )). Then S ⊆ U(f ,µ) ∪ U(f ,µ⁰) = U(f ,µ⁰), so Ch(U(f ,µ⁰),P(f )) R(f ) S. But Ch(U(f ,µ⁰),P(f )) ⊆ Ch(U(f ,µ⁰),P(f )) ∪ Ch(U(f ,µ),P(f )), so we have established (39).

Now,(T µ⁰)(f ) = Ch(U(f ,µ⁰),P(f )) and (T µ)(f ) = Ch(U(f ,µ),P(f )), so (39) implies that

(T µ⁰)(f ) = Ch([(T µ⁰)(f ) ∪ (T µ)(f )],P(f ))).

The proof for(T µ⁰)(w ) is analogous. ƒ

NowT|V⁰:V⁰→ V⁰is increasing andV⁰is a lattice (Remark 9.1). We haveT(V ) ⊆ V⁰ soE (P) ⊆ V⁰, andE (P) equals the set of fixed points of T |V⁰. So Tarski’s fixed point theo-rem implies that〈E (P), ≤^B〉 and 〈E (P), ≤〉 are nonempty lattices. Item (ii) inTheorem 9.7 follows from standard results (Topkis 1998, Chapter 4).

This finishes the proof ofTheorem 9.7.

14.2 Proof ofTheorem 9.8 We first prove item (i).

Let ν, ν⁰ ∈ E (P) be such that ν⁰(w ) R(w ) ν(w ) for all w ∈ W . Suppose, by way of contradiction, that there is some f ∈ F such that ν⁰(f ) P(f ) ν(f ). Let C = Ch(ν(f ) ∪ ν⁰(f ),P(f )), so C R(f ) ν⁰(f ) P(f ) ν(f ). But ν ∈ E (P) implies that ν(f ) = Ch(ν(f ),P(f )) (Lemma 11.4), so C * ν(f ). Hence there is w ∈ C \ ν(f ); note that w∈ ν⁰(f ). Now

w∈ C = Ch (ν(f ) ∪ ν⁰(f ) ∪ w ,P(f ))

and the substitutability ofP(f ) imply that w ∈ Ch(ν(f ) ∪ w ,P(f )). So f ∈ V (w ,ν).

Now,w ∈ ν⁰(f ) \ ν(f ) implies f ∈ ν⁰(w ) \ ν(w ). Then ν⁰(w ) R(w ) ν(w ) implies that ν⁰(w ) P(w ) ν(w ), as P(w ) is strict. But ν⁰(w ) = Ch(ν⁰(w ),P(w )) = Ch(ν⁰(w ) ∪ f ,P(w )) by Lemma 11.4. So strong substitutability implies that f ∈ Ch (ν(w ) ∪ f , P(w )). Since f /∈ ν(w ), we obtain ν(w ) ∪ f P(w ) ν(w ). This contradicts ν ∈ E (P), since we showed f ∈ V (w , ν) and ν ∈ E (P) imply ν(w ) = Ch (V (w , ν), P(w )).

To prove item (ii) in the theorem, note that whenP(F ) is strongly substitutable the model is symmetric, and the argument above holds with firms in place of workers, and

workers in place of firms. ƒ

14.3 Proof ofTheorem 9.11

We first prove that〈E (P), ≤〉 is a sublattice of 〈V , ≤〉. That 〈E (P), ≤〉 is distributive follows then immediately. We need to verify that the lattice operations∨ and ∧ in V are the lattice operations in〈E (P), ≤〉.

Letν¹,ν²∈ E (P). Let ν = ν¹∨ν²inV . We prove that ν is the join of ν¹,ν²in〈E (P), ≤〉.

The proof forν¹∧ ν²is analogous.

By hypothesisν is a matching; so

w∈ ν(f ) → f ∈ ν(w ).

We prove that ν ∈ E (P). Suppose, by way of contradiction, that there is f such that (T ν)(f ) 6= ν(f ). Without loss of generality, say that ν(f ) = ν¹(f ) R(f ) ν²(f ). Since ν¹∈ E (P), ν¹is individually rational (Lemma 11.4), sof ∈ Ch (ν¹(w ),P(w )) = Ch(ν¹(w ) ∪ f , P(w )) for all w ∈ ν¹(f ). For all w , on the other hand, ν¹(w ) R(w ) ν(w ). So strong sub-stitutability gives f ∈ Ch (ν(w ) ∪ f , P(w )) for all w ∈ ν¹(f ). Thus ν¹(f ) ⊆ U(f ,ν). Since (T ν)(f ) = Ch(U(f ,ν),P(f )) and ν¹is individually rational,(T ν)(f ) \ ν(f ) 6= ;.

Letw ∈ (T ν)(f ) \ ν(f ). By substitutability, w ∈ Ch (ν¹(f ) ∪ w ,P(f )). Strong substi-tutability andν¹(f ) R(f ) ν²(f ) then imply w ∈ Ch(ν²(f ) ∪ w ,P(f )). So

f ∈ V (w , νⁱ) (40)

fori= 1,2.

On the other handw∈ (T ν)(f ) implies w ∈ U (f , ν), so

f ∈ Ch (ν(w ) ∪ f , P(w )). (41)

Leti be such thatν(w ) = νⁱ(w ). Then (40) andνⁱ ∈ E (P) imply νⁱ(w ) ∪ f ∈ V (w ,νⁱ).

But we assumedw /∈ ν(f ), so f /∈ νⁱ(w ), as ν is a matching. Then νⁱ(w ) ∪ f 6= νⁱ(w ), which contradictsνⁱ ∈ E (P), given (41) andν(w ) ∪ f ∈ V (w ,νⁱ).

For the rest of the theorem, we need a lemma.

LEMMA14.4. Let P be strongly substitutable. For all f and w , for any ν and ν⁰ in V , U(f ,ν ∨ ν⁰) = U(f ,ν) ∪U(f ,ν⁰), U(f ,ν ∧ ν⁰) = U(f ,ν) ∩U(f ,ν⁰), V (w,ν ∨ ν⁰) = V (w,ν) ∩ V(w,ν⁰), and V (w,ν ∧ ν⁰) = V (w,ν) ∪ V (w,ν⁰).

PROOF. We prove only thatU(f ,ν ∨ ν⁰) = U(f ,ν) ∪ U(f ,ν⁰) and that V (w,ν ∨ ν⁰) = V(w,ν) ∩ V (w,ν⁰). The proof of the other statements is symmetric.

We first prove thatU(f ,ν ∨ν⁰) ⊆ U(f ,ν)∪U(f ,ν⁰). Let w ∈ U(f ,ν ∨ν⁰), so f ∈ Ch((ν ∨ ν⁰)(w ) ∪ f ,P(w )). Now, (ν ∨ ν⁰)(f ) equals either ν(f ) or ν⁰(f ). If (ν ∨ ν⁰)(w ) = ν(w ), then f ∈ Ch (ν(w )∪ f , P(w )); so w ∈ U (f , ν). Similarly, if (ν ∨ν⁰)(w ) = ν⁰(w ), then w ∈ U(f ,ν⁰).

This proves thatU(f ,ν ∨ ν⁰) ⊆ U(f ,ν) ∪U(f ,ν⁰).

Second, we prove thatU(f ,ν) ∪ U(f ,ν⁰) ⊆ U(f ,ν ∨ ν⁰). Let w ∈ U(f ,ν), so f ∈ Ch(ν(w ) ∪ f ,P(w )). Now ν(w ) R(w ) (ν ∨ ν⁰)(w ), so strong substitutability implies f ∈ Ch((ν ∨ ν⁰)(w ) ∪ f ,P(w )). Hence w ∈ U(f ,ν ∨ ν⁰). This proves that U(f ,ν) ∪ U(f ,ν⁰) ⊆ U(f ,ν ∨ ν⁰). So, U(f ,ν ∨ ν⁰) = U(f ,ν) ∪U(f ,ν⁰).

We now prove thatV(w,ν ∨ ν⁰) = V (w,ν) ∩ V (w,ν⁰). First we prove V (w,ν ∨ ν⁰) ⊆ V(w,ν) ∩ V (w,ν⁰). Let f ∈ V (w,ν ∨ ν⁰), so

w∈ Ch ((ν ∨ ν⁰)(f ) ∪ w,P(f )). (42) Without loss of generality, say(ν ∨ν⁰)(f ) = ν(f ) R(f ) ν⁰(f ). Then (ν ∨ν⁰)(f ) = ν(f ) implies thatf ∈ V (w, ν). Statement (42) and strong substitutability imply thatw∈ Ch (ν⁰(f ) ∪ w , P(f )), as (ν ∨ ν⁰)(f ) R(f ) ν⁰(f ). Thus f ∈ V (w,ν), and we obtain V (w,ν ∨ ν⁰) ⊆ V (w,ν) ∩ V(w,ν⁰).

Finally, we prove thatV(w,ν) ∩ V (w,ν⁰) ⊆ V (w,ν ∨ ν⁰). Let f ∈ V (w,ν) ∩ V (w,ν⁰), so w ∈ Ch (ν(f ) ∪ w, P(f )) and w ∈ Ch (ν⁰(f ) ∪ w,P(f )). Now, (ν ∨ ν⁰)(w ) equals either ν(w ) orν⁰(w ), so either way w ∈ Ch((ν ∨ ν⁰)(f ) ∪ w,P(f )). Hence f ∈ V (w,ν ∨ ν⁰). ƒ Lemma 14.4implies immediately thatψ is a lattice homomorphism: Let ν⁰,ν ∈ V . For anyf and w ,

(ψ(ν ∨ ν⁰))(f ) = U(f ,ν ∨ ν⁰) = U(f ,ν) ∪U(f ,ν⁰) = (ψν)(f ) ∪ (ψν⁰)(f ) (ψ(ν ∨ ν⁰))(w ) = V (w,ν ∨ ν⁰) = V (w,ν) ∩ V (w,ν⁰) = (ψν)(f ) ∩ (ψν⁰)(f ).

Soψ(ν ∨ ν⁰) = ψν t ψν⁰. Thatψ(ν ∧ ν⁰) = ψν u ψν⁰is also trivial fromLemma 14.4.

We now show thatψ|E (P) is an isomorphism onto its range. Letν, ν⁰ ∈ E (P). Let ψν = ψν⁰. Then for all f , U(f ,ν) = U(f ,ν⁰) so (T ν)(f ) = (T ν⁰)(f ). Similarly (T ν)(w ) = (T ν⁰)(w ) for all w . So T ν = T ν⁰. Thenν, ν⁰∈ E (P) implies ν = ν⁰, asv= T ν and v⁰= T ν⁰. Henceψ is one-to-one, as ψν = ψν⁰impliesν = ν⁰.

REFERENCES

Adachi, Hiroyuki (2000), “On a characterization of stable matchings.”Economic Letters, 68, 43–49. [236]

Alcalde, José, David Pérez-Castrillo, and Antonio Romero-Medina (1998), “Hiring proce-dures to implement stable allocations.”Journal of Economic Theory, 82, 469–480. [237] Alcalde, José and Antonio Romero-Medina (2000), “Simple mechanisms to implement the core of college admissions problems.”Games and Economic Behavior, 31, 294–302.

[237]

Alkan, Ahmet (2001), “On preferences over subsets and the lattice structure of stable matchings.”Review of Economic Design, 6, 99–111. [237]

Alkan, Ahmet (2002), “A class of multipartner matching models with a strong lattice structure.”Economic Theory, 19, 737–746. [237]

Barberà, Salvador, Hugo Sonnenschein, and Lin Zhou (1991), “Voting by committees.”

Econometrica, 59, 595–609. [247]

Blair, Charles (1988), “The lattice structure of the set of stable matchings with multiple partners.” Mathematics of Operations Research, 13, 619–628. [236,238,243,248,250, 253,254,266,268]

Crawford, Vincent P. and Elsie M. Knoer (1981), “Job matching with heterogeneous firms and workers.”Econometrica, 49, 437–450. [247]

Dutta, Bhaskar and Jordi Massó (1997), “Stability of matchings when individuals have preferences over colleagues.”Journal of Economic Theory, 75, 464–475. [247]

Echenique, Federico (2004), “Counting combinatorial choice rules.” Social Science Working Paper 1199, California Institute of Technology. Forthcoming inGames and Eco-nomic Behavior. [247]

Echenique, Federico and Jorge Oviedo (2004), “Core many-to-one matchings by fixed point methods.”Journal of Economic Theory, 115, 358–376. [236,243,245]

Fleiner, Tamas (2003), “A fixed-point approach to stable matchings and some applica-tions.”Mathematics of Operations Research, 28, 103–126. [236]

Gale, David and Lloyd S. Shapley (1962), “College admissions and the stability of mar-riage.”American Mathematical Monthly, 69, 9–15. [245]

Hatfield, John and Paul Milgrom (2005), “Auctions, matching and the law of auctions matching and the law of aggregate demand.”American Economic Review, 95, 913–935.

[236,240]

Kelso, Alexander S. and Vincent P. Crawford (1982), “Job matching, coalition formation, and gross substitutes.”Econometrica, 50, 1483–1504. [235,247]

Klijn, Flip and Jordi Massó (2003), “Weak stability and a bargaining set for the marriage model.”Games and Economic Behavior, 42, 91–100. [237,243]

Knuth, Donald E. (1976), Marriages Stables et leurs relations avec d’autres problèmes combinatoires. Les Presses de l’Université de Montréal, Montréal. Translated as Stable Marriage and Its Relation to Other Combinatorial Problems by Martin Goldstein, Amer-ican Mathematical Society, Providence, Rhode Island, 1997. [253,254,255]

Konishi, Hideo and M. Utku Ünver (2005), “Credible group-stability in many-to-many matching problems.” Unpublished paper, Boston College. Forthcoming,Journal of Eco-nomic Theory. [237,241,246]

Martínez, Ruth, Jordi Massó, Alejandro Neme, and Jorge Oviedo (2000), “Single agents and the set of many-to-one stable matchings.”Journal of Economic Theory, 91, 91–105.

[247]

Martínez, Ruth, Jordi Massó, Alejandro Neme, and Jorge Oviedo (2001), “On the lattice structure of the set of stable matchings for a many-to-one model.” Optimization, 50, 439–457. [255]

Martínez, Ruth, Jordi Massó, Alejandro Neme, and Jorge Oviedo (2004), “An algorithm to compute the full set of many-to-many stable matchings.”Mathematical Social Sciences, 47, 187–210. [237,245]

Roth, Alvin and Marilda Sotomayor (1988), “Interior points in the core of two-sided matching markets.”Journal of Economic Theory, 45, 85–101. [236]

Roth, Alvin and Marilda Sotomayor (1990), Two-sided Matching: A Study in Game-Theoretic Modelling and Analysis, volume 18 of Econometric Society Monographs. Cam-bridge University Press, CamCam-bridge. [234,238,240,247,248]

Roth, Alvin E. (1984), “Stability and polarization of interests in job matching.” Econo-metrica, 52, 47–57. [234,236]

Roth, Alvin E. (1985), “Conflict and coincidence of interest in job matching: Some new results and open questions.”Mathematics of Operations Research, 10, 379–389. [236, 237,253,254]

Roth, Alvin E. and Elliott Peranson (1999), “The redesign of the matching market for American physicians: Some engineering aspects of economic design.”American Eco-nomic Review, 89, 748–780. [234]

Sönmez, Tayfun (1996), “Strategy-proofness in many-to-one matching problems.” Eco-nomic Design, 1, 365–380. [247]

Sotomayor, Marilda (1999), “Three remarks on the many-to-many stable matching problem.”Mathematical Social Sciences, 38, 55–70. [234,236,238,242,246]

Sotomayor, Marilda (2004), “Implementation in the many-to-many matching market.”

Games and Economic Behavior, 46, 199–212. [237]

Topkis, Donald M. (1998),Supermodularity and Complementarity. Princeton University Press, Princeton, New Jersey. [250,269]

Zhou, Lin (1994), “A new bargaining set of an N-person game and endogenous coalition formation.”Games and Economic Behavior, 6, 512–526. [243]

Submitted 2005-9-1. Final version accepted 2005-12-16. Available online 2005-12-16.