Supplementary material for “Complementary inputs and the existence of stable outcomes in large trading networks” Ravi Jagadeesan August 24, 2017 This supplement proves maximal domain results asserted in the paper and extends the model to incorporate multilateral contracts and continuous heterogeneity. Section S1 proves Theorems 2 and 3. Section S2 extends the basic model to capture multilateral contracts. Section S3 incorporates continuous heterogeneity into the model. Section S4 proves the existence of seller-initiated-stable outcomes in settings with continuous heterogeneity.

Contents S1 Proofs of Theorems 2 and 3

2

S2 Multilateral matching

13

S3 Modeling continuous heterogeneity

18

S4 Proof of Theorem S1

29

References

39

1

b =I U b ′ x

y

bO

f

f

f (a) Case of b = b ′ .

f z

x

x ′ =z

y

x′

/ b′

(b) Case of b ̸= b ′ .

Figure S1: The network structures produced in the proof of Theorem 3. Undirected edges denote contracts that could go in either direction. The exact network structure depends on whether b = b ′ , where b and b ′ are auxiliary firms to be defined in the course of the proofs. In either case f has complementary preferences over {x , x ′ }.

S1

Proofs of Theorems 2 and 3

The proofs of Theorems 2 and 3 are similar. The strategy is to embed Example 2 into the economy in the form of Figure S1 on page 2.

S1.1

Proof of Theorem 2

I begin by describing the preferences of firms other than f and the open set 𝑈 in the statement of Theorem 2, and then complete the proof. Defining the preferences and the open set 𝑈 . Because 𝑐f is not substitutable in the sale-direction, there exist a set of contracts Z ⊆ 𝑋f and contracts x , x ′ ∈ 𝑋f → rZ such that x ∈ / 𝑐f (Z∪{x }) and x ∈ 𝑐f (Z∪{x , x ′ }). The irrelevance of rejected contracts condition for 𝑐f implies that x ′ ∈ 𝑐f (Z ∪ {x , x ′ }). Let b = b(x ) and let b ′ = b(x ′ ). Let f ∈ 𝐹 r {b , b ′ , f } be arbitrary—such a firm type exists because |𝐹 | ≥ 4. Let y ∈ 𝑋b ∩ 𝑋f be arbitrary, and let Z′ = Z ∪ {y }. Define the preferences of firm types 𝑓 ∈ 𝐹 r {b , b ′ , f } as follows: (︁ )︁ Case 1: 𝑓 ∈ 𝐹large . Let 𝐶 𝑓 (𝜇) = 𝜇Z′𝑓 , 0𝑋𝑓 rZ′𝑓 for all 𝜇 ∈ X𝑓 . Case 2: 𝑓 ∈ 𝐹small . Let 𝑐𝑓 (𝑌 ) = 𝑌 ∩ Z′𝑓 for all 𝑌 ⊆ 𝑋𝑓 . Note that the preferences defined above are substitutable, hence in particular substitutable in the sale-direction.

2

We divide into cases based on whether b = b ′ to define a contract z ∈ 𝑋b ∩ 𝑋b ′ as in Figure S1 on page 2. Case 1: b = b ′ . Let z = x ′ (as in Figure S1(a) on page 2). Case 2: b ̸= b ′ . Let z ∈ 𝑋b ∩ 𝑋b ′ be arbitrary (as in Figure S1(b) on page 2). {︀ }︀ Let 𝐾 = min Mx , Mx ′ , My , Mz . Define the preference of b as follows. Case 1: b ∈ 𝐹large . For 𝜇 ∈ Xb , let (︃ 𝐶 b (𝜇) =

)︃ {︀ }︀ {︀ }︀ 𝜇Zb , min 𝜇x , 𝜇y + 𝜇z , 𝐾 x , min 𝜇x , 𝜇y , 𝐾 y , . {︀ {︀ }︀ }︀ min min {𝜇x , 𝐾} − min 𝜇x , 𝜇y , 𝐾 , 𝜇z z , 0𝑋b rZb r{x ,y ,z }

Case 2: b ∈ 𝐹small . For 𝑌 ⊆ 𝑋b , let ⎧ ⎪ ⎪ 𝑌 ∩ Zb if x ∈ / 𝑌 or 𝑌 ∩ {y , z } = ∅ ⎪ ⎨ . 𝑐𝑓 (𝑌 ) = (𝑌 ∩ Zb ) ∪ {x , y } if {x , y } ⊆ 𝑌 ⎪ ⎪ ⎪ ⎩(𝑌 ∩ Z ) ∪ {x , z } if 𝑌 ∩ {x , y , z } = {x , z } b Note that the choice function of b exhibits complementarities only between x and y , and between x and z . Since b = b(x ), it follows that the choice function defined above is substitutable in the sale-direction. We divide into cases based on whether b = b ′ to define the preference of b ′ . Case 1: b = b ′ . In the previous paragraph, we already defined the preference of b , which is substitutable in the sale-direction by construction. Case 2: b ̸= b ′ . We further divide into cases based on whether b ′ is a large firm or a small firm type. Subcase 2.1: b ′ ∈ 𝐹large . For 𝜇 ∈ Xb ′ , let (︁ )︁ ′ 𝐶 b (𝜇) = 𝜇Zb ′ , min {𝜇x ′ , 𝜇z , 𝐾}{x ′ ,z } , 0𝑋b ′ rZb ′ r{x ′ ,z } . Subcase 2.2: b ′ ∈ 𝐹small . For 𝑌 ⊆ 𝑋b ′ , let ⎧ ⎨𝑌 ∩ Z ′ if |𝑌 ∩ {x ′ , z }| ≤ 1 ′ b 𝑐b (𝑌 ) = . ⎩(𝑌 ∩ Z ′ ) ∪ {x ′ , z } if {x ′ , z } ⊆ 𝑌 b 3

Note that the choice function of b ′ exhibits complementarities only between x ′ and z . Since b ′ = b(x ′ ), it follows that the choice function defined above is substitutable in the sale-direction. Define an open set 𝑈 ⊆ R𝐹>0small by {︃ 𝑈=

𝜁 ∈ R𝐹>0small

⃒ }︃ ⃒ 𝜁 𝑓 < M for all 𝑥 ∈ 𝑋 and 𝑓 ∈ 𝐹 ⃒ 𝑥 . ⃒ ⃒ and 𝜁 f < 𝜁 𝑓 for all 𝑓 ∈ 𝐹small r {f }

The open set 𝑈 is non-empty because M𝑥 > 0 for all 𝑥 ∈ 𝑋. Completion of the proof of Theorem 2. It remains to prove that the economy has neither a seller-initiated-stable outcome nor a stable outcome whenever 𝜁 ∈ 𝑈 . The first claim shows some basic arithmetical properties of individually rational outcomes. (︂ (︁ )︁ )︂ 𝑓^ Claim S1.1. If 𝒪 = 𝜇, 𝐷 is an individually rational outcome, then: 𝑓^∈𝐹small

(a) 𝜇𝑥 = 0 for all 𝑥 ∈ / Z ∪ {x , x ′ , y , z }; and (b) 𝜇x ′ = 𝜇z . Proof. First, we prove Part (a). Let 𝑥 ∈ 𝑋 r Z r {x , x ′ , y , z } and 𝑓 ∈ {b(𝑥), s(𝑥)} r {f }—such an 𝑓 exists because b(𝑥) ̸= s(𝑥). If 𝑓 ∈ 𝐹large , then we have 𝐶 𝑓 (𝜇)𝑥 = 0 by construction. If 𝑓 ∈ 𝐹small , then we have 𝑥 ∈ / 𝑐𝑓 (𝑌 ) for all 𝑌 ⊆ 𝑋𝑓 by construction. In either case, the individual rationality of 𝒪 implies that 𝜇𝑥 = 0. It remains to prove Part (b). This assertion is vacuously true if b = b ′ , and thus we can assume without loss of generality that b ̸= b ′ . If b ′ ∈ 𝐹large , then we have ′ ′ ′ 𝐶 b (𝜇)x ′ = 𝐶 b (𝜇)z by construction. If b ′ ∈ 𝐹small , then we have 𝑐b (𝑌 ) ∩ {x ′ , z } ∈ {∅, {x ′ , z }} for all 𝑌 ⊆ 𝑋b ′ by construction. In either case, the individual rationality of 𝒪 implies that 𝜇x ′ = 𝜇z . The second claim exploits the definitions of the choice functions to show that there are many rational deviations. When 𝜁 ∈ 𝑈, let {︀ }︀ min min𝑓 ̸=f 𝜁 𝑓 , min𝑥∈𝑋 M𝑥 − 𝜁 f 𝜖= , |𝑋| · 2|𝑋| which is positive due to the definition of 𝑈 . 4

Claim S1.2. Suppose that 𝜁 ∈ 𝑈 . If 𝒪 is an individually rational outcome and 𝑓 ̸= f , then 𝛽 ∈ R𝑋𝑓 is rational for 𝑓 at 𝒪 whenever: ∙ 𝑍(𝛽) ⊆ Z𝑓 ∪ {x , x ′ , z }; ∙ ‖𝛽‖∞ < 𝜖; ∙ and 𝑓 ∈ / {b , b ′ } or 𝛽z = 𝛽𝑥 for 𝑥 ∈ {x , x ′ }𝑓 . Proof. As 𝜁 ∈ 𝑈 and 𝒪 is an outcome, we have 𝜇𝑧 ≤ 𝜁 f for all 𝑧 ∈ Z ∪ {x , x ′ }. It follows that 𝜇𝑓 +𝛽 ≤ M. Because 𝒪 is individually rational, Claim S1.1(b) guarantees that 𝜇x ′ = 𝜇z . It follows that 𝛽𝑧 + 𝜇𝑧 ≤ 𝐾 for all 𝑧 ∈ {x , y , z }𝑓 . We divide into cases based on whether 𝑓 is a large firm or a small firm type to prove the claim. Case 1: 𝑓 ∈ 𝐹large . Note that 𝐶 𝑓 (𝜇𝑓 + 𝛽) = 𝜇𝑓 + 𝛽 by construction, since 𝐶 𝑓 (𝜇𝑓 ) = 𝜇𝑓 . Thus, 𝛽 is rational for 𝑓 at 𝒪. Case 2: 𝑓 ∈ 𝐹small . For this case, we divide further into cases based on whether z ∈ 𝑍(𝛽). Subcase 2.1: z ∈ / 𝑍(𝛽). Let 𝑍(𝛽) = {𝑧1 , . . . , 𝑧𝑘 }, and let 𝑍 𝑗 = {𝑧𝑗 } and i𝑗 = 𝛽𝑗 . For each 𝑗, since 𝜁 𝑓 −𝜇𝑧𝑗 ≥ |𝑋|·2|𝑋| ·𝜖 there exists 𝑌 𝑗 ⊆ 𝑋𝑓 r{𝑧𝑗 } with 𝐷𝑌𝑓 𝑗 ≥ |𝑋| · 𝛽𝑗 . By construction, we have 𝑘 ∑︁

(︀ )︀ i𝑗 1{𝑌 𝑗 } , 0𝒫(𝑋𝑓 )r{𝑌 𝑗 } ≤ 𝐷𝑓

and

𝑗=1

𝑘 ∑︁ (︀

)︀ 1𝑍 𝑗 , 0𝑋𝑓 r𝑍 𝑗 = 𝛽.

𝑗=1

Thus, 𝛽 is rational for 𝑓 at 𝒪. Subcase 2.2: z ∈ 𝑍(𝛽). Let 𝑍(𝛽) r {x , y , z } = {𝑧1 , . . . , 𝑧𝑘−1 }. Define 𝑌 𝑗 , 𝑍 𝑗 , and i𝑗 for 1 ≤ 𝑗 ≤ 𝑘 − 1 as in the previous subcase. Let 𝑍 𝑘 = {x , x ′ , z }𝑓 . Let 𝑥 = x if 𝑓 = b and 𝑥 = x ′ if 𝑓 = b ′ , so that 𝑍 𝑘 = {𝑥, z }. As 𝜁 𝑓 − 𝜇𝑥 ≥ |𝑋| · 2|𝑋| · 𝜖, there exists 𝑌 𝑘 ⊆ 𝑋𝑓 r {𝑥} with 𝐷𝑌𝑓 𝑘 ≥ |𝑋| · 𝛽𝑥 . Since 𝒪 is individually rational, we must have z ∈ / 𝑌 𝑘. Moreover, we have 𝑍 𝑘 ⊆ 𝑐𝑓 (𝑌 𝑘 ∪ 𝑍 𝑘 ). By construction, we have 𝑘 ∑︁

(︀ )︀ i𝑗 1{𝑌 𝑗 } , 0𝒫(𝑋𝑓 )r{𝑌 𝑗 } ≤ 𝐷𝑓

𝑗=1

and

𝑘 ∑︁ (︀ 𝑗=1

5

)︀ 1𝑍 𝑗 , 0𝑋𝑓 r𝑍 𝑗 = 𝛽.

Thus, 𝛽 is rational for 𝑓 at 𝒪. The cases clearly exhaust all possibilities, completing the proof of the claim. The remaining two claims show that 𝜇x ′ > 0 and 𝜇z = 0 must hold, respectively, in any stable or seller-initiated-stable outcome. These facts will easily to seen to imply the non-existence of stable outcomes and of seller-initiated-stable outcomes. )︂ (︂ (︁ )︁ 𝑓^ is a stable or sellerClaim S1.3. Suppose that 𝜁 ∈ 𝑈 . If 𝒪 = 𝜇, 𝐷 𝑓^∈𝐹small

initiated-stable outcome, then 𝜇x ′ =

𝜁f .

Proof. We prove the contrapositive. Assume that 𝜇x ′ < 𝜁 f and that 𝒪 is individually f rational. Because 𝒪 is an outcome, there exists 𝑌 ⊆ 𝑋f such that 𝐷𝑌 > 0 and x′ ∈ / 𝑌 . Claim S1.1(a) implies that 𝑌 ⊆ Z ∪ {x }. Let 𝜉 = min{𝜖, 𝐷𝑌𝑓 }. We divide into cases based on whether x ∈ 𝑌 to prove that 𝒪 is not seller-initiated-stable or stable. Case 1: x ∈ 𝑌 . By individual rationality, we have 𝑐f (𝑌 ) = 𝑌. Claim S1.1(a) implies that 𝑌 ⊆ Z ∪ {x }. Let 𝑍 = 𝑐𝑓 (Z ∪ {x }) r 𝑌 ⊆ Z, which is non-empty by the irrelevance of rejected contracts condition (because x ∈ / 𝑐𝑓 (Z ∪ {x }). Let 𝛽 = (𝜉𝑍 , 0𝑋r𝑍 ). The mass 𝛽 is rational for f at 𝒪 by construction. For all 𝑓 ̸= f , the mass 𝛽 is rational for 𝑓 at 𝒪 by Claim S1.2. Thus, 𝛽 blocks 𝒪, so that 𝒪 is not stable. Consider the set 𝑍 = 𝑍(𝛽). Let s(𝑍) r {f } = {𝑓1 , . . . , 𝑓𝑘 }, and, for 𝑖 = ⋃︀ 1, . . . , 𝑘, let 𝛽𝑖 = 𝛽𝑓𝑖 → . Let 𝑍𝑘+1 = 𝑍 r 𝑘𝑖=1 𝑍𝑖 and let 𝑓𝑘+1 = 𝑓, so that ((𝛽1 , 𝑓1 ) , . . . , (𝛽𝑘+1 , 𝑓𝑘+1 )) is a seller-initiated proposal sequence. We claim that ((𝛽1 , 𝑓1 ) , . . . , (𝛽𝑘+1 , 𝑓𝑘+1 )) blocks 𝒪. For all 1 ≤ 𝑖 ≤ 𝑘, the f mass 𝛽𝑖 is rational for 𝑓 at 𝒪 by Claim S1.2. The mass 𝛽f = 𝛽≤𝑘 ∨ 𝛽𝑘+1 is rational for f at 𝒪 by construction. If 𝑍f → = ∅, then we have shown that ((𝛽1 , 𝑓1 ) , . . . , (𝛽𝑘+1 , 𝑓𝑘+1 )) blocks 𝒪 (by Lemma 6). If 𝑧 ∈ 𝑍f → , then Claim S1.2 (︀ )︀ guarantees that 𝛽𝑧 , 0𝑍r{𝑧} is rational for b(𝑧) at 𝒪. By Lemma 6, the proposal sequence ((𝛽1 , 𝑓1 ) , . . . , (𝛽𝑘+1 , 𝑓𝑘+1 )) blocks 𝒪, so that 𝒪 is not seller-initiatedstable either. (︀ )︀ Case 2: x ∈ / 𝑌. Let 𝑍 = 𝑐𝑓 (Z ∪ {x , x ′ }) r 𝑌 ∪ {z }, and note that {x , x ′ } ⊆ 𝑍. Let 𝛽 = (𝜉𝑍 , 0𝑋r𝑍 ). 6

The mass 𝛽f is rational for f at 𝒪 by construction. For all 𝑓 ̸= f , the mass 𝛽𝑓 is rational for 𝑓 at 𝒪 by Claim S1.2. Thus, 𝛽 blocks 𝒪, so that 𝒪 is not stable. Let s(𝑍) r {f } = {𝑓1 , . . . , 𝑓𝑘 }, and, for 𝑖 = 1, . . . , 𝑘, let 𝑍𝑖 = 𝑍𝑓𝑖 → ∩ Z. Let ⋃︀ 𝑍𝑘+1 = 𝑍 r 𝑘𝑖=1 𝑍𝑖 , and let 𝑓𝑘+1 = f . Let 𝑓𝑘+2 = s(z ) and let

𝑍𝑘+2

⎧ ⎨{z } if b ̸= b ′ . = ⎩∅ if b = b ′ ,

Let 𝛽𝑖 = 𝛽𝑍𝑖 for 1 ≤ 𝑖 ≤ 𝑘 + 2, so that ((𝛽1 , 𝑓1 ) , . . . , (𝛽𝑘+2 , 𝑓𝑘+2 )) is a sellerinitiated proposal sequence. 𝑓 We claim that ((𝛽1 , 𝑓1 ) , . . . , (𝛽𝑘+2 , 𝑓𝑘+2 )) blocks 𝒪. The mass 𝛽f = 𝛽≤𝑘 ∨ 𝛽𝑘+1 is rational for f at 𝒪 by construction. For all 1 ≤ 𝑖 ≤ 𝑘, the mass 𝛽𝑖 is rational 𝑓𝑘+2 for 𝑓 at 𝒪 by Claim S1.2. The mass 𝛽𝑘+2 is rational for 𝑓𝑘+2 at 𝒪 given 𝛽≤(𝑘+1) b(z ) and rational for b(z ) at 𝒪 given 𝛽≤(𝑘+2) by Claim S1.2 and Lemma 6. Thus, ((𝛽1 , 𝑓1 ) , . . . , (𝛽𝑘+1 , 𝑓𝑘+1 )) blocks 𝒪, so that 𝒪 is not seller-initiated-stable either.

The cases clearly exhaust all possibilities, and thus we have proved the claim. (︂ (︁ )︁ )︂ 𝑓^ Claim S1.4. If 𝜁 ∈ 𝑈 and 𝜇, 𝐷 is stable or seller-initiated-stable, then 𝑓^∈𝐹small

𝜇z = 0. (︂ (︁ )︁ ^ Proof. We prove the contrapositive. Suppose that 𝒪 = 𝜇, 𝐷𝑓

)︂

is an (︀ )︀ individually rational outcome with 𝜇z > 0 and that 𝜁 ∈ 𝑈. Let 𝛽 = 𝜉y , 0𝑋r{y } where {︃ }︃ 𝜉 = min 𝜖, 𝜇z , inf 𝐷𝑌b b >0 𝐷𝑌

𝑓^∈𝐹small

.

We claim that 𝛽 blocks 𝒪. To prove that 𝛽 is rational for b at 𝒪, we divide into cases based on whether b is a large firm or a small firm type. Case 1: b ∈ 𝐹large . Since 𝜉 ≤ 𝜇z , we have 𝐶 b (𝜇b + 𝛽)y = 𝜇y + 𝛽y by construction. Thus, 𝛽 is rational for b at 𝒪.

7

Case 2: b ∈ 𝐹small . Because 𝜇z > 0, there exists 𝑌 ⊆ 𝑋b such that 𝐷𝑌b > 0 and z ∈ 𝑌 . We have 𝜉 ≤ 𝐷𝑌b by construction. The individual rationality of 𝒪 implies that x ∈ 𝑌 . As y ∈ 𝑐b (𝑌 ∪ {y }), it follows that 𝛽 is rational for b at 𝒪. The cases clearly exhaust all possibilities, and thus 𝛽 is rational for b at 𝒪. The proof that 𝛽 is rational for f at 𝒪 is similar to the proof of Claim S1.2. Note that 𝜇y ≤ 𝜁 f because 𝜇 is an outcome. It follows that 𝜇y + 𝛽y ≤ My and 𝜇y + 𝛽y ≤ 𝜁 f if f ∈ 𝐹small . To prove that 𝛽 is rational for f at 𝒪, we divide into cases based on whether f is a large firm or a small firm type. Case 1: f ∈ 𝐹large . As 𝐶 f (𝜇f + 𝛽) = 𝜇f + 𝛽 by construction, 𝛽 is rational for 𝑓 at 𝒪. Case 2: f ∈ 𝐹small . As 𝜁 f − 𝜇y ≥ 2|𝑋| · 𝜖, there exists 𝑌 ⊆ 𝑋f with 𝐷𝑌f ≥ 𝜖 and y∈ / 𝑌 . Since y ∈ 𝑐f (𝑌 ∪ {y }), the mass 𝛽 is rational for f at 𝒪. The cases clearly exhaust all possibilities, completing the proof that 𝛽 blocks 𝒪. Clearly, 𝜇z = 0 must hold in any stable outcome. Note that 𝜇z = 0 must also hold in any seller-initiated outcome—if 𝜇z > 0, then the seller-initiated proposal sequence (𝛽, s(y )) blocks 𝒪 by the discussion of the previous two paragraphs. Claims S1.3 and S1.4 together imply that, if 𝜁 ∈ 𝑈, then 𝜇z < 𝜇x ′ must hold in any stable or seller-initiated-stable outcome. But Claim S1.1(b) guarantees that 𝜇z = 𝜇x ′ in any individually rational outcome. Thus, no seller-initiated-stable or stable outcome can exist when 𝜁 ∈ 𝑈.

S1.2

Proof of Theorem 3

The argument is similar to the proof of Theorem 2. I begin by describing the preferences of firms other than f , and then complete the proof. Note that since 𝐹small = ∅, outcomes are uniquely determined by their associated allocations. By abuse of notation, I do not distinguish between allocations and outcomes. Defining the preferences of firms other than f . Since 𝐶 f is continuous and satisfies the irrelevance of rejected contracts condition, Lemma 5 guarantees that if (︀ )︀ 𝑥 ∈ 𝑋𝑓 , 𝜇 ∈ X𝑓 , and 𝐶 𝑓 (𝜇)𝑥 < 𝐶 𝑓 𝜇𝑋𝑓 r{𝑥} , 𝜇′𝑥 , then 𝐶 𝑓 (𝜇)𝑥 = 𝜇𝑥 . Thus, since 𝐶 f is not substitutable in the sale-direction, there exist 𝜈 ≤ 𝜂 ∈ Xf and x , x ′ ∈ 𝑋f → 8

such that 𝜈𝑋f r{x ′ } = 𝜂𝑋f r{x ′ } and 𝐶 f (𝜈)x < 𝐶 f (𝜂)x . The irrelevance of rejected contracts condition for 𝐶 f implies that 𝐶 f (𝜂)x ′ > 𝜇x ′ . The irrelevance of rejected contracts condition also ensures that we can assume that 𝐶 f (𝜂){x ,x ′ } = 𝜂{x ,x ′ } and that 𝐶 f (𝜈)x ′ = 𝜈x ′ without loss of generality. Let b = b(x ), and let b ′ = b(x ′ ). Let f ∈ 𝐹 r {b , b ′ , f } be arbitrary—such a firm type exists because |𝐹 | ≥ 4. Let y ∈ 𝑋b ∩ 𝑋f be arbitrary. By rescaling the contractual units of x , y , and z , we can assume that 𝜂x − 𝐶 f (𝜈)x = 𝜂x ′ − 𝜈x ′ and that My > Mz > max{Mx , Mx ′ }. (︀ )︀ Define the preferences of firms 𝑓 ∈ 𝐹 r {b , b ′ , f } by 𝐶 𝑓 (𝜇) = 𝜇 ∧ 𝜂𝑋𝑓 r{y } , My for 𝜇 ∈ X𝑓 . Note that 𝐶 𝑓 is substitutable, hence in particular substitutable in the saledirection, for all 𝑓 ∈ 𝐹 r {b , b ′ , f }. We divide into cases based on whether b = b ′ to define a contract z ∈ 𝑋b ∩ 𝑋b ′ . Case 1: b = b ′ . Let z = x ′ (as in Figure S1(a) on page 2). Case 2: b ̸= b ′ . Let z ∈ 𝑋b ∩ 𝑋b ′ be arbitrary (as in Figure S1(b) on page 2). Define the preference of b by ⎛

{︀ }︀ ⎞ (𝜇 ∧ 𝜈b )𝑋b r{x ,y ,z } , min 𝜇x , 𝜇y + 𝜇z − min {𝜇z , 𝜈x ′ } + 𝐶 f (𝜈)x , 𝜂x x , ⎜ ⎟ {︀ {︀ }︀ }︀ f ⎟ 𝐶 b (𝜇) = ⎜ x , 𝜂x } − min 𝜇x , 𝐶 (𝜈)x , 𝜇y y , ⎝ min {︀min {𝜇 ⎠ {︀ }︀ }︀ f min max min {𝜇x , 𝜂x } − 𝐶 (𝜈)x − 𝜇y , 0 + 𝜈x ′ , 𝜇z z for 𝜇 ∈ Xb . Note that 𝐶 b exhibits complementarities only between x and y , and between x and z . Since b = b(x ), it follows that 𝐶 b is substitutable in the saledirection. We divide into cases based on whether b = b ′ to define the preference of b ′ . Case 1: b = b ′ . In the previous paragraph, we already defined the preference of b , which is substitutable in the sale-direction by construction. Case 2: b ̸= b ′ . For 𝜇 ∈ Xb ′ , let (︁ )︁ ′ 𝐶 b (𝜇) = (𝜇 ∧ 𝜈b ′ )𝑋b ′ r{x ′ ,z } , min {𝜇x ′ , 𝜇z , 𝜂x ′ }{x ′ ,z } .

9

′

Note that 𝐶 b exhibits complementarities only between x ′ and z . Since b = b(x ), it follows that the preference defined above is substitutable in the sale-direction. Completion of the proof of Theorem 3. It remains to prove that the economy has neither a seller-initiated-stable outcome nor a stable outcome. The first claim is analogous to Claim S1.1. Claim S1.5. If 𝜇 ∈ X is an individually rational outcome, then: (a) 𝜇𝑋r{y ,z }∪{x ′ } ≤ 𝜂𝑋r{y ,z }∪{x ′ } ; (b) 𝜇x ′ = 𝜇z ; (c) and max{𝜇x − 𝐶 f (𝜈)x , 0} = 𝜇y + max{𝜇z − 𝜈x ′ , 0}. Proof. First, we prove Part (a). Let 𝑥 ∈ 𝑋 r {y , z } ∪ {x ′ } be arbitrary, and let 𝑓 ∈ {b(𝑥), s(𝑥)} r {f } be arbitrary—such an 𝑓 exists because b(𝑥) ̸= s(𝑥). We have 𝐶 𝑓 (𝜇𝑓 )𝑥 ≤ 𝜂𝑥 by construction. The individual rationality of 𝜇 implies that 𝜇𝑥 ≤ 𝜂𝑥 . We now prove Part (b). This assertion is vacuously true if b = b ′ , and thus we ′ ′ can assume without loss of generality that b ̸= b ′ . We have 𝐶 b (𝜇b ′ )x ′ = 𝐶 b (𝜇b ′ )z by construction. The individual rationality of 𝜇 implies that 𝜇x ′ = 𝜇z . It remains to prove Part (c). Note that {︁ }︁ {︀ }︀ max 𝐶 b (𝜇)x − 𝐶 f (𝜈)x , 0 = 𝐶 b (𝜇)y + max 𝐶 𝑓 (𝜇)z − 𝜈x ′ , 0 by construction. The individual rationality of 𝜇 implies the claim. The second claim is analogous to Claim S1.2. 𝑋

Claim S1.6. If 𝜇 ∈ X is an individually rational outcome, 𝑓 ̸= f , and 𝛽 ∈ R≥0𝑓 satisfies ∙ 𝛽f + 𝜇f ≤ 𝜈; ∙ 𝛽x + 𝜇x ≤ 𝐶 f (𝜈)x ; ∙ 𝛽z + 𝜇z ≤ 𝜈x ′ ; ∙ 𝑍(𝛽) ⊆ 𝑋f ∪ {z };

10

∙ and 𝑓 ̸= b ′ or 𝛽x ′ = 𝛽z , then 𝛽 is rational for 𝑓 at 𝜇. Proof. When 𝑓 ̸= b ′ , we have 𝐶 𝑓 (𝜇𝑓 + 𝛽) = 𝜇𝑓 + 𝛽. Note that 𝜇x ′ = 𝜇z by Claim S1.5(b). When 𝑓 = b ′ , the hypotheses of the claim guarantee that 𝜇x ′ + 𝛽x ′ = 𝜇z + 𝛽z ≤ 𝜈z . By construction, it follows that 𝐶 𝑓 (𝜇𝑓 + 𝛽) = 𝜇𝑓 + 𝛽. The remaining two claims are analogous to Claims S1.3 and S1.4, respectively. Claim S1.7. If 𝜇 ∈ X is a stable or seller-initiated-stable outcome, then 𝜇x ′ > 𝜈x ′ . Proof. We prove the contrapositive. Assume that 𝜇 is individually rational and that 𝜇x ′ ≤ 𝜈x ′ . Claim S1.5(a) implies that 𝜇𝑋r{y ,z }∪{x ′ } ≤ 𝜈𝑋r{y ,z }∪{x ′ } . In particular, we have 𝜇f ≤ 𝜈. Note also that 𝜇z ≤ 𝜈x ′ by Claim S1.5(b). We divide into cases based on whether 𝜇f = 𝐶 f (𝜈) to prove that 𝜇 is not seller-initiated-stable or stable. Case 1: 𝜇f ̸= 𝐶 f (𝜈). Let (︁

f

)︁

𝛽 = 𝐶 (𝜈) − 𝜇 ∨ 0 and let (︁ )︁ 𝛽 ′ = 𝛽𝑋𝑓 r{x ′ } , (𝛽x ′ ){x ′ ,z } , 0𝑋r𝑋𝑓 r{z } . The irrelevance of rejected contracts condition for 𝐶 f ensures that 𝛽 ̸= 0. The mass 𝛽 = 𝛽f′ is rational for f at 𝜇 by construction. For all 𝑓 ̸= f , the mass 𝛽𝑓′ is rational for 𝑓 at 𝜇 by Claim S1.6. Thus, 𝛽 ′ blocks 𝜇, so that 𝜇 is not stable. Consider the sets 𝑍 = 𝑍(𝛽) and 𝑍 ′ = 𝑍(𝛽 ′ ). Let s(𝑍) r {f } = {𝑓1 , . . . , 𝑓𝑘 }, ⋃︀ and, for 𝑖 = 1, . . . , 𝑘, let 𝑍𝑖 = 𝑍𝑓𝑖 → . Let 𝑍𝑘+1 = 𝑍 r 𝑘𝑖=1 𝑍𝑖 r{z } and let 𝑓𝑘+1 = 𝑓. Let 𝑍𝑘+2 = 𝑍 ′ r 𝑍 and let 𝑓𝑘+2 = s(z ). Let 𝛽𝑖 = 𝛽𝑍′ 𝑖 for 1 ≤ 𝑖 ≤ 𝑘 + 2 so that ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑘+2 , 𝑓𝑘+2 )) is a seller-initiated proposal sequence. We claim that ((𝛽1 , 𝑓1 ) , . . . , (𝛽𝑘+2 , 𝑓𝑘+2 )) blocks 𝜇. For all 1 ≤ 𝑖 ≤ 𝑘, the mass f 𝛽𝑖 is rational for 𝑓 at 𝜇 by Claim S1.6. The mass 𝛽 = 𝛽≤𝑘 ∨ 𝛽𝑘+1 is rational for 𝑓𝑘+2 f at 𝜇 by construction. The mass 𝛽𝑘+2 is rational for 𝑓𝑘+2 at 𝜇 given 𝛽≤(𝑘+1) by Claim S1.6 and Lemma 6. We divide into cases to prove that there exists 𝑧 ∈ 𝑍 ′ (︀ )︀ b(𝑧) such that 𝛽𝑧′ , 0𝑋r{𝑧} is rational for b(𝑧) at 𝜇 given 𝛽≤(𝑘+2) . Subcase 1.1: 𝑍f → = ∅. We can take any 𝑧 ∈ 𝑍f (by Lemma 6). 11

Subcase 1.2: x ′ ∈ 𝑍f → . Note that z ∈ 𝑍𝑘+2 . It follows from Claim S1.6 and (︀ )︀ b(z ) Lemma 6 that 𝛽z′ , 0𝑋r{𝑧} is rational for b(z ) at 𝜇 given 𝛽≤(𝑘+2) . Subcase 1.3: 𝑍f → ̸= ∅ and x ′ ∈ / 𝑍f → . If 𝑧 ∈ 𝑍f → , then Claim S1.6 guarantees (︀ ′ )︀ (︀ )︀ that 𝛽𝑧 , 0𝑋r{𝑧} is rational for b(𝑧) at 𝜇, so that 𝛽𝑧′ , 0𝑋r{𝑧} is rational b(𝑧) for b(𝑧) at 𝜇 given 𝛽≤(𝑘+2) by Lemma 6. In all cases, we have shown that there exists 𝑧 ∈ 𝑍 ′ such that {𝑧} is rational for b(𝑧) b(𝑧) at 𝜇 given 𝛽≤(𝑘+2) . Since the cases exhaust all possibilities, proposal sequence ((𝛽1 , 𝑓1 ) , . . . , (𝛽𝑘+2 , 𝑓𝑘+2 )) blocks 𝜇, so that 𝜇 is not seller-initiated-stable either. Case 2: 𝜇f = 𝐶 f (𝜈). Let (︁ )︁ 𝛽 = 𝐶 f (𝜂) − 𝜇 ∨ 0 and let ′

(︁

)︁

𝛽 = 𝛽𝑋𝑓 r{x ′ } , (𝛽x ′ ){x ′ ,z } , 0𝑋r𝑋𝑓 r{z } . Note that 𝛽 ̸= 0 because 𝛽x , 𝛽x ′ > 0. The mass 𝛽 = 𝛽f′ is rational for f at 𝜇 by construction. For all 𝑓 ∈ / {b , b ′ , f }, the mass 𝛽𝑓 = 𝛽𝑓′ is rational for 𝑓 at 𝜇 by Claim S1.6. For 𝑓 = b , b ′ , the mass 𝛽𝑓′ is clearly rational for 𝑓 at 𝜇. Thus, 𝛽 ′ blocks 𝜇, so that 𝜇 is not stable. Consider the set 𝑍 = 𝑍(𝛽). Let s(𝑍) r {f } = {𝑓1 , . . . , 𝑓𝑘 }, and, for 𝑖 = 1, . . . , 𝑘, ⋃︀ let 𝑍𝑖 = 𝑍𝑓𝑖 → . Let 𝑍𝑘+1 = 𝑍 r 𝑘𝑖=1 𝑍𝑖 , and let 𝑓𝑘+1 = f . Let 𝑓𝑘+2 = s(z ) and let

𝑍𝑘+2

⎧ ⎨{z } if b ̸= b ′ = . ⎩∅ if b = b ′

Let 𝛽𝑖 = 𝛽𝑍′ 𝑖 for 1 ≤ 𝑖 ≤ 𝑘 + 2, so that ((𝛽1 , 𝑓1 ) , . . . , (𝛽𝑘+2 , 𝑓𝑘+2 )) is a sellerinitiated proposal sequence. f

We claim that ((𝛽1 , 𝑓1 ) , . . . , (𝛽𝑘+2 , 𝑓𝑘+2 )) blocks 𝜇. The mass 𝛽 = 𝛽≤𝑘 ∨ 𝛽𝑘+1 is rational for f at 𝜇 by construction. For all 1 ≤ 𝑖 ≤ 𝑘, the mass 𝛽𝑖 is rational for 𝑓 𝑓𝑘+2 at 𝜇 by Claim S1.6. The mass 𝛽𝑘+2 is clearly rational for 𝑓𝑘+2 at 𝜇 given 𝛽≤(𝑘+1) b(z ) and rational for b(z ) at 𝜇 given 𝛽≤(𝑘+2) . Thus, ((𝛽1 , 𝑓1 ) , . . . , (𝛽𝑘+2 , 𝑓𝑘+2 )) blocks 𝜇, so that 𝜇 is not seller-initiated-stable either. The cases clearly exhaust all possibilities, and thus we have proved the claim. 12

Claim S1.8. If 𝜇 ∈ X is a stable or seller-initiated-stable outcome, then 𝜇z ≤ 𝜈x ′ . Proof. We prove the contrapositive. Suppose that (︁ )︁ 𝜇 is an individually rational outcome with 𝜇z > 𝜈x ′ . Let 𝛽 = (𝜇z − 𝜈x ′ )y , 0𝑋r{y } . We claim that 𝛽 blocks 𝜇. Because 𝜇 is individually rational, we have 𝐶 b (𝜇b + 𝛽)y = 𝜇y + 𝛽y by construction and Claim S1.5(c). Thus, 𝛽 is rational for b at 𝒪. By construction, we have 𝐶 f (𝜇f + 𝛽)y = 𝜇y + 𝛽y , so that the mass 𝛽 is rational for b at 𝜇. Thus, 𝛽 blocks 𝜇. Clearly, 𝜇z = 0 must hold in any stable outcome. Note that 𝜇z = 0 must also hold in any seller-initiated outcome—if 𝜇z > 0, then the seller-initiated proposal sequence (𝛽, s(y )) blocks 𝒪 due to the discussion of the previous paragraph. Claims S1.7 and S1.8 together imply that 𝜇x ′ > 𝜈x ′ ≥ 𝜇z in any seller-initiatedstable or stable outcome. But Claim S1.5(b) guarantees that 𝜇x ′ = 𝜇z for any individually rational 𝜇. Thus, no seller-initiated-stable or stable outcome can exist.

S2

Multilateral matching

This section extends the existence results to multilateral matching. Section S2.1 adapts the model to allow for multilateral contracts. Section S2.2 shows that tree stable outcomes exist, and Section S2.3 shows that stable outcomes exist if all firms’ preferences are substitutable. Section S2.4 relates the existence results to Azevedo and Hatfield (2013). Section S2.5 proves the existence results.

S2.1

Model

I model multilateral matching with multi-unit demand following Appendix B in Azevedo and Hatfield (2013). There is a finite set 𝐹 of firm types (with the same structure as in Section 4.1) and a finite set of contracts 𝑋. There is also a finite set of roles R , and each role r ∈ R is associated to a firm type a(r ) ∈ 𝐹. Each contract 𝑥 ∈ 𝑋 involves a non-empty finite set of roles r(𝑥) ⊆ R . I always assume that roles are 13

contract-specific—i.e., that r(𝑥) ∩ r(𝑦) = ∅ whenever 𝑥 ̸= 𝑦 ∈ 𝑋. It can be assumed without loss of generality that R = r(𝑋). Firms have preferences over roles.1 When a(r(𝑥)) ̸⊆ 𝐹large , mass 𝑚 of contract 𝑥 specifies that mass 𝑚 of type a(𝑥) ∩ 𝐹small will trade with a(𝑥) ∩ 𝐹large . When a(r(𝑥)) ⊆ 𝐹large , mass 𝑚 of contract 𝑥 specifies that 𝑚 units of the continuously divisible contract 𝑥 will be traded among firms a(r(𝑥)). As in Section 4.2, there is an upper bound M𝑥 on the amount of 𝑥 that can be traded for each 𝑥 ∈ 𝑋. I assume that M𝑥 ≥ 𝜁 𝑓 for all 𝑓 ∈ a(r(𝑥)) ∩ 𝐹small . By abuse of notation, I write Mr = M𝑥 whenever r ∈ r(𝑥). For S ⊆ R , 𝑌 ⊆ 𝑋, and 𝑓 ∈ 𝐹, let

S𝑓 = {r ∈ S | a(r ) = 𝑓 } 𝑌𝑓 = r(𝑌 )𝑓 denote the set of roles in S that are associated to 𝑓 and the set of roles involved in contracts in 𝑌 that are associated to 𝑓 , respectively. Given 𝜇 ∈ X , define 𝜇𝑓 ∈ X𝑓 by (𝜇𝑓 )r = 𝜇𝑥 for all r ∈ r(𝑥)𝑓 . Allocations, preferences, outcomes are exactly as in Section 4, and individual rationality, rational deviations, blocks, and (tree) stable outcomes are exactly as in Section 5.

S2.2

Tree stability in multilateral matching

As in settings with bilateral contracts, continuity and convexity alone ensure the existence of tree stable outcomes in large multilateral matching markets. In order to define tree stability, I need a notion of a tree in a multilateral economy. Trees are sets of contracts that are acyclic in the sense of hypergraphs. Definition S1. A set 𝑍 ⊆ 𝑋 of contracts is a tree if there do not exist distinct firms 𝑓1 , . . . , 𝑓𝑛 , distinct contracts 𝑧1 , . . . , 𝑧𝑛 ∈ 𝑍, and distinct roles r1 , . . . , r2𝑛 ∈ R such that 𝑓𝑖 = a(r2𝑖−1 ) = a(r2𝑖 ) and {r2𝑖 , r2𝑖+1 } ⊆ a(r(𝑧𝑖 )) for all 𝑖 = 1, . . . , 𝑛, where r2𝑛+1 = r1 . Corollary S1. If 𝐶 𝑓 (resp. 𝑐𝑓 ) satisfies the irrelevance of rejected contracts condition for all 𝑓 ∈ 𝐹large (𝑓 ∈ 𝐹small ) and 𝐶 𝑓 is continuous for all 𝑓 ∈ 𝐹large , then a tree stable outcome exists. 1

In contrast, Hatfield and Kominers (2015) and Rostek and Yoder (2017) assume that each contract contains at most one role associated to each firm type The distinction between these two approaches is relevant when there are multiple agents of each type, such as in my model.

14

To prove Corollary S1, I consider an auxiliary two-sided economy in which firms are matched to contracts. Every blocking tree in the original economy gives rise to a blocking tree in the auxiliary economy. Applying Corollary 1 in the auxiliary market yields Corollary S1. See Section S2.5 for the details of the argument.

S2.3

Existence of stable outcomes

As in settings with bilateral contracts, a substitutability condition is needed to ensure the existence of stable outcomes. Because there are no “directions” in multilateral matching, the relevant condition is substitutability itself. In the context of multilateral matching, substitutability requires that no role makes a firm want another role more. I recall Hatfield and Milgrom’s (2005) definition as the definition of substitutability for small firms and extend the definition to the setting of continuously divisible contracts for large firms. Definition S2. ∙ (Hatfield and Milgrom, 2005) For 𝑓 ∈ 𝐹small , choice function 𝑐𝑓 is substitutable if for all 𝑌 ⊆ 𝑋𝑓 and 𝑥, 𝑦 ∈ 𝑋𝑓 such that 𝑦 ∈ / 𝑐𝑓 (𝑌 ∪ {𝑦}), we have 𝑦 ∈ / 𝑐𝑓 (𝑌 ∪ {𝑥, 𝑦}). ∙ For 𝑓 ∈ 𝐹large , choice function 𝐶 𝑓 is substitutable if for all 𝜇 ≤ 𝜇′ ∈ X𝑓 we have 𝐶 𝑓 (𝜇′ ) ∧ 𝜇 ≤ 𝐶 𝑓 (𝜇).2 Substitutability, in addition to continuity and the irrelevance of rejected contracts condition, ensures the existence of stable outcomes in large, multilateral matching markets. Thus, a form of Corollary 4 generalizes to multilateral matching. Corollary S2. Suppose that: ∙ 𝐶 𝑓 (resp. 𝑐𝑓 ) satisfies the irrelevance of rejected contracts condition for all 𝑓 ∈ 𝐹large (𝑓 ∈ 𝐹small ); ∙ 𝐶 𝑓 is continuous for all 𝑓 ∈ 𝐹large ; ∙ and 𝐶 𝑓 (resp. 𝑐𝑓 ) is substitutable for all 𝑓 ∈ 𝐹large (𝑓 ∈ 𝐹small ). Then, a stable outcome exists. 2

Che et al. (2013) have defined substitutability in settings with continuously divisible contracts in terms of the monotonicity of the rejection function (after Fleiner, 2003). My definition of substitutability is stronger than theirs.

15

To prove Corollary S2, I consider with the auxiliary economy described in Section S2.2, in which firms are matched to contracts. The substitutability of the preferences of all firms in the original market ensure the substitutability of the preferences of one side of the auxiliary market. Applying Corollary 4 in the auxiliary economy yields Corollary S2. See Section S2.5 for the details of the argument. Corollary S2 proves a novel result even in matching with bilateral contracts. There, stable outcomes exist even in networks with cycles as long as all firms’ preferences are substitutable. Intuitively, contracts can always be redirected to remove cycles. Substitutability in the original economy ensures that all firms’ preferences are substitutable in the sale-direction in the acyclic economy.

S2.4

Relationship to Azevedo and Hatfield (2013)

Corollary S2 generalizes Theorem 2 in Azevedo and Hatfield (2013), as the core coincides with the set of stable outcomes when all firms have unit demand. In order to prove Theorem 2 in Azevedo and Hatfield (2013) (or more generally to prove the case of Corollary S2 where 𝐹large = ∅), one can apply Theorem 1 in Azevedo and Hatfield (2013) instead of the full generality of Corollary 4 in the proof of Corollary S2.

S2.5

Proofs of Corollaries S1 and S2

To prove Corollaries S1 and S2, I apply Corollaries 1 and 4 in a two-sided auxiliary ′ ′ economy. Let 𝐹small = 𝐹small ∪ 𝑋 and let 𝐹large = 𝐹large . Let 𝑋 ′ = R , and define s : 𝑋 ′ → 𝐹 by s(r ) = a(r ). For r ∈ R , let b(r ) = 𝑥 when r ∈ r(𝑥). Define the choice functions of firm types 𝑓 ∈ 𝐹 to be exactly as in the original economy—note that 𝑋𝑓 = 𝑋𝑓′ for all 𝑓 ∈ 𝐹. For 𝑥 ∈ 𝑋, let 𝜁 𝑥 = M𝑥 and, for 𝑌 ⊆ 𝑋𝑥′ , let 𝑥

𝑐 (𝑌 ) =

⎧ ⎨𝑋 ′

if 𝑌 = 𝑋𝑦′

⎩∅

otherwise

𝑦

.

The following claim shows that each buyer in the auxiliary economy trades the same amount of all of its contracts in any individually rational outcome. (︂ (︁ )︁ )︂ ′ 𝑓^ Claim S2.1. Let 𝜇 , 𝐷 be an individually rational outcome in the aux′ 𝑓^∈𝐹small

iliary economy and let 𝑥 ∈ 𝑋. We have 𝜇′r = 𝜇′r ′ for all r , r ′ ∈ a(𝑥). 16

Proof. Note that r(𝑥) and ∅ are the only individually rational sets of contracts for 𝑥 in the auxiliary economy. )︂ (︂ (︁ )︁ ′ 𝑓^ in the auxiliary econGiven an individually rational outcome 𝜇 , 𝐷 ′ 𝑓^∈𝐹small

omy, let (︂(︂ 𝜏

′

(︁

𝜇, 𝐷

)︂)︂

)︁

𝑓^

′ 𝑓^∈𝐹small

(︂ (︁ )︁ ^ = 𝜇, 𝐷𝑓

𝑓^∈𝐹small

)︂ ,

where 𝜇𝑥 = 𝜇′r for all r ∈ a(𝑥). The above formula yields a well-defined outcome by Claim S2.1. For 𝛽 ∈ X , define r* 𝛽 ∈ X ′ by (r* 𝛽)r = 𝛽r−1 (r ) . The following claim shows that blocks of 𝜏 (𝒪′ ) induce blocks of 𝒪′ . Claim S2.2. Let 𝒪′ be an individually rational outcome in the auxiliary economy. Then, 𝒪 = 𝜏 (𝒪′ ) is an individually rational outcome. If 𝛽 ∈ X blocks 𝒪, then r* 𝛽 blocks 𝒪′ . Proof. We have already shown that 𝒪 is an outcome (by Claim S2.1). The individual rationality of 𝒪 is immediate from the definitions of the choice functions in the auxiliary economy. Suppose that 𝛽 ∈ X blocks 𝜏 (𝒪′ ) . Let 𝛽 ′ = r* 𝛽. It is clear that 𝛽𝑓′ is rational for 𝑓 at 𝒪 for all 𝑓 ∈ 𝐹 . Note that 𝛽𝑥′ = 0 for 𝑥 ∈ 𝑋 r 𝑍(𝛽), so that in particular 𝛽𝑥′ is rational for 𝑥 at 𝒪 for all 𝑥 ∈ 𝑋 r 𝑍(𝛽). It remains to prove that 𝛽𝑧′ is rational for 𝑧 at 𝒪 for all 𝑧 ∈ 𝑍(𝛽). Let 𝑧 ∈ 𝑍(𝛽) be arbitrary. Let r ∈ r(𝑧) be arbitrary and let 𝑓 = a(r ). We divide into cases based on whether (︂ (︁ 𝑓)︁ is a large )︂ firm or a small firm type to prove that ^ 𝜇𝑧 + 𝛽𝑧 ≤ M𝑧 , where 𝒪 = 𝜇, 𝐷𝑓 . 𝑓^∈𝐹small

Case 1: 𝑓 ∈ 𝐹large . Because 𝛽𝑓 is rational for 𝑓 at 𝒪, there exists 𝜇𝑓 ≤ 𝜅 ≤ 𝜇𝑓 +𝛽𝑓 such that 𝐶 𝑓 (𝜅)r ≥ 𝜇𝑧 + 𝛽𝑧 . Since 𝐶 𝑓 (𝜅) ∈ X𝑓 , we have M𝑧 ≥ 𝐶 𝑓 (𝜅)r ≥ 𝜇𝑧 + 𝛽𝑧 . Case 2: 𝑓 ∈ 𝐹small . Because 𝛽𝑓 is rational for 𝑓 at 𝒪, there exists 𝑌 1 , . . . , 𝑌 𝑘 ⊆ ∑︀ 𝑋𝑓 r{r } with 𝑘𝑖=1 𝐷𝑌𝑓 𝑖 ≥ 𝛽𝑧 . Since 𝒪 is an outcome, it follows that 𝜇𝑧 ≤ 𝜁 𝑓 −𝛽𝑧 ≤ M 𝑧 − 𝛽𝑧 . The cases clearly exhaust all possibilities, and thus we (︂have proved that)︂ 𝜇𝑧 < M𝑧 . (︁ )︁ ^ ′ 𝑧 ′ ′ . Because It follows that 𝜇r < 𝜁 for all r ∈ a(𝑧), where 𝒪 = 𝜇 , 𝐷𝑓 ′ 𝑋𝑧′

=

𝑍𝑧′

𝑓^∈𝐹small

and ∅ are the only individually rational sets for 𝑧 in the auxiliary economy, 17

we have (𝐷)𝑧∅ ≥ 𝛽𝑧 , so that 𝛽𝑧′ is rational for 𝑧 at 𝒪′ (in the auxiliary economy). Since 𝑧 ∈ 𝑍 was arbitrary, it follows that 𝛽 ′ blocks 𝒪′ . Note that r(𝑍) is a tree in the auxiliary economy whenever 𝑍 is a tree in the original economy by Definitions 7 and S1. Thus, Corollaries S1 and S2 follow from Corollaries 1 and 4, respectively, and Claim S2.2.

S3

Modeling continuous heterogeneity

This section incorporates continuous heterogeneity in small firms into the basic model. Section S3.1 discusses mathematical preliminaries. Section S3.2 describes the structure of firms, Section S3.3 describes the set of contracts, and Section S3.4 describes outcomes. Section S3.5 defines preferences, Section S3.6 defines rational deviations, and Section S3.7 defines substitutability in the sale-direction. Section S3.8 presents the existence results. Section S3.9 relates the results to Che et al. (2013). The technical existence proof is deferred to Section S4.

S3.1

Mathematical preliminaries

Given a measure space (𝑋, 𝐺), let ℳ(𝑋)≤𝐺 = {measures 𝜇 on 𝑋 | 𝜇 ≤ 𝐺}. When 𝐺 is finite, note that the Radon-Nikodym derivative defines a linear bijection 𝑑 : ℳ(𝑋)≤𝐺 → {𝑓 ∈ 𝐿∞ (𝑋, 𝐺) | 0 ≤ 𝑓 ≤ 1}. 𝑑𝐺

(S1)

The Riesz Representation Theorem gives a canonical duality 𝐿∞ (𝑋, 𝐺) ∼ = 𝐿1 (𝑋, 𝐺)* , which induces a weak-* topology on 𝐿∞ (𝑋, 𝐺). I topologize ℳ(𝑋)≤𝐺 via the topology induced by the weak-* topology and (S1). Let − ∨ − and − ∧ − denote the join and meet of two measures, respectively. The function − ∧ − is not weak-* continuous is general (it is continuous when 𝑋 is finite set). As a result, the aggregate demand functions defined by Azevedo and Hatfield (2013) are discontinuous in general in the presence of continuous heterogeneity in small firms. My Gale-Shapley operator does not rely on the meet operation and hence

18

remains upper hemi-continuous even in the presence of continuous heterogeneity. Che et al. (2013) have used the topology of weak convergence of measures (on compact metric spaces) instead of the weak-* topology defined above. When 𝑋 is a compact metric space equipped with its Borel 𝜎-algebra, the topology defined is actually induced by the topology of weak convergence of measures.3 To see this, assume without loss of generality that 𝐺 has full support on 𝑋. Let ℳ(𝑋) denote the space of finite, signed Borel measures on 𝑋 and let 𝒞(𝑋) denote the space of continuous functions from 𝑋 to R. The Riesz Representation Theorem gives a canonical duality ℳ(𝑋) ∼ = 𝒞(𝑋)* , which induces the topology of weak convergence of measures. Since 𝒞(𝑋) is a subspace of 𝐿1 (𝑋, 𝐺) as a vector space, the topology of weak convergence of measures induces a coarser topology on ℳ(𝑋)≤𝐺 than the weak-* topology induced by 𝐿∞ (𝑋, 𝐺) ∼ = 𝐿1 (𝑋, 𝐺)* . But the latter topology is compact (by the Banach– Alaoglu theorem) and the former topology is Hausdorff, which implies that the two topologies coincide (by, e.g., Theorem 26.6 in Munkres, 2000). On the other hand, the literature on general equilibrium literature on markets with infinitely many commodities has proposed the Mackey topology to model settings with uncertainty (Bewley, 1972). However, the weak-* topology and the Mackey topology agree on bounded subsets of 𝐿∞ (see, e.g., Noguchi, 1997a,b).

S3.2

Firms

There is a finite set 𝐹large of “large” firms and a a finite set 𝐹small of “small” firm contractual types. For each small firm type 𝑓 ∈ 𝐹small , there is measurable space Θ𝑓 of firm types of contractual type 𝑓 and a finite measure G𝑓 on Θ𝑓 . The measure G𝑓 specifies the distribution of firm types among small firms of contractual type 𝑓 . The firm structure presented in Section 4.1 is the special case when the contractual type of a small firm determines its type, i.e., when |Θ𝑓 | = 1 for all 𝑓 ∈ 𝐹small . When Θ𝑓 is a compact metric space (equipped with its Borel 𝜎-algebra), the extended model incorporates continuous heterogeneity in the (types of) small firms. The distinction between contractual types and types is important, because small firms will be assumed to care about the contractual types of counterparties in trades with other small firms. This assumption parallels standard assumptions in the clubs literature on the finiteness of type spaces (Ellickson et al., 1999; Scotchmer and Shan3

This assertion is true even if 𝑋 is a Polish space, by essentially identical logic.

19

non, 2015). However, the types of small firms enter into the preferences of large firms, thereby capturing the many-to-one matching settings with rich continuous heterogeneity that have been modeled in the literature (Abdulkadiro˘glu et al., 2015; Azevedo and Leshno, 2016; Che et al., 2013).

S3.3

Contracts

There is a measurable space Ω of contract terms, equipped with measurable functions b, s : Ω → 𝐹 . Contract terms specify the contractual types of the counterparties, what is being traded, transfers, and non-pecuniary contract terms, but not the type (in Θ𝑓 ) of any involved small firm. In the case of matching without contracts, there is at most one contract term between any two firm types. Assume that Ω𝑓 is a finite set (equipped with the discrete 𝜎-algebra) for all 𝑓 ∈ 𝐹small . Given a set Ξ ⊆ Ω of contract terms, let Ξlg−lg = {𝜔 ∈ Ξ | |{b(𝑥), s(𝑥)} ∩ 𝐹small | = 0} Ξlg−sm = {𝜔 ∈ Ξ | |{b(𝑥), s(𝑥)} ∩ 𝐹small | = 1} Ξsm−sm = {𝜔 ∈ Ξ | |{b(𝑥), s(𝑥)} ∩ 𝐹small | = 2}, so that Ξlg−lg , Ξlg−sm , and Ξsm−sm are the set of contract terms in Ξ that are between two large firms, a large firm and a small firm, and two small firms, respectively. Contracts specify contract terms as well as the type of the involved small firm in contracts between a large firm and a small firm type. Because small firms do not care about the types of counterparties (beyond the information carried by contractual types), contracts between small firms do not need to specify types (beyond contractual types). Thus, the set of contracts is ⋃︁ (︁

X = Ωlg−lg ∪ Ωsm−sm ∪

)︁ Ωlg−sm × Θ 𝑓 . 𝑓

𝑓 ∈𝐹small

Let 𝜑(𝑥) ∈ Ω denote the contract terms associated to a contract 𝑥 ∈ 𝑋. Given 𝑓 ∈ 𝐹

20

and Z ⊆ X, let Z𝑓 → = 𝜑−1 (Ω𝑓 → ) ∩ Z Z→𝑓 = 𝜑−1 (Ω𝑓 → ) ∩ Z Z𝑓 = Z𝑓 → ∪ Z→𝑓 . Define 𝜋 :

⋃︀

𝑓 ∈𝐹small

(Ω𝑓 × Θ𝑓 ) → X by ⎧ ⎨(𝜔, 𝜃) if 𝜔 ∈ Ωlg−sm 𝜋(𝜔, 𝜃) = . ⎩𝜔 if 𝜔 ∈ Ωsm−sm

There is fixed upper bound on the amount of each contract that can be traded, which is described by a finite measure M on X. Paralleling the assumption in Section 4.2 that M𝑥 ≥ 𝜁 𝑓 for all 𝑓 ∈ 𝐹small with 𝑓 ∈ {b(𝑥), s(𝑥)}, assume that, for any 𝑓 ∈ 𝐹small , we have M({𝜔}, −) ≥ G𝑓 for all 𝜔 ∈ Ωlg−sm and M({𝜔}) ≥ G𝑓 (Θ𝑓 ) for 𝑓 all 𝜔 ∈ Ω𝑓sm−sm . The lower bound on M can be written more compactly as ⎛ M ≥ marg ⎝ 𝜋

⎞ ∑︁

𝛿𝜔 ⊗ G𝑓 ⎠

𝜔∈Ω𝑓

for all 𝑓 ∈ 𝐹small ,4 where 𝛿𝜔 is a Dirac mass and − ⊗ − is used to denote product measures (using Fubini’s Theorem). Small firms can only trade finitely many contracts, but large firms trade masses of infinitely many contracts. Thus, my model encapsulates discrete-price versions of general equilibrium models with infinitely many goods (Bewley, 1972; Zame, 1987; Ostroy and Zame, 1994),5 (︀ )︀ Here, the measure marg𝜋 𝜇 is the marginal of 𝜇 along 𝜋, defined by (marg𝜋 𝜇) (Z) = 𝜇 𝜋 −1 (Z) for all measurable sets Z. The marginal is also called the pushforward of 𝜇 along 𝜋. 5 In settings with infinitely many commodities, the choice of the space of commodity bundles and the topology on it are both relevant (Zame, 1987; Ostroy and Zame, 1994). I use an interval with the weak-* topology as the commodity space, like Bewley (1972), Zame (1987), and Ostroy and Zame (1994). Other choices induce spaces of measures on a compact metric space (Mas-Colell, 1975; Jones, 1984), spaces of square-integrable functions (Harrison and Kreps, 1979; Duffie and Huang, 1985; Duffie, 1986), and Banach lattice settings (Aliprantis and Brown, 1983; Yannelis and Zame, 1986; Aliprantis et al., 1987a,b). My methods work with these alternative spaces as long as the space of consumption bundles is bounded, compact (with respect to a chosen locally convex topology on an ambient vector space), and convex. 4

21

The set of allocations is X = ℳ(X )≤M . Thus, an allocation is a mass 𝜇 of contracts that satisfies the exogenously-specified bound 𝜇 ≤ M. The set of masses that can be traded by 𝑓 (if 𝑓 ∈ 𝐹large ) or in the aggregate by firms of contractual type 𝑓 (if 𝑓 ∈ 𝐹small ) is X𝑓 = ℳ (X𝑓 )≤ M| . Let X𝑓 → = ℳ (X𝑓 )≤ M| . Given a mass X𝑓

X𝑓 →

𝜇 ∈ X and contractual types 𝑓, 𝑓 ′ ∈ 𝐹, let 𝜇𝑓 → = 𝜇|X𝑓 → 𝜇→𝑓 = 𝜇|X→𝑓 𝜇𝑓 = 𝜇𝑓 → ∨ 𝜇→𝑓 𝜇𝑓 →𝑓 ′ = 𝜇𝑓 → ∧ 𝜇→𝑓 ′ .

S3.4

Outcomes

To define outcomes, I must define distributions for small firm contractual types 𝑓 ∈ 𝐹small . Similarly to Section 4.3, a distribution for 𝑓 assigns each possible set of contract terms Ξ ⊆ Ω𝑓 is assigned to a (potentially zero) mass 𝐷𝑓 of firms. 𝒫(Ω )

Definition S3. Let 𝑓 ∈ 𝐹small . An distribution for 𝑓 is a vector 𝐷𝑓 ∈ ℳ(Θ𝑓 )≤G𝑓𝑓 satisfying ∑︁ 𝑓 𝐷Ξ = G𝑓 . Ξ⊆Ω𝑓

Each distribution 𝐷𝑓 induces an allocation A(𝐷𝑓 ) that describes the total mass of contracts traded by firms of contractual type 𝑓 under 𝐷𝑓 . Definition S4. A distribution 𝐷𝑓 for 𝑓 induces allocation A(𝐷𝑓 ) ∈ X𝑓 by ∑︁

A(𝐷𝑓 ) = marg 𝜋

𝛿𝜔 ⊗ 𝐷Ξ𝑓 .

𝜔∈Ξ⊆Ω𝑓

Outcomes are exactly as Definition 3 in Section 4.3: an outcome consists of an allocation 𝜇 ∈ X and a distribution 𝐷𝑓 for each 𝑓 ∈ 𝐹small satisfying A(𝐷𝑓 ) = 𝜇𝑓 .

S3.5

Preferences and individual rationality

Large firms have preferences over masses of contracts. Thus, each large firm 𝑓 ∈ 𝐹large has a choice correspondence 𝐶 𝑓 : X𝑓 ⇒ X𝑓 , which is assumed to satisfy 𝜈 ≤ 𝜇 for all 𝜈 ∈ 𝐶 𝑓 (𝜇) and 𝜇 ∈ X𝑓 . 22

Because small firms trade discrete contracts and only care about the contractual types of their counterparties, small firms have preferences over sets of contract terms. Formally, each small firm contractual type 𝑓 ∈ 𝐹small has a choice correspondence 𝑐𝑓 : 𝒫(Ω𝑓 ) → 𝒫(Ω𝑓 ), which is assumed to be non-empty-valued and satisfy Λ ⊆ Ξ for all Λ ∈ 𝑐𝑓 (Ξ). Individual rationality requires that no firm strictly prefers dropping some of the contracts assigned to it. In discrete matching with contracts with indifferences, a set 𝐴 is individually rational for 𝑓 if 𝐴 ∈ 𝐶 𝑓 (𝐴) (see, e.g., Hatfield et al., 2013; Che et al., 2013). This definition extends naturally to large-market limits. )︂ (︂ (︁ )︁ ^ is individually rational if: Definition S5. An outcome 𝒪 = 𝜇, 𝐷𝑓 𝑓^∈𝐹small

∙ 𝜇𝑓 ∈ 𝐶 𝑓 (𝜇𝑓 ) for all 𝑓 ∈ 𝐹large ; and ∙ Ξ ∈ 𝑐𝑓 (Ξ) for all 𝑓 ∈ 𝐹small and Ξ ⊆ Ω𝑓 with 𝐷Ξ𝑓 > 0.

S3.6

Rational deviations and blocks

I adapt the definition of rational deviations given in Section 5.2 to settings with indifferences. In discrete matching with indifferences, Hatfield et al. (2013) (effectively) call 𝑍 a rational for 𝑓 at 𝐴 if 𝑓 always demands all of 𝑍 when given access to 𝐴 ∪ 𝑍— i.e., if 𝑍 ⊆ 𝑌 for all 𝑌 ∈ 𝐶 𝑓 (𝐴 ∪ 𝑍). I modify Hatfield et al.’s (2013) definition analogously to Definition 5.6 (︂ (︁ )︁ )︂ 𝑓^ Definition S6. Let 𝑓 ∈ 𝐹 be a firm type and let 𝒪 = 𝜇, 𝐷 be an 𝑓^∈𝐹small

outcome. A mass 𝛽 ∈ X is rational for 𝑓 at 𝒪 given 𝛾 ∈ X if 𝜇𝑓 + 𝛽 ∈ X𝑓 and: Case 1: 𝑓 ∈ 𝐹large . There exists 𝜅 ≤ (𝜇+𝛽 ∨𝛾)𝑓 such that 𝜅 ≥ 𝜇 and (𝜈 −𝜇)∨0 ≥ 𝛽 for all 𝜈 ∈ 𝐶 𝑓 (𝜅). Case 2: 𝑓 ∈ 𝐹small . There exist sets Ξ1 , Λ1 , Γ1 , . . . , Ξ𝑘 , Λ𝑘 , Γ𝑘 ⊆ Ω𝑓 and measures 6

Che et al. (2013) have taken a different approach to matching with indifferences, saying essentially that a deviation 𝑍 is rational for 𝑓 at 𝐴 if 𝑓 strictly prefers having some of 𝑍 over having only 𝐴 and sometimes demands all 𝑍 when given access to 𝐴 ∪ 𝑍—i.e., if 𝐴 ∈ / 𝐶 𝑓 (𝐴 ∪ 𝑍) and there 𝑓 exists 𝑌 ∈ 𝐶 (𝐴 ∪ 𝑍) with 𝑍 ⊆ 𝑌 . In Section S3.9, I show that, under the existence assumptions made by Che et al. (2013), my notion of stability is equivalent to theirs.

23

𝐽 1 , . . . , 𝐽 𝑘 ∈ ℳ(Θ𝑓 )≤G𝑓 with Γ𝑗 ⊆ ϒ𝑗 r Ξ𝑗 for all 𝑗 and ϒ𝑗 ∈ 𝑐𝑓 (Ξ𝑗 ∪ Λ𝑗 ∪ Γ𝑗 ), 𝑘 (︁ ∑︁

)︁ 𝑗 𝑓 𝑗 𝐽{𝑌 , 0 𝑗} 𝒫(Ω𝑓 )r{𝑌 } ≤ 𝐷 ,

(S2)

𝑗=1

and 𝛽 ≤ marg 𝜋

𝑘 ∑︁ ∑︁ 𝑗=1

𝛿𝜔 ⊗ 𝐽 𝑗 ≤ marg 𝜋

𝜔∈Γ𝑗

𝑘 ∑︁ ∑︁ 𝑗=1

𝛿𝜔 ⊗ 𝐽 𝑗 ≤ 𝛽 ∨ 𝛾.

(S3)

𝜔∈Λ𝑗 ∪Γ𝑗

Blocking sets, proposal sequences, and seller-initiated-stability are exactly as in Definitions 6, 9, and 10, respectively. In order to define acyclic stability (as in Definition 8), I need to define what it means for a mass 𝛽 ∈ X to be acyclic. Definition S7. A mass 𝛽 ∈ X is acyclic if there do not exist firm contractual types 𝑓1 , . . . , 𝑓𝑛 ∈ 𝐹 such that 𝛽𝑓𝑖 →𝑓𝑖+1 ̸= 0 for all 𝑖 = 1, . . . , 𝑛, where 𝑓𝑛+1 = 𝑓1 . Lemma 3 persists in the extended model. Lemma S1. Every seller-initiated-stable outcome is acyclically stable. Proof. In light of the proof of Lemma 3, it suffices to prove that Lemma 6 persists in the extended model. But the proof of Lemma 6 given in Appendix B itself persists in the extended model.

S3.7

Substitutability in the sale-direction

The existence of seller-initiated-stable outcomes relies on some form of substitutability in the sale-direction. Hatfield et al. (2013, 2015) and Fleiner, Jagadeesan, Jank´o, and Teytelboym (2017) have studied substitutability in settings with continuous prices and indifferences. One of Hatfield et al.’s (2013) substitutability conditions, choicelanguage expansion full substitutability (CEFS), focuses on retaining substitutability as the set of available contracts grows. Expansion-substitutability in the sale-direction, which is a weakening of one part of Hatfield et al.’s (2015) CEFS condition, requires that sales are substitutable as the set of possible sales expands (holding the set of available buys fixed).7 7

Expansion-substitutability in the sale-direction is a weakening of one part of Hatfield et al.’s (2015) indicator-language increasing-price full substitutability (IIFS) and demand-language expansion full substitutability conditions. Jagadeesan (2017) shows that IIFS is the key condition to ensure existence results in discrete trading networks with indifferences.

24

Definition S8. ∙ For 𝑓 ∈ 𝐹small , choice correspondence 𝑐𝑓 is expansion-substitutable in the sale-direction if for all Ξ ⊆ Ω𝑓 and 𝜔, 𝜓 ∈ 𝑋𝑓 → such that there exists Γ ∈ 𝑐𝑓 (Ξ∪{𝜓}) with 𝜓 ∈ / Γ, there exists Γ′ ∈ 𝑐𝑓 (Ξ∪{𝜔, 𝜓}) with 𝜓 ∈ / 𝑊 ′. ∙ For 𝑓 ∈ 𝐹large , choice correspondence 𝐶 𝑓 is expansion-substitutable in the saledirection if for all 𝜇 ≤ 𝜇′ ∈ X𝑓 with 𝜇→𝑓 = 𝜇′→𝑓 and all 𝜈 ∈ 𝐶 𝑓 (𝜇), there exists 𝜈 ′ ∈ 𝐶 𝑓 (𝜇′ ) with 𝜈 ′ ∧ 𝜇𝑓 → ≤ 𝜈.

S3.8

Existence results

In order to ensure the existence of stable outcomes, I need to impose a form of the irrelevance of rejected contracts condition. It turns out that the right condition is a form of Sen’s 𝛼—Sen’s 𝛽 plays no role in my existence results. Since I do not impose Sen’s 𝛽, I allow for incomplete preferences and intransitive indifferences, which to my knowledge have not been previously considered in matching models. Thus, the results of this section are matching-theoretic analogues of results on existence of general equilibrium with incomplete preferences.8 Definition S9. ∙ For 𝑓 ∈ 𝐹small , choice correspondence 𝑐𝑓 satisfies the irrelevance of rejected contracts condition if for all Λ ⊆ Ξ ⊆ Ω𝑓 and all Γ ∈ 𝑐𝑓 (Ξ) with Γ ⊆ Λ, we have Γ ∈ 𝑐𝑓 (Λ). ∙ For 𝑓 ∈ 𝐹large , choice correspondence 𝐶 𝑓 satisfies the irrelevance of rejected contracts condition if for all 𝜇 ≤ 𝜇′ ∈ X𝑓 and all 𝜈 ∈ 𝐶 𝑓 (𝜇′ ) with 𝜈 ≤ 𝜇, we have 𝜈 ∈ 𝐶 𝑓 (𝜇). Since individual contracts may be negligible when there is continuous heterogeneity in firms or contracts, sequential stability is not a reasonable solution concept. The first result shows the existence of seller-initiated-stable outcomes under expansionsubstitutability in the sale-direction, extending Corollary 2. Theorem S1 (Extension of Corollary 2). Suppose that: ∙ 𝐶 𝑓 (resp. 𝑐𝑓 ) satisfies the irrelevance of rejected contracts condition for all 𝑓 ∈ 𝐹large (𝑓 ∈ 𝐹small ); 8

See, for example, Schmeidler (1969), Mas-Colell (1974), Shafer and Sonnenschein (1975), Yamazaki (1978), and Aliprantis and Brown (1983).

25

∙ 𝐶 𝑓 is upper hemi-continuous and non-empty compact convex-valued for all 𝑓 ∈ 𝐹large ; ∙ 𝐶 𝑓 (resp. 𝑐𝑓 ) is expansion-substitutable in the sale-direction for all 𝑓 ∈ 𝐹large (𝑓 ∈ 𝐹small ). Then, a seller-initiated-stable outcome exists. Proof. See Section S4. Extensions of Corollaries 3 and 4 follow, as in Section 6.1. Corollary S3 (Extension of Corollary 3). Under the hypotheses of Theorem S1, acyclically stable outcomes exist. Proof. Follows from Theorem S1 and Lemma S1. Corollary S4 (Main Theorem; Extension of Corollary 4). Suppose that: ∙ 𝐶 𝑓 (resp. 𝑐𝑓 ) satisfies the irrelevance of rejected contracts condition for all 𝑓 ∈ 𝐹large (𝑓 ∈ 𝐹small ); ∙ 𝐶 𝑓 is upper hemi-continuous and non-empty compact convex-valued for all 𝑓 ∈ 𝐹large ; ∙ 𝐶 𝑓 (resp. 𝑐𝑓 ) is expansion-substitutable in the sale-direction for all 𝑓 ∈ 𝐹large (𝑓 ∈ 𝐹small ); ∙ and the trading network is acyclic. Then, a stable outcome exists. Proof. Follows from Corollary S3 and Lemma 1.9 9

Lemma 1 asserts that acyclic stability implies stability in acyclic networks. Although Lemma 1 is stated formally for the basic model (with single-valued choice functions and discrete type spaces) identical logic shows that the result is true in the extended model.

26

S3.9

Relationship to Che et al. (2013)

Corollary S4 generalizes Theorems 4 and S5 in Che et al. (2013). Indeed, the Che et al. (2013) model is the case of the model presented in this section in which unit-supply small sellers are matched with large buyers. As the preferences of unit-supply firms are always substitutable and two-sided markets are acyclic, Corollary S4 guarantees that stable outcomes exist. To be precise regarding the relationship between Corollary S4 and Che et al.’s (2013) existence results, I restrict consideration to matching without contracts for the sake of simplicity. Suppose that Ω = 𝐹small × 𝐹large with b(𝑠, 𝑏) = 𝑏 and s(𝑠, 𝑏) = 𝑠 for all (𝑠, 𝑏) ∈ 𝑋. Suppose furthermore that ≻𝑠 is complete for all 𝑠 ∈ 𝐹small and that ∑︀ ∅ ≻𝑠 𝑌 whenever 𝑌 ⊆ 𝑋𝑠 satisfies |𝑌 | > 1, and that M = 𝜔∈Ω 𝛿𝜔 ⊗ Gs(𝜔) . These assumptions yield the model described in Section 6 in Che et al. (2013). One subtlety is that Che et al. (2013) use a slightly different definition of blocking sets than in this paper. Formally, Definition 5 in Che et(︂al. (2013) says)︂that an (︁ )︁ ^ allocation 𝛽 ∈ X𝑏 (for some 𝑏 ∈ 𝐵) blocks an outcome 𝒪 = 𝜇, 𝐷𝑓 if 𝑓^∈𝐹small

∙

∑︀

{𝜔}≻s(𝜔) {𝜔 ′ }

𝑠 𝐷{𝜔 ′ } ≥ 𝛽 ({𝜔} × −) for all 𝜔 ∈ Ω𝑏 and

∙ 𝛽 ∈ 𝐶 𝑏 (𝛽 ∨ 𝜇𝑏 ) and 𝜇𝑏 ∈ / 𝐶 𝑏 (𝛽 ∨ 𝜇𝑏 ) , where − ∨ − denotes componentwise maximum. Stability in the sense of Definition 5 in Che et al. (2013) clearly implies stability in the sense of this paper as long as choice correspondences are non-empty-valued. However, under the assumptions of Che et al.’s (2013) existence results, the definition of stability considered in this paper is equivalent to Che et al.’s (2013) definition. More formally, the following proposition shows that, assuming that 𝐶 𝑓 is nonempty compact-valued and satisfies Che et al.’s (2013) revealed preference property, stability implies stability in the sense of Definition 5 in Che et al. (2013). Recall that 𝐶 𝑓 : X𝑓 ⇒ X𝑓 satisfies the revealed preference property if 𝐶 𝑓 satisfies the irrelevance of rejected contracts condition and 𝐶 𝑓 (𝜇) ⊆ 𝐶 𝑓 (𝜇′ ) whenever 0 ≤ 𝜇 ≤ 𝜇′ and there exists 𝜈 ′ ∈ 𝐶 𝑓 (𝜇′ ) with 𝜈 ′ ≤ 𝜇. Proposition S1. Suppose that 𝑋 = 𝐹small × 𝐹large with b(𝑓, 𝑓 ′ ) = 𝑓 ′ and s(𝑓, 𝑓 ′ ) = 𝑓 for all (𝑓, 𝑓 ′ ) ∈ 𝑋. Suppose furthermore that ≻𝑓 is a strict preference for all 𝑓 ∈ 𝐹 and that |𝑐𝑓 (Ξ)| ≤ 1 for all 𝑓 ∈ 𝐹small and Ξ ⊆ Ω𝑓 . If 𝐶 𝑓 is non-empty compact-valued 27

and satisfies Che et al.’s (2013) revealed preference property for all 𝑓 ∈ 𝐹large , every outcome that is stable is also stable in the sense of Definition 5 in Che et al. (2013). In light of Proposition S1, Corollary S4 implies Theorem 4 in Che et al. (2013). )︂ (︂ (︁ )︁ 𝑓^ be an outcome. Proof. We prove the contrapositive. Let 𝒪 = 𝜇, 𝐷 𝑓^∈𝐹small

If 𝒪 is not individually rational, then it is not stable. Thus, we can assume that 𝒪 is individually rational and that there exists 𝑏 ∈ 𝐹large and 𝛽 ∈ X𝑏 such that ∑︀ 𝛽 ∈ 𝐶 𝑏 (𝛽 ∨ 𝜇𝑏 ), 𝜇𝑏 ∈ / 𝐶 𝑏 (𝛽 ∨ 𝜇𝑏 ), and {(𝑠,𝑏)}≻𝑌 𝐷𝑌𝑠 ≥ 𝛽 for all 𝑠 ∈ 𝐹small . We will need the following technical claim, whose proof is deferred to after the proof of the proposition. Claim S3.1. The set 𝑇 = {(𝜈 − 𝜇) ∨ 0 | 𝜈 ∈ 𝐶 𝑏 (𝛽 ∨ 𝜇𝑏 )} has a minimal element. Let 𝛾 be a minimal element of 𝑇 . As 𝜇𝑏 ∈ / 𝐶 𝑏 (𝛽 ∨ 𝜇𝑏 ), we have 𝜇𝑏 ∈ / 𝐶 𝑏 (𝛾 + 𝜇𝑏 ) (by the revealed preference property), so that 𝛾 ̸= 0. Che et al.’s (2013) revealed preference property implies that (𝜈 − 𝜇𝑏 ) ∨ 0 ≥ 𝛾 for all 𝜈 ∈ 𝐶 𝑏 (𝛾 + 𝜇𝑏 ). Because ∑︀ 𝑠 {(𝑠,𝑏)}≻𝑌 𝐷𝑌 ≥ 𝛽 for all 𝑠 ∈ 𝐹small , the mass 𝛾 is rational for 𝑠 at 𝒪 for all 𝑠 ∈ 𝐹small . Thus, 𝛾 blocks 𝒪, which implies that 𝒪 is not stable. Proof of Claim S3.1. Let (𝛾𝑖 )𝑖∈𝐼 be a chain in 𝑇, where 𝐼 is a totally ordered set. For each 𝑖 ∈ 𝐼, let 𝜈𝑖 ∈ 𝐶 𝑏 (𝛽 ∨ 𝜇𝑏 ) be such that 𝛾𝑖 = (𝜈𝑖 − 𝜇) ∨ 0. By the Banach–Alaoglu Theorem, ℳ(X )≤M is (weak-*) compact. We assumed that 𝐶 𝑏 (𝛽 ∨ 𝜇𝑏 ) is compact. As a result, ℳ(X )≤M × 𝐶 𝑏 (𝛽 ∨ 𝜇𝑏 ) is compact. Regarding the transfinite sequence ((𝛾𝑖 , 𝛽𝑖 ))𝑖∈𝐼 as a net in ℳ(X )≤𝐺 × 𝐶 𝑏 (𝛽 ∨ 𝜇𝑏 ), there must be a convergent subnet ((𝛾𝑗 , 𝛽𝑗 ))𝑗∈𝐽 (by Theorem 24 in Kelley, 1950).10 Suppose that 𝛾𝑗 → 𝛾 and 𝜈𝑗 → 𝜈 ∈ 𝐶 𝑏 (𝛽 ∨ 𝜇𝑏 ). We have 𝛾𝑗 ≥ 𝜈𝑗 − 𝜇 for all 𝑗 by construction, which implies that 𝛾 ≥ 𝜈 − 𝜇. It follows that 𝛾 ≥ (𝜈 − 𝜇) ∨ 0. Since ((𝛾𝑗 , 𝛽𝑗 ))𝑗∈𝐽 is a subnet of the chain ((𝛾𝑖 , 𝛽𝑖 ))𝑖∈𝐼 , the mass (𝜈 − 𝜇) ∨ 0 ∈ 𝑇 is a lower bound for (𝛾𝑖 )𝑖∈𝐼 . As (𝛾𝑖 )𝑖∈𝐼 is an arbitrary chain, we have proved that every chain in 𝑇 has a lower bound. By the Kuratowski–Zorn Lemma, 𝑇 must have a minimal element. 10

For an introduction to the theory of nets, see Section 2.1 in Megginson (1998). Proposition 2.1.37 in Megginson (1998) reproduces Theorem 24 in Kelley (1950).

28

S4

Proof of Theorem S1

The proof of Theorem S1 is similar to the proof of Theorem 1. For the sake of notational simplicity in defining the Gale-Shapley operator, I first consider markets in which every contract is between a large firm and a small firm type. Proposition S2. Theorem S1 holds if Ωlg−lg = Ωsm−sm = ∅ and M ({𝜔} × −) = G𝑓 whenever 𝑓 ∈ {b(𝜔), s(𝜔)} ∩ 𝐹small . Section S4.1 reduces Theorem S1 to Proposition S2. Section S4.2 formally defines the Gale-Shapley operator based on Section 6.3. Section S4.3 shows that fixed points of the Gale-Shapley operator give rise to sequentially stable outcomes. Section S4.4 concludes the proof of Proposition S2. Throughout, I use the notation of Appendix A.

S4.1

Proof of Theorem S1 assuming Proposition S2

We construct an auxiliary economy satisfying the hypotheses of Proposition S2. Let Ω′ = Ωlg−sm ∪

(︀(︀

)︀ )︀ 2 𝐹large ∪ Ωsm−sm × {𝑏, 𝑠} .

′ ′ 2 ′ Define sets 𝐹large = 𝐹large ∪ Ωsm−sm and 𝐹small = 𝐹small ∪ 𝐹large , and let 𝐹 ′ = 𝐹large ∪ ′ ′ ′ 𝐹small . Define b, s : Ω → 𝐹 by

⎧ ⎪ ⎪ b(𝜔 ′ ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 𝜔 ⎪ ⎨ b(𝜔 ′ ) = (𝑓1 , 𝑓2 ) ⎪ ⎪ ⎪ ⎪ ⎪b(𝜔) ⎪ ⎪ ⎪ ⎪ ⎩𝑓 1 ⎧ ⎪ ⎪ s(𝜔 ′ ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 𝜔 ⎪ ⎨ s(𝜔 ′ ) = (𝑓1 , 𝑓2 ) ⎪ ⎪ ⎪ ⎪ ⎪ s(𝜔) ⎪ ⎪ ⎪ ⎪ ⎩𝑓 2

if 𝜔 ′ ∈ 𝑋 lg−sm if 𝜔 ′ = (𝜔, 𝑠) ∈ Ωsm−sm × {𝑠} 2 if 𝜔 ′ = (𝑓1 , 𝑓2 , 𝑠) ∈ 𝐹large × {𝑠}

if 𝜔 ′ = (𝜔, 𝑏) ∈ Ωsm−sm × {𝑏} 2 if 𝜔 ′ = (𝑓1 , 𝑓2 , 𝑏) ∈ 𝐹large × {𝑏, 𝑠}

if 𝜔 ′ ∈ 𝑋 lg−sm if 𝜔 ′ = (𝜔, 𝑏) ∈ Ωsm−sm × {𝑏} 2 if 𝜔 ′ = (𝑓1 , 𝑓2 , 𝑠) ∈ 𝐹large × {𝑏}

if 𝜔 ′ = (𝜔, 𝑠) ∈ Ωsm−sm × {𝑠} 2 if 𝜔 ′ = (𝑓1 , 𝑓2 , 𝑏) ∈ 𝐹large × {𝑏, 𝑠}

29

.

, and let M′(𝑓1 ,𝑓2 ) = For 𝑓1 , 𝑓2 ∈ 𝐹large , let Θ(𝑓1 ,𝑓2 ) = Ωlg−lg 𝑓1 →𝑓2 , let G(𝑓1 ,𝑓2 ) = M|Ωlg−lg 𝑓1 →𝑓2 (︀ )︀ 𝛿(𝑓1 ,𝑓2 ,𝑏) + 𝛿(𝑓1 ,𝑓2 ,𝑠) ⊗ G(𝑓1 ,𝑓2 ) . For 𝜔 ∈ Ωlg−lg , let M′ ({(𝜔, 𝑏)} × −) = Gb(𝜔) M′ ({(𝜔, 𝑠)} × −) = Gs(𝜔) . , let M′ ({𝜔} × −) = G𝑓 . For 𝑓 ∈ 𝐹small and 𝜔 ∈ Ωlg−sm 𝑓 Firms 𝑓 ∈ 𝐹 face essentially the same decision problems in the original and auxiliary economies. To see this 𝑓 ∈ 𝐹large , note that the projection 𝜛 : X ′ → X given by ⎧ ⎪ ⎪ 𝑥′ if 𝜑(𝑥′ ) ∈ Ωlg−sm ⎪ ⎨ 𝜛(𝑥′ ) = 𝜔 if 𝑥 ∈ X′𝜔 with 𝜔 ∈ Ωsm−sm ⎪ ⎪ ⎪ ⎩𝑥 if 𝑥′ = (𝑓 , 𝑓 , 𝑥) with 𝑥 ∈ Ωlg−lg 1

2

𝑓1 →𝑓2

defines a measurable bijection from X′𝑓 to X𝑓 . For 𝑓 ∈ 𝐹small , note that the projection 2 𝜛 : Ω′ r (𝐹large × {𝑏, 𝑠}) → Ω given by 𝜛(𝜔 ′ ) =

⎧ ⎨𝜔 ′

if 𝜔 ∈ Ωlg−sm

⎩𝜔

if 𝜔 ∈ Ωsm−sm × {𝑏, 𝑠}

defines a bijection from Ω′𝑓 → Ω𝑓 . Choice correspondences in the auxiliary economy are defined as follows. For 𝜔 ∈ Ωsm−sm and 𝜇 ∈ X𝑓 , let ⃒ (︀ }︃ ⃒ 𝜈 {(𝜔, 𝑏)} × Θb(𝜔) )︀ = 𝜈 (︀{(𝜔, 𝑠)} × Θs(𝜔) )︀ ⃒ 𝜈≤𝜇⃒ {︀ (︀ )︀ (︀ )︀}︀ , ⃒ = min 𝜇 {(𝜔, 𝑏)} × Θb(𝜔) , 𝜇 {(𝜔, 𝑠)} × Θs(𝜔)

{︃ 𝐶 𝜔 (𝜇) =

so that 𝐶 𝜔 is upper hemi-continuous and non-empty compact convex-valued. Let 𝑐(𝑓1 ,𝑓2 ) maximize ≻(𝑓1 ,𝑓2 ) : {(𝑓1 , 𝑓2 , 𝑏), (𝑓1 , 𝑓2 , 𝑠)} ≻(𝑓1 ,𝑓2 ) ∅. For firm types 𝑓 ∈ 𝐹, choice correspondences in the auxiliary economy can be obtained directly from choice correspondences in the original economy using the projections 𝜛. It is straightforward that expansion-substitutability in the sale-direction is satisfied in the auxiliary economy. Proposition S2 (which is proved independently) guarantees that the auxiliary economy has a seller-initiated-stable outcome 𝒪′ =

30

(︂ (︁ )︁ ^ 𝜇′ , 𝐷 𝑓

′ 𝑓^∈𝐹small

)︂ . Since 𝒪′ is individually rational, we have 𝜇′ ({(𝑓1 , 𝑓2 , 𝑏)} × −) = 𝜇′ ({(𝑓1 , 𝑓2 , 𝑠)} × −)

for all 𝑓1 , 𝑓2 ∈ 𝐹large and (︀ )︀ (︀ )︀ 𝜇′ {(𝜔, 𝑏)} × Θb(𝜔) = 𝜇′ {(𝜔, 𝑠)} × Θs(𝜔) for all 𝜔 ∈ Ωsm−sm . Define an allocation 𝜇 in the original economy by 𝜇|Xlg−sm = 𝜇′ |Xlg−sm , 𝜇|Ω𝑓 →𝑓 = 𝜇′ ({(𝑓1 , 𝑓2 , 𝑏)} × −) = 𝜇′ ({(𝑓1 , 𝑓2 , 𝑠)} × −) 1

2

for all 𝑓1 , 𝑓2 ∈ 𝐹large , and (︀ )︀ (︀ )︀ 𝜇({𝜔}) = 𝜇′ {𝜔} × Θb(𝜔) = 𝜇′ {𝜔} × Θs(𝜔) sm−sm

for all 𝜔 ∈ Ω

(︂ (︁ )︁ ^ . We obtain an outcome 𝒪 = 𝜇, 𝐷𝑓

)︂ 𝑓^∈𝐹small

in the original

economy by using the fact that 𝜛 induces a bijection Ω′𝑓 to Ω𝑓 for 𝑓 ∈ 𝐹small to obtain distributions in the original economy from distributions in the auxiliary economy. It remains to prove that 𝒪 is seller-initiated-stable in the original economy. The fact that 𝒪′ is individually rational (in the auxiliary economy) implies that 𝒪 is individually rational (in the original economy). Suppose for the sake of deriving a contradiction that a seller-initiated proposal sequence ((𝛽1 , 𝑓1 ) , . . . , (𝛽𝑛 , 𝑓𝑛 )) blocks 𝒪 in the original economy. We will construct a seller-initiated blocking proposal sequence in the auxiliary economy similarly to the proof of Theorem 1. The idea is to make agents propose via intermediaries and to verify that the new proposal sequence blocks in the auxiliary economy, an approach that is notationally complicated due to the presence of continuous heterogeneity. When 𝑓𝑖 ∈ 𝐹small , let Ξ1𝑖 , Γ1𝑖 , Λ1𝑖 , . . . , Ξ𝑘𝑖 𝑖 , Γ𝑘𝑖 𝑖 , Λ𝑘𝑖 𝑖 ⊆ Ω𝑓 and 𝐽𝑖1 , . . . , 𝐽𝑖𝑘𝑖 be such that (︀ )︀ Λ𝑗𝑖 ⊆ ϒrΞ1𝑖 for all 𝑗 and ϒ ∈ 𝑐𝑓𝑖 Ξ𝑗𝑖 ∪ Γ𝑗𝑖 ∪ Λ𝑗𝑖 and (S2) and (S3) are satisfied for 𝛽 = 𝑓𝑖 𝛽𝑖 and 𝛾 = 𝛽≤(𝑖−1) . Write 𝐹large = {f1 , . . . , f|𝐹large | } and Ωsm−sm = {𝜓1 , . . . , 𝜓|Ωsm−sm | }. Let ⌉︂ ⌈︂ max𝑓 ∈𝐹small G𝑓 (Θ𝑓 ) 𝐾= min{𝛽𝑖 ({𝜔}) | 𝜔 ∈ 𝐹small and 𝛽𝑖 ({𝜔}) > 0} and let 𝐿 = 1 + max{|𝐹large |, 𝐾 · |Ωsm−sm |}. Define a seller-initiated proposal sequence 31

′ ((𝛾1 , 𝑓1′ ) , . . . , (𝛾𝐿𝑛 , 𝑓𝐿𝑛 )) in the auxiliary economy as follows. If 𝑓𝑖 ∈ 𝐹large and 𝑗 = 𝐿(𝑖 − 1) + 𝑟 with 1 ≤ 𝑟 ≤ 𝐿, let

⎧ ∑︀ ⎪ ⎪ 𝛿 ⊗ (𝛽𝑖 )→𝑓^ + 𝛽𝑖 |X lg−sm ⎪ ⎨ 𝑓^∈𝐹large (𝑓𝑖 ,𝑓^,𝑠) (𝛾𝑗 , 𝑓𝑗′ ) = 𝛿(𝑓𝑖 ,f𝑟−1 ,𝑏) ⊗ (𝛽𝑖 )→f𝑟−1 ⎪ ⎪ ⎪ ⎩(0, 𝑓 ) 𝑖

if 𝑟 = 1 if 𝑟 ∈ [2, |𝐹large | + 1] . if 𝑟 > |𝐹large | + 1

For 1 ≤ 𝑖 ≤ 𝑘 and 1 ≤ 𝑞 ≤ |Ωsm−sm |, let (︀ )︀ 1 𝐾 𝜂𝑖,𝑞 , . . . , 𝜂𝑖,𝑞 ∈ ℳ Θb(𝜓𝑞 ) ≤G

b(𝜓𝑞 ) −𝜇({𝜓𝑞 }×−)

(︀ )︀ 𝑟′ be such that 𝜂𝑖,𝑞 Θb(𝜓𝑞 ) ≤ 𝛽𝑖 ({𝜓𝑞 }) for 1 ≤ 𝑟′ ≤ 𝐾 and 𝜅≤

𝐾 ⋁︁

(︀ )︀ 𝑟′ 𝜂𝑖,𝑞 if 𝜅 ∈ ℳ Θb(𝜓𝑞 ) ≤G

𝑟′ =1

b(𝜓𝑞 ) −𝜇({𝜓𝑞 }×−)

(︀ )︀ is such that 𝜅 Θb(𝜓𝑞 ) ≤ 𝛽𝑖 ({𝜓𝑞 })

′

𝑟 —such 𝜂𝑖,𝑞 exist due to the definition of 𝐾. If 𝑓𝑖 ∈ 𝐹small and 𝑗 = 𝐿(𝑖 − 1) + 𝑟 = 𝐿(𝑖 − 1) + 𝐾(𝑞 − 1) + 1 + 𝑟′ with 1 ≤ 𝑟 ≤ 𝐿 and 1 ≤ 𝑟′ ≤ 𝐾, let

⎧(︁∑︀ (︁∑︀ )︁ )︁ 𝑘𝑖 𝑗 ⎪ ⎪ if 𝑟 = 1 𝑗=1 ⎪ 𝜔∈Λ𝑗𝑖 𝛿𝜔 ⊗ 𝛿𝑠 ⊗ 𝐽𝑖 , 𝑓𝑖 ⎨ )︀ (︀ 𝑟′ (𝛾𝑗 , 𝑓𝑗′ ) = , 𝜓𝑞 if 𝑟 ∈ [2, 𝐾 · |Ωsm−sm | + 1] . 𝛿(𝜓𝑞 ,𝑠) ⊗ 𝜂𝑖,𝑞 ⎪ ⎪ ⎪ ⎩(0, 𝑓 ) if 𝑟 > 𝐾 · |Ωsm−sm | + 1 𝑖 𝑓𝑖 Since 𝛽𝑖 is rational for 𝑓𝑖 at 𝒪 given 𝛽≤(𝑖−1) in the original economy for all 𝑖, the mass 𝑓′

𝑗 whenever 𝑗 ≡ 1 (mod 𝐿). When 𝑓 ∈ 𝐹large 𝛾𝑗 is rational for 𝑓𝑗′ at 𝒪 given 𝛾≤(𝑗−1) and 𝑗 = 𝐿(𝑖 − 1) + 𝑟 with 𝑟 ∈ [2, |𝐹large | + 1], note that

𝛾𝐿(𝑖−1)+1 ≥ 𝛿(𝑓𝑖 ,f𝑟−1 ,𝑠) ⊗ (𝛽𝑖 )→f𝑟−1 and 𝛾𝑗 = 𝛿(𝑓𝑖 ,f𝑟−1 ,𝑏) ⊗ (𝛽𝑖 )→f𝑟−1 , 𝑓′

𝑗 so that 𝛾𝑗 is rational for 𝑓𝑗′ at 𝒪 given 𝛾≤(𝑗−1) . When 𝑓 ∈ 𝐹small and 𝑗 = 𝐿(𝑖 − 1) +

32

𝐾(𝑞 − 1) + 1 + 𝑟′ with 1 ≤ 𝑟′ ≤ 𝐾 and 1 ≤ 𝑞 ≤ |Ωsm−sm |, note that (︀ )︀ {︀ (︀ )︀ (︀ )︀ }︀ 𝑟′ Θb (𝜓𝑞 ) ≤ 𝑚𝑖𝑛 𝛾𝑖 {𝜓𝑞 } × Θs(𝜓𝑞 ) , Gb(𝜓𝑞 ) Θb(𝜓𝑞 ) − 𝜇 ({𝜓𝑞 }) , 𝜂𝑖,𝑞 𝑓′

𝑗 . Thus, we have proved that 𝛾𝑗 is so that 𝛾𝑗 is rational for 𝑓𝑗′ at 𝒪 given 𝛾≤(𝑗−1)

𝑓′

𝑗 rational for 𝑓𝑗′ at 𝒪 given 𝛾≤(𝑗−1) for all 𝑗. The existence of a firm type that eventually accepts a proposal in the original economy (under ((𝛽1 , 𝑓1 ) , . . . , (𝛽𝑛 , 𝑓𝑛 ))) implies the existence of a firm type ′ ))). that accepts a proposal in the modified economy (under ((𝛾1 , 𝑓1′ ) , . . . , (𝛾𝐿𝑛 , 𝑓𝐿𝑛 ′ Thus, ((𝛾1 , 𝑓1′ ) , . . . , (𝛾𝐿𝑛 , 𝑓𝐿𝑛 )) blocks 𝒪′ , contradicting the fact that 𝒪′ is sequentially stable. We can conclude that ((𝛽1 , 𝑓1 ) , . . . , (𝛽𝑛 , 𝑓𝑛 )) does not block 𝒪. Since ((𝛽1 , 𝑓1 ) , . . . , (𝛽𝑛 , 𝑓𝑛 )) was arbitrary, 𝒪 must be seller-initiated-stable.

S4.2

The Gale-Shapley operator

For the remainder of Section S4, assume that the hypotheses of Proposition S2 are satisfied. The only notational difference is that I will track the pair of the set that is offered to a small firm and the pair that a small firm chooses, since the former does not uniquely determine the latter in settings with indifferences. Formally, let

H=

⎧ ⎨ ⎩

H ∈

×

𝒫(Ω )2

ℳ(Θ𝑓 )≤G𝑓𝑓

𝑓 ∈𝐹small

⃒ ⎫ ⃒ ⎬ ∑︁ ⃒ 𝑓 𝑓 𝑓 ⃒ H = G and Γ ∈ / 𝑐 (Ξ) =⇒ H = 0 . 𝑓 Ξ,Γ Ξ,Γ ⃒ ⎭ ⃒ Ξ,Γ⊆Ω𝑓

Intuitively, an element of H assigns a pair of sets of contracts to every small firm so that the second set could be chosen from the first—the first set in the pair will be set of contracts that the mediator offers. Define functions p1 , p2 : H → D that recover the distribution of offered sets and the distribution of chosen sets, respectively, by 𝑓

p1 (H )Ξ =

∑︁

𝑓 𝑓 HΞ,Γ and p2 (H )Γ =

Γ⊆Ω𝑓

∑︁

𝑓 HΞ,Γ .

Ξ⊆Ω𝑓

I formalize the steps of the t^atonnement process described in Section 6.3 as follows. 𝑓 𝑓 𝑓 ̂︀ lg : X → X by Φ ̂︀ lg (𝛼) = {M − 𝛼 + ∑︀ Step 1: Define Φ 𝑓 ∈𝐹 𝜈 | 𝜈 ∈ 𝐶 (𝛼)}.

{︀ (︀ )︀ }︀ ˇ ∈D|A 𝐷 ˇ =𝜍 . Step 2: As in Appendix A.2, define 𝒟 : X ⇒ D by 𝒟 (𝜍) = 𝐷

33

{︀ }︀ ˇ . Step 3: Define 𝒞̂︀ : D → H by 𝒞̂︀ = H ∈ H | p1 (H ) = 𝐷 ̂︀ sm : H → X by Φ ̂︀ sm (H ) = M − A (p1 (H )) + A (p2 (H )) . Step 4: Define Φ ̂︀ : X 2 × D × H ⇒ X 2 × D × H by Define a Gale-Shapley operator Φ }︁ (︀ )︀ {︁ sm {︀ (︀ )︀}︀ ̂︀ ˇ ̂︀ ̂︀ lg (𝛼) × 𝒟 (𝜍) × 𝒞 𝐷 ˇ . Φ 𝛼, 𝜍, 𝐷, H = Φ (H ) × Φ ̂︀ (H ) by Given H ∈ H, define an outcome Ψ ̂︀ (H ) = Ψ (p2 (H )) = (A (p2 (H )) , p2 (H )) . Ψ ̂︀ give rise to stable The key to the proof of Proposition S2 is that fixed points of Φ ̂︀ as the following proposition shows formally. outcomes under the projection Ψ, Proposition S3. If 𝐶 𝑓 (resp. 𝑐𝑓 ) is expansion-substitutable in the sale-direction and satisfies the irrelevance of rejected contracts condition for all 𝑓 ∈ 𝐹large (𝑓 ∈ 𝐹small ) )︀ (︀ )︀ (︀ ˇ H ∈Φ ̂︀ 𝛼, 𝜍, 𝐷, ˇ H , then Ψ ̂︀ (H ) is seller-initiated-stable. and 𝛼, 𝜍, 𝐷,

S4.3

Proof of Proposition S3

The following claim is analogous to Claim A.1. (︀ )︀ ˇ H is a good triple and 𝐶 𝑓 (resp. 𝑐𝑓 ) satisfies the irrelevance Claim S4.1. If 𝛼, 𝐷, of rejected contracts condition for all 𝑓 ∈ 𝐹large (𝑓 ∈ 𝐹small ), then: (︀ )︀ ˇ ; and (a) we have M + 𝜇 = 𝛼 + A 𝐷 (b) we have 𝜇𝑓 ∈ 𝐶 𝑓 (𝛼𝑓 ) for all 𝑓 ∈ 𝐹large . (︀ )︀ ˇ , we have 𝐷 ˇ = p1 (H ) . The definition of Ψ ̂︀ yields that Proof. Since H ∈ H 𝐷 𝜇 = A (p2 (H )). To prove Part (a), note that (︀ )︀ ̂︀ sm (H ) = M − A (p1 (H )) + A (p2 (H )) = M − A 𝐷 ˇ + 𝜇. 𝛼=Φ ˇ ∈Φ ̂︀ lg (𝛼), there exist 𝜐 𝑓 ∈ 𝐶 𝑓 (𝛼𝑓 ) such that To prove Part (b), note that since 𝐷 ∑︁ (︀ )︀ ˇ +𝛼−M= A 𝐷 𝜐𝑓 . 𝑓 ∈𝐹large

Part (a) yields that 𝜇 =

∑︀

𝑓 ∈𝐹large

𝜐 𝑓 , so that 𝜇𝑓 ∈ 𝐶 𝑓 (𝛼𝑓 ) for all 𝑓 ∈ 𝐹large . 34

The following claim is analogous to Claim A.2. (︀ )︀ ˇ H ∈ Claim S4.2. Let 𝑓 ∈ 𝐹small , let Ξ, Λ ⊆ X𝑓 , and let 𝐽 ∈ ℳ (Θ𝑓 )≤G𝑓 . If 𝛼, 𝜍, 𝐷, (︀ )︀ ̂︀ 𝛼, 𝜍, 𝐷, ˇ H , (𝛼 − A (p2 (H ))) ∧ 𝛾 = 0, 𝛾 ≥ ∑︀ ˇ𝑓 Φ 𝜔∈Λ 𝛿𝜔 ⊗ 𝐽, and 𝐷Ξ ∧ 𝐽 > 0, then Ξ ⊇ Λ. Proof. Suppose for the sake of deriving a contradiction that Λ ̸⊆ Ξ. Let 𝜔 ∈ Λ r Ξ. We must have (︀ (︀ 𝑓 )︀)︀ ˇ ˇ 𝑓 ∧ 𝐽 > 0. M𝑓 − A 𝐷 ({𝜔} × −) ∧ 𝐽 ≥ 𝐷 Ξ (︀ (︀ 𝑓 )︀)︀ ∑︀ ˇ Since 𝛾 ≥ 𝜔∈Λ 𝛿𝜔 ⊗ 𝐽, it follows that M𝑓 − A 𝐷 ∧ 𝛾 > 0. Claim S4.1(a) implies that (︀ (︀ 𝑓 )︀)︀ ˇ (𝛼 − A (p2 (H ))) ∧ 𝛾 ≥ M𝑓 − A 𝐷 ∧ 𝛾 > 0, contradicting the hypothesis that (𝛼 − 𝜇) ∧ 𝛾 = 0. We now begin the proof of Proposition S3 in earnest. Let 𝐷 = p2 (H ). ̂︀ (H ) is individually rational. Note that Ξ ∈ 𝑐𝑓 (Ξ) whenever We first prove that Ψ 𝐷Ξ𝑓 > 0 because H ∈ H and 𝑐𝑓 satisfies the irrelevance of rejected contracts condition for all 𝑓 ∈ 𝐹small . Claim S4.1(b) and the irrelevance of rejected contracts condition for the choice functions of large firms together imply that A (𝐷)𝑓 ∈ 𝐶 𝑓 (A(𝐷)𝑓 ) for all 𝑓 ∈ 𝐹large . ̂︀ (H ) is not blocked by any seller-initiated proposal It remains to prove that Ψ sequence. Consider a seller-initiated proposal sequence ((𝛽1 , 𝑓1 ) , . . . , (𝛽𝑛 , 𝑓𝑛 )) and ̂︀ (H ) given 𝛽 𝑓𝑖 suppose that 𝛽𝑖 is rational for 𝑓𝑖 at Ψ ≤(𝑖−1) for all 𝑖. 𝑓 for all 𝑓 ∈ 𝐹large and We proceed by induction on 𝑖 to prove that 𝛼 ≥ 𝜇 + 𝛽≤𝑖 𝑓 (𝛼 − 𝜇) ∧ 𝛽≤𝑖 = 0 for all 𝑓 ∈ 𝐹small . The base case of 𝑖 = 0 is obvious. For the 𝑓 𝑓 inductive step, assume that 𝛼 ≥ 𝜇 + 𝛽≤𝑘 for all 𝑓 ∈ 𝐹large and that (𝛼 − 𝜇) ∧ 𝛽≤𝑘 =0 for all 𝑓 ∈ 𝐹small . We divide into cases based on whether 𝑓𝑘+1 is a large firm or a small firm contractual type to prove the inductive step. Case 1: 𝑓𝑗+1 ∈ 𝐹large . Suppose for the sake of deriving a contradiction that (𝛼 − ̂︀ (H ) given 𝜇)∧𝛽𝑗+1 ̸= 0. Let 𝜐 = (𝛼−A(𝐷))∧𝛽𝑗+1 . Since 𝛽 is rational for 𝑓𝑗+1 at Ψ 𝑓𝑗+1 𝛽≤𝑗 , there exists A(𝐷)𝑓 ≤ 𝜅 ≤ A(𝐷)𝑓 + 𝛽 ∨ 𝛽 𝑓𝑗+1 ≤ 𝑗 such that (𝜈 − 𝜇) ∨ 0 ≥ 𝛽 for all 𝜈 ∈ 𝐶 𝑓 (𝜅). Since 𝛽 ∈ X𝑓 , expansion-substitutability in the sale-direction (︁ (︁ )︁)︁ 𝑓 implies that (𝜈 − 𝜇) ∨ 0 ≥ 𝜐 > 0 for all 𝜈 ∈ 𝐶 𝑓 𝜅 ∧ A(𝐷)𝑓 + 𝜐 ∨ 𝛽≤𝑗𝑗+1 .

35

(︁ (︁ )︁)︁ 𝑓 In particular, we have A(𝐷)𝑓 ∈ / 𝐶 𝑓 𝜅 ∧ A(𝐷)𝑓 + 𝜐 ∨ 𝛽≤𝑗𝑗+1 . The inductive hypothesis yields that (︁ )︁ 𝑓 𝑓 𝛼 ≥ A(𝐷)𝑓 + 𝜐 ∨ 𝛽≤𝑗𝑗+1 ≥ 𝜅 ∧ A(𝐷)𝑓 + 𝜐 ∨ 𝛽≤𝑗𝑗+1 . The irrelevance of rejected contracts condition guarantees that 𝜇𝑓 ∈ / 𝐶 𝑓 (𝛼𝑓 ) , contradicting Claim S4.1(b). Thus, we can conclude that (𝛼 − 𝜇) ∧ 𝛽𝑗+1 ̸= 0. 𝑓

Case 2: 𝑓𝑗+1 ∈ 𝐹small . Let 𝛾 = 𝛽≤𝑗𝑗+1 . Let Ξ1 , Λ1 , Γ1 , . . . , Ξ𝑘 , Λ𝑘 , Γ𝑘 ⊆ Ω𝑓 and 𝐽 1 , . . . , 𝐽 𝑘 ∈ ℳ (Θ𝑓 )≤G𝑓 be such that Γ𝑗 ⊆ ϒ𝑗 r Ξ𝑗 for all 𝑗 and ϒ𝑗 ∈ 𝑐𝑓 (Ξ𝑗 ∪ Λ𝑗 ∪ Γ𝑗 ), and (S2) and (S3) are satisfied. 𝑓 𝑗 Let 1 ≤ 𝑗 ≤ 𝑘 be arbitrary and suppose that HΔ,Ξ > 0. In particular, 𝑗 ∧ 𝐽 ∑︀ 𝑗 𝑗 ˇ 𝑓 ∧ 𝐽 𝑗 > 0. (S3) implies that 𝛾 ≥ we have 𝐷 Δ 𝜔∈Λ𝑗 𝛿𝜔 ⊗ 𝐽 , so that Δ ⊇ Λ by Claim S4.2. As 𝑐𝑓 is substitutable in the sale-direction and Γ𝑗 ⊆ Ω𝑓 → , we must have Π𝑗 ⊆ ϒ𝑗 for all Π𝑗 ⊆ Γ𝑗 and ϒ𝑗 ∈ 𝑐𝑓 (Ξ𝑗 ∪ Λ𝑗 ∪ Π𝑗 ). Since H is in the range of H, we must have Ξ𝑗 ∈ 𝑐𝑓 (Ξ𝑗 ∪ Δ). By the irrelevance of rejected contracts condition for 𝑐𝑓 and since Γ𝑗 ∩ Ξ𝑗 = ∅, it follows that Δ ∩ Γ𝑗 = ∅.

By reordering the indices 𝑗, we can assume that ̃︀ 1 Ξ1 = Ξ2 = · · · = Ξ𝑗1 = Ξ ̃︀ 2 Ξ𝑗1 +1 = Ξ𝑗1 +2 = · · · = Ξ𝑗2 = Ξ .. . ̃︀ ℓ , Ξ𝑗ℓ−1 +1 = Ξ𝑗ℓ−1 +2 = · · · = Ξ𝑘 = Ξ ̃︀ 1 , . . . , Ξ ̃︀ ℓ are pairwise distinct and where Ξ 1 ≤ 𝑗1 < 𝑗2 < · · · < 𝑗ℓ−1 < 𝑘. For notational convenience, let 𝑗0 = 0 and 𝑗ℓ = 𝑘. The previous paragraph implies that ⎛ ⎞ 𝑗𝑖 ∑︁ 𝑓 ⎝ HΔ, 𝐽 𝑖⎠ = 0 ̃︀ 𝑖 ∧ Ξ 𝑗=𝑗𝑖−1 +1

36

when Δ ∩ ∑︁ 𝜔∈Ω𝑓

⋃︀𝑗𝑖

𝛿𝜔 ⊗

𝑗=𝑗𝑖−1

𝐷Ξ𝑓̃︀ 𝑗

Γ𝑗 ̸= ∅. (S2) ensures that 𝐷Ξ𝑓̃︀ 𝑗 ≥ (︁

−A H

𝑓 ̃︀ 𝑖 −,Ξ

)︁

𝑗𝑖 ∑︁

∑︁

≥

⋃︀𝑗𝑖 𝜔∈ 𝑗=𝑗

𝑖−1

∑︀𝑗𝑖

𝑗=𝑗𝑖−1 +1

𝑗

𝛿𝜔 ⊗ 𝐽 ≥

Γ𝑗 𝑗=𝑗𝑖−1 +1

𝐽 𝑗 , so that 𝑗𝑖 ∑︁

∑︁

𝛿𝜔 ⊗ 𝐽 𝑗 .

𝑗=𝑗𝑖−1 +1 𝜔∈Γ𝑗

Summing over 1 ≤ 𝑖 ≤ ℓ, we have 𝑘 ∑︁ ∑︁ (︀ )︀ ˇ M𝑓 − A 𝐷 = M𝑓 − A (p1 (H )) ≥ 𝛿𝜔 ⊗ 𝐽 𝑗 ≥ 𝛽, 𝑗=1 𝜔∈Γ𝑗

where the last inequality is by (S3). Claim S4.1(a) implies that (︀ )︀ ˇ ≥ 𝛽. 𝛼𝑓 − 𝜇𝑓 = M𝑓 − A 𝐷 Since Ωlg−lg = Ωsm−sm = ∅, the inductive step follows in both cases, completing the 𝑓 for all 𝑓 ∈ 𝐹large and inductive argument. Taking 𝑖 = 𝑛, we have 𝛼 ≥ 𝜇 + 𝛽≤𝑛 𝑓 (𝛼 − 𝜇) ∧ 𝛽≤𝑛 = 0 for all 𝑓 ∈ 𝐹small . 𝑓 ̂︀ (H ) given 𝛽 𝑓 . We divide into cases Suppose that 𝛽 ′ ≤ 𝛽≤𝑛 is rational for 𝑓 at Ψ ≤𝑛 based on whether 𝑓 is large firm or a small firm contractual type to show that 𝛾 = 0. Case 1: Claim S4.1(b) guarantees that 𝜇𝑓 ∈ 𝐶 𝑓 (𝛼𝑓 ) for all 𝑓 ∈ 𝐹large . By the irrelevance of rejected contracts condition, it follows that 𝜇𝑓 ∈ 𝐶 𝑓 (𝜅) for all 𝑓 ∈ 𝑓 . Thus, we must have 𝛽 ′ = 0. 𝐹large and 𝜇𝑓 ≤ 𝜅 ≤ 𝜇𝑓 + 𝛽≤𝑛 Case 2: Let Ξ, Λ, Γ ⊆ Ω𝑓 and let 𝐽 ∈ ℳ (Θ𝑓 )≤G𝑓 be such that Γ ⊆ ϒ r Ξ for all ϒ ∈ 𝑐𝑓 (Ξ ∪ Λ ∪ Γ), 0 < 𝐽 ≤ 𝐷Ξ𝑓 , and ∑︁

𝑓 𝛿𝜔 ⊗ 𝐽 ≤ 𝛽≤𝑛 .

𝜔∈Γ∪Λ 𝑓 Let Δ ⊆ Ω𝑓 be such that HΔ,Ξ ∧ 𝐽 ̸= 0—such a Δ exists because 𝐷Ξ𝑓 ≥ 𝐽 and 𝐷 = p2 (H ). Claim S4.2 guarantees that Δ ⊇ Λ ∪ Γ. Since H is the in range of H, we must have Ξ ∈ 𝑐𝑓 (Δ), which implies that Γ = ∅ by the irrelevance of rejected contracts condition. Since Ξ, Λ, Γ, and 𝐽 were arbitrary, it follows that 𝛽 ′ = 0.

̂︀ (H ). The cases exhaust all possibilities, so that ((𝛽1 , 𝑓1 ) , . . . , (𝛽𝑛 , 𝑓𝑛 )) cannot block Ψ ̂︀ (H ) must be seller-initiated-stable. Since ((𝛽1 , 𝑓1 ) , . . . , (𝛽𝑛 , 𝑓𝑛 )) was arbitrary, Ψ 37

S4.4

Proof of Proposition S2

By the Banach–Alaoglu Theorem, X is compact with respect to the weak-* topology, which is Hausdorff and locally convex. Similarly, D𝑓 is compact as a subset 𝐿∞ (Θ𝑓 , G𝑓 )𝒫(Ω𝑓 ) for each 𝑓 ∈ 𝐹, so that D is compact. Claim S4.3. The space H is compact as a subset of

×

𝒫(Ω𝑓 )

𝑓 ∈𝐹small

D𝑓

.

Proof. By the Banach–Alaoglu Theorem, it suffices to show that H is closed in the 𝒫(Ω ) product 𝑓 ∈𝐹small D𝑓 𝑓 . For 𝑓 ∈ 𝐹small , let

×

A𝑓 = {(Ξ, Γ) ∈ 𝒫(Ω𝑓 )2 | Γ ∈ 𝑐𝑓 (Ξ)}. Define Σ :

×

𝑓 ∈𝐹small

𝑓

𝐿∞ (𝑋, G𝑓 )A →

×

𝑓 ∈𝐹small

𝐿∞ (𝑋, G𝑓 ) by

∑︁

𝑓

Σ (H )Ξ =

𝑓 HΞ,Γ .

Γ⊆Ω𝑓

The function Σ is clearly weak-* continuous. Note that −1

H=Σ Thus, H is closed in

×

𝑓 ∈𝐹small

(︁

)︁

𝑓

(G𝑓 )𝑓 ∈𝐹small ∩ 𝐿∞ (𝑋, G𝑓 )A≥0 . 𝑓

𝐿∞ (𝑋, G𝑓 )A , hence closed in

×

𝑓 ∈𝐹small

𝒫(Ω𝑓 )

D𝑓

.

̂︀ is upper hemi-continuous and non-empty compact convex-valued, To show that Φ ̂︀ lg , 𝒟, 𝒞, ̂︀ and Φ ̂︀ sm each have those properties. The continuity it suffices to show that Φ ̂︀ lg . It is clear that Φsm of the choice functions of large firms implies the continuity of Φ is continuous. Note that the graph of 𝒟 is {︀(︀

)︀ (︀ )︀ }︀ ˇ |A 𝐷 ˇ =𝜍 , 𝜍, 𝐷

which is the graph of A. Since D is compact and X is Hausdorff, the graph of A is closed, and it follows that 𝒟 is upper hemi-continuous and compact-valued. The hypothesis that M ({𝜔} × −) = G𝑓 whenever 𝑓 ∈ {b(𝜔), s(𝜔)} ∩ 𝐹small ensures that 𝒟 is non-empty-valued. Since A is linear and D is convex, 𝒟 must be convex-valued. Similarly, H is inverse to the continuous, linear function p1 , which has compact domain and codomain by Claim S4.3. As a result, H is upper hemi-continuous and compact convex-valued. Since 𝑐𝑓 is non-empty valued for all 𝑓 ∈ 𝐹small , the correspondence H is 38

̂︀ is upper hemi-continuous and non-empty non-empty-valued as well. It follows that Φ compact convex-valued. The space X 2 × D × H is compact by Claim S4.3 and is clearly convex. The ̂︀ has a fixed point Kakutani–Fan–Glicksberg Fixed Point Theorem guarantees that Φ (︀ )︀ ˇ H . Proposition S3 ensures that Ψ ̂︀ (H ) is a seller-initiated-stable outcome. 𝛼, 𝜍, 𝐷,

References Abdulkadiro˘glu, A., Y.-K. Che, and Y. Yasuda (2015). Expanding “choice” in school choice. American Economic Journal: Microeconomics 7 (1), 1–42. Alaoglu, L. (1940). Weak topologies of normed linear spaces. Annals of Mathematics 41 (1), 252–267. Aliprantis, C. D. and D. J. Brown (1983). Equilibria in markets with a Riesz space of commodities. Journal of Mathematical Economics 11 (2), 189–207. Aliprantis, C. D., D. J. Brown, and O. Burkinshaw (1987a). Edgeworth equilibria. Econometrica 55 (5), 1109–1137. Aliprantis, C. D., D. J. Brown, and O. Burkinshaw (1987b). Edgeworth equilibria in production economies. Journal of Economic Theory 43 (2), 252–291. Azevedo, E. M. and J. W. Hatfield (2013). Existence of equilibrium in large matching markets with complementarities. Working paper. Azevedo, E. M. and J. D. Leshno (2016). A supply and demand framework for twosided matching markets. Journal of Political Economy 124 (5), 1235–1268. Banach, S. (1932). Th´eorie des op´erations lin´eaires, Volume 1 of Monografie Matematyczne. Subwencji Funduszu Kultury Narodowej. Bewley, T. F. (1972). Existence of equilibria in economies with infinitely many commodities. Journal of Economic Theory 4 (3), 514–540. Che, Y.-K., J. Kim, and F. Kojima (2013). Stable matching in large economies. Working paper. Duffie, D. (1986). Competitive equilibria in general choice spaces. Journal of Mathematical Economics 15 (1), 1–23. Duffie, D. and C.-F. Huang (1985). Implementing Arrow-Debreu equilibria by continuous trading of few long-lived securities. Econometrica 53, 1337–1356.

39

Ellickson, B., B. Grodal, S. Scotchmer, and W. R. Zame (1999). Clubs and the market. Econometrica 67 (5), 1185–1217. Fan, K. (1952). Fixed-point and minimax theorems in locally convex topological linear spaces. Proceedings of the National Academy of Sciences 38 (2), 121–126. Fleiner, T. (2003). A fixed-point approach to stable matchings and some applications. Mathematics of Operations Research 28 (1), 103–126. Fleiner, T., R. Jagadeesan, Z. Jank´o, and A. Teytelboym (2017). Trading networks with frictions. Working paper. Glicksberg, I. L. (1952). A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points. Proceedings of the American Mathematical Society 3 (1), 170–174. Harrison, J. M. and D. M. Kreps (1979). Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory 20 (3), 381–408. Hatfield, J. W. and S. D. Kominers (2015). Multilateral matching. Journal of Economic Theory 156, 175–206. Hatfield, J. W., S. D. Kominers, A. Nichifor, M. Ostrovsky, and A. Westkamp (2013). Stability and competitive equilibrium in trading networks. Journal of Political Economy 121 (5), 966–1005. Hatfield, J. W., S. D. Kominers, A. Nichifor, M. Ostrovsky, and A. Westkamp (2015). Full substitutability. Working paper. Hatfield, J. W. and P. R. Milgrom (2005). Matching with contracts. American Economic Review 95 (4), 913–935. Jagadeesan, R. (2017). A general theory of matching with indifferences. In preparation. Jones, L. E. (1984). A competitive model of commodity differentiation. Econometrica 52 (2), 507–530. Kakutani, S. (1941). A generalization of Brouwer’s Fixed Point Theorem. Duke Mathematical Journal 8 (3), 457–459. Kelley, J. L. (1950). Convergence in topology. Duke Mathematical Journal 17 (3), 277–283. Kuratowski, C. (1922). Une m´ethode d’´elimination des nombres transfinis des raisonnements math´ematiques. Fundamenta Mathematicae 3 (1), 76–108.

40

Mas-Colell, A. (1974). An equilibrium existence theorem without complete or transitive preferences. Journal of Mathematical Economics 1 (3), 237–246. Mas-Colell, A. (1975). A model of equilibrium with differentiated commodities. Journal of Mathematical Economics 2 (2), 263–295. Megginson, R. E. (1998). An Introduction to Banach Space Theory, Volume 183 of Graduate Texts in Mathematics. Springer-Verlag New York. Munkres, J. R. (2000). Topology (Second ed.). Prentice Hall. Noguchi, M. (1997a). Economies with a continuum of agents with the commodityprice pairing (𝑙∞ , 𝑙1 ). Journal of Mathematical Economics 28 (3), 265–287. Noguchi, M. (1997b). Economies with a continuum of consumers, a continuum of suppliers and an infinite dimensional commodity space. Journal of Mathematical Economics 27 (1), 1–21. Ostroy, J. M. and W. R. Zame (1994). Nonatomic economies and the boundaries of perfect competition. Econometrica 62 (3), 593–633. Rostek, M. and N. Yoder (2017). Matching with multilateral contracts. Working paper. Schmeidler, D. (1969). Competitive equilibria in markets with a continuum of traders and incomplete preferences. Econometrica 37 (4), 578–585. Scotchmer, S. and C. Shannon (2015). Verifiability and group formation in markets. Working paper. Shafer, W. and H. Sonnenschein (1975). Equilibrium in abstract economies without ordered preferences. Journal of Mathematical Economics 2 (3), 345–348. Yamazaki, A. (1978). An equilibrium existence theorem without convexity assumptions. Econometrica 46 (3), 541–555. Yannelis, N. C. and W. R. Zame (1986). Equilibria in Banach lattices without ordered preferences. Journal of Mathematical Economics 15 (2), 85–110. Zame, W. R. (1987). Competitive equilibria in production economies with an infinitedimensional commodity space. Econometrica 55 (5), 1075–1108. Zorn, M. (1935). A remark on method in transfinite algebra. Bulletin of the American Mathematical Society 41 (10), 667–670.

41