Complementary inputs and the existence of stable
outcomes in large trading networks*
Ravi Jagadeesan†
June 25, 2017
Abstract
This paper studies a model of large trading networks with bilateral contracts. Contracts capture exchange, production, and prices, as well as frictions,
such as market incompleteness, price regulation, and taxes. In my setting,
a stable outcome exists in any acyclic network, as long as firms regard sales
as substitutes and standard continuity and convexity conditions are satisfied.
Thus, complementarities between inputs do not preclude the existence of stable
outcomes in large markets, unlike in discrete markets. Even when sales are not
substitutable, tree stable outcomes exist in my setting.
The model presented in this paper generalizes and unifies versions of general equilibrium models with divisible and indivisible goods, matching models with continuously divisible contracts, models of large (two-sided) matching
with complementarities, and club formation models. Additional results explain
what kinds of equilibria are guaranteed to exist when substitutability in the
sale-direction and acyclicity are relaxed.
JEL codes: C62, C78, D47, D51, D52, L14
Keywords: Trading networks; Supply chains; Complementary inputs; Frictions; Matching with contracts; Large markets; Stability; Substitutability
*
An extended abstract of this paper appeared in Proceedings of the 2017 ACM Conference of
Economics and Computation (EC’17). I would like to thank Sandro Ambuehl, Eduardo Azevedo,
Eric Budish, Kevin Chen, Jeremy Fox, Jerry Green, Ben Golub, John Hatfield, Fuhito Kojima,
Michael Ostrovsky, Ross Rheingans-Yoo, Rachit Singh, Alex Teytelboym, Sahana Vasudevan, seminar audiences at Harvard, and, especially, Scott Kominers for helpful comments. This research
was conducted in part while the author was an Economic Design Fellow at the Harvard Center of
Mathematical Sciences and Applications.
†
Department of Mathematics, Harvard University. Email: [email protected].
1
Contents
1 Introduction
3
2 Illustrative examples
2.1 Continuity helps ensure that stable outcomes exist . . . . . . . . . . .
2.2 Complications of complementarities . . . . . . . . . . . . . . . . . . .
2.3 Setbacks from cycles . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
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10
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3 Relationship to the literature
3.1 Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 General equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
11
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4 Model
4.1 Firms . . .
4.2 Contracts .
4.3 Outcomes .
4.4 Preferences
4.5 Stability . .
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5 Other stability properties
5.1 Blocking sets and strong stability . . . . . . . .
5.2 Strong tree stability and strong acyclic stability
5.3 Proposal sequences and sequential stability . . .
5.4 Seller-initiated-stability . . . . . . . . . . . . . .
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6 Existence results
6.1 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Relationship to Azevedo and Hatfield (2013) and Che et al. (2013) . .
6.3 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
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7 Maximal domain results
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8 Conclusion
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A Multilateral matching
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B Incorporating indifferences
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Supplementary appendix
42
References
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1
Introduction
Micro-level financial imperfections can have interesting consequences for the aggregate behavior of markets. Imperfections limit the financial market’s ability to equalize
agents’ valuations of payments that involve several currencies, trade credit, or other
non-cash financial instruments. Yet, the standard general equilibrium approach assumes that complex payments can be summarized as transfers in a single numéraire.
For example, small firms sometimes finance purchases with trade credit, which is
subject to imperfectly-insurable idiosyncratic default risk. Trade credit is a relevant
feature of trading networks since it makes defaults propagate (Kiyotaki and Moore,
1997)1 —a channel that explains a significant fraction of bankruptcies (Jacobson and
von Schedvin, 2015). However, creditors and debtors may value debt differently when
it is costly or impossible to obtain perfect insurance against default. Thus, amounts
paid in cash and amounts paid in trade credit cannot be combined into a single price.
As another example, imperfections in financial markets limit the currency market’s
ability to absorb country-specific shocks (Gabaix and Maggiori, 2015).2 However,
financial imperfections also cause agents to value currencies differently in equilibrium,
making it impossible to consolidate payments in several currencies into a transfer in
a single numéraire. A similar issue arises whenever financial markets are imperfect
or otherwise incomplete, as agents may have different marginal rates of substitution
between forms of transfer in equilibrium.3 Thus, modeling the market incompleteness
discussed above requires a departure from the standard general equilibrium approach.
This paper analyzes trading networks with complex frictions using a matching
model. Modern matching theory captures many frictions (including market incompleteness) in a unified framework by modeling contracts instead of goods. From a
conceptual perspective, contracts specify what goods or services are being traded as
well as pecuniary and non-pecuniary contract terms.4 Non-transferabilities and other
bounds on prices can be incorporated into the set of contract terms.5 Allowing con1
See also Battiston et al. (2007). Related analyses of contagion in financial markets include Allen
and Gale (2000) and Acemoglu et al. (2015).
2
Related work includes Alvarez et al. (2002) and Maggiori (2017).
3
That is, the pricing kernel, a distribution that is used to price assets when arbitrage is impossible,
is non-unique in incomplete markets (Harrison and Kreps, 1979; Hansen and Richard, 1987).
4
See Roth (1984b, 1985), Hatfield and Milgrom (2005), Ostrovsky (2008), Klaus and Walzl (2009),
Hatfield and Kominers (2012, 2015b, 2017), and Hatfield et al. (2013) for interpretations of contracts.
5
See, for example, Roth (1984a,b, 1985), and Hatfield and Milgrom (2005).
3
tract terms to specify multiple prices captures many forms of market incompleteness,
such as imperfectly-insurable default risk, imperfect financial markets, and imperfect convertibility. Taking agents to have preferences over contracts instead of goods
incorporates bargaining frictions and transaction taxes.6 Letting preferences (over
contracts) implicitly incorporate technological constraints allows matching to model
production.7
After the seminal contribution of Gale and Shapley (1962), stability is the standard solution concept in matching theory. Stability requires that no agent wants
to drop any of the contracts assigned to it and that no group of agents would like
to sign new contracts among themselves (while possibly dropping some previouslysigned contracts). Quite generally, stable outcomes behave similarly to competitive
equilibria.8 Thus, as Hatfield et al. (2013) have argued, matching may substitute for
general equilibrium analysis in settings in which frictions or market incompleteness
prevent the use of general equilibrium methods.9 In typical matching models, however, complementarities between inputs preclude the existence of stable outcomes10
unless strong restrictions are imposed on the structure of the market.11 On the other
6
See Jaffe and Kominers (2014), Galichon et al. (2016), and Fleiner, Jagadeesan, Jankó, and
Teytelboym (2017). See also Nöldeke and Samuelson (2015) for related interpretations of taxes.
7
See Ostrovsky (2008), Hatfield et al. (2013, 2015), and Hatfield and Kominers (2015b).
8
See Crawford and Knoer (1981), Kelso and Crawford (1982), Hatfield et al. (2013), Fleiner,
Jagadeesan, Jankó, and Teytelboym (2017), Rostek and Yoder (2017), and Jagadeesan (2017d).
9
For example, in one-to-one matching with bounded prices, stable outcomes are essentially equivalent to Drèze (1975) equilibria (Herings, 2015). See also Hatfield, Plott, and Tanaka (2012, 2016).
10
Hatfield and Kominers (2012) and Hatfield et al. (2013) have shown that full substitutability—
which requires that inputs are substitutable for each other, that sales are substitutable for each other,
and that inputs and sales are complementary to one another—is necessary in a maximal domain
sense for stable outcomes to exist. Full substitutability can also be regarded as the requirement that
all goods are substitutable (Gul and Stacchetti, 1999; Sun and Yang, 2006; Hatfield and Kominers,
2012; Baldwin and Klemperer, 2015; Hatfield et al., 2015). Some form of substitutability is also
necessary in a maximal domain sense for the existence of stable outcomes in two-sided matching
markets (Kelso and Crawford, 1982; Roth, 1984a; Echenique and Oviedo, 2004, 2006; Hatfield and
Kojima, 2008; Klaus and Walzl, 2009; Hatfield and Kominers, 2017; Schlegel, 2016).
11
Some papers have shown the existence of (pairwise) stable outcomes in many-to-one matching
with complementarities (Hatfield and Kojima, 2010; Hatfield and Kominers, 2015a; Hatfield, Kominers, and Westkamp, 2015; Schlegel, 2016; Alva, 2017; Alva and Teytelboym, 2017), but these papers
do not consider many-to-many matching. Other papers have shown the existence of equilibria for
certain domains of preferences that allow for complementarities (Danilov et al., 2001; Pycia, 2012;
Baldwin and Klemperer, 2015), but these papers impose strong conditions on the permissible forms
of complementarity and substitutability. Recently, Rostek and Yoder (2017) have shown that stable
outcomes exist in discrete, multilateral matching when all contracts are complementary, but they
rule out any substitutability between contracts. Hatfield and Kominers (2015b) do not assume that
contracts are complementary or substitutable, but instead require that utility is transferable and
4
hand, inputs are complementary whenever the production of a good requires multiple
commodities or intermediates.12 Similarly, complementarities between technological
inputs are a key feature of modern manufacturing.13 Thus, the requirement that
inputs are substitutable prevents matching from being applied to the analysis of
complex, real-world markets.
Discreteness is partially responsible for the non-existence of stable outcomes in the
presence of complementarities (Azevedo and Hatfield, 2013). On the other hand, contracts are often discrete in real-world markets once one looks closely enough (Ostrovsky, 2008). For example, because workers are quite heterogeneous, the discreteness
of labor contracts is relevant to small firms. Micro-level discreteness can cause aggregate discontinuities, which is how discreteness obstructs the existence of equilibrium
in standard matching models. Such discontinuities may very well be an important
feature of markets in which few agents are present or few units of each contract are
traded. However, these discontinuities are less likely to be relevant in large markets.
For example, the discreteness of smartphones is quite salient to individual consumers
but irrelevant to global markets.
This paper analyzes trading networks in a large-market limit setting in which
the market is continuous in the aggregate, although micro-level discreteness may be
present. I show that complementarities between inputs do not obstruct the existence
of stable outcomes in such markets. Even under continuity, a mild substitutability
condition is necessary (in a maximal domain sense) for the existence of stable outcomes. Since my model is based on matching with contracts, this paper allows for
frictions, market incompleteness, and discreteness, as discussed above.
The model studied in this paper features “large” and “small” firms that interact
in a trading network via an exogenously specified set of bilateral contracts. As in
Aumann’s (1964) classic model of large markets, “small” firms each constitute an
infinitesimal portion of the economy since there is a continuum of small firms of each
type. For example, some small firms could represent individual workers supplying
labor. Like Edgeworth (1881) and Scarf (1962), I assume that there are finitely many
types of firms. As in Shapiro and Shapley (1978), Milnor and Shapley (1978), and
Hildenbrand (1970), I allow the economy to contain finitely many “large” firms, which
that contracts are continuously divisible.
12
Fox (2010, 2016) has shown empirically that inputs are complementary in auto manufacturing.
13
See, for example, Milgrom and Roberts (1990, 1995).
5
are not negligible in size. Large firms trade continuously divisible contracts/goods
with one another. On the other hand, small firms trade discrete contracts/goods,
both with one another and with large firms. Large firms view the contracts/goods
traded by small firms as divisible, just as consumers count bananas while distributors
measure quantities of bananas in pounds. Thus, the presence of a continuum of small
firms of each type substitutes for the divisibility of the contracts/goods traded by
large firms in ensuring aggregate continuity.
The existence of stable outcomes in my model relies on three key assumptions. The
first assumption is classical: it requires that large firms’ preferences are continuous
and convex.14,15 The second assumption, substitutability in the sale-direction, requires
that all firms regard sales as substitutes. This assumption allows complementarities
between inputs and is likely to be satisfied when no disassembly occurs during production. The third assumption requires that the trading network is acyclic, so that
the network forms a vertical supply chain (Ostrovsky, 2008). Under these three assumptions, stable outcomes exist. Unlike in two-sided markets, substitutability in
the sale-direction and acyclicity are both necessary in a maximal domain sense for
the existence of stable outcomes in my model.16
In large, complete markets without frictions, substitutability in the sale-direction
and acyclicity are not needed to ensure the existence of stable outcomes.17 Indeed,
classical results in general equilibrium theory guarantee the existence of competitive
equilibria in markets with continuously divisible goods/contracts under continuity
and convexity assumptions, while mild finiteness restrictions ensure the existence of
14
Competitive equilibria exist in large markets even with non-convexities because the presence of
a large number of small firms convexifies the aggregate excess demand correspondence (Aumann,
1966; Shapley and Shubik, 1966; Starr, 1969; Hildenbrand, 1970). However, one needs to impose
convexity conditions on the preferences of large firms, such as in my first assumption, because they
each play a non-negligible role in the economy even in the large-market limit (Hildenbrand, 1970).
15
The basic model presented in this paper implicitly assumes that large firms have complete,
continuous, and strictly convex preferences, as each large firm is assumed to have a (single-valued)
continuous choice function. Appendix B allows for weakly convex and incomplete preferences by
assuming that each large firm has an upper hemi-continuous non-empty compact convex-valued
choice correspondence.
16
Azevedo and Hatfield (2013) have shown that the substitutability of the preferences of one side
of a two-sided market ensures the existence of stable outcomes in settings with a continuum of firms.
However, Azevedo and Hatfield’s (2013) conditions do not define a maximal domain for the existence
of stable outcomes in large two-sided markets, as I show in Proposition 4 in Section 7.
17
Acyclicity is not necessary for the existence of stable outcomes in discrete trading networks with
complete markets (Hatfield et al., 2013; Fleiner, Jagadeesan, Jankó, and Teytelboym, 2017).
6
competitive equilibria in large markets with transferable utility.18 Recently, Fleiner,
Jagadeesan, Jankó, and Teytelboym (2017) have shown that competitive equilibria
give rise to stable outcomes whenever financial instruments are rich enough to equalize
marginal rates of substitution across agents—i.e., when the financial market is complete.19,20 Therefore, continuity, convexity, and mild finiteness restrictions together
ensure that stable outcomes exist in large markets with transferable utility.
The existence of stable outcomes is more subtle when there are frictions or the
financial market is incomplete. Competitive equilibria may not yield stable outcomes
(Fleiner, Jagadeesan, Jankó, and Teytelboym, 2017). Furthermore, transfers may be
multidimensional, making it unclear how to define competitive equilibrium. In order
to capture frictions, this paper does not construct stable outcomes from competitive
equilibria but instead builds them directly by imposing substitutability in the saledirection and acyclicity.
In some settings, it may be difficult for agents to identify and implement complex
recontracting opportunities (blocks). Thus, the observed market outcome may not
be stable—while simple blocks are unlikely to exist in equilibrium, complex blocking
opportunities may persist.21 As a complement to the main result on the existence
of stable outcomes, I relax the definition of stability to obtain existence results that
require weaker hypotheses than the main result. When the network has cycles, there
are outcomes that cannot be blocked by a sequence of proposals of sets of contracts
by their sellers. I call such outcomes seller-initiated-stable. Seller-initiated-stable
18
See Azevedo et al. (2013) Azevedo and Hatfield (2013). Competitive equilibria may not exist in
large markets with indivisibilities and strong income effects due to discontinuities in the aggregate
excess demand correspondence (Broome, 1972; Mas-Colell, 1977).
19
See also Hatfield et al. (2013), Hatfield and Kominers (2015b), and Rostek and Yoder (2017).
Hatfield et al. (2013), Hatfield and Kominers (2015b), and Rostek and Yoder (2017) assume that
utility is transferable. Although Fleiner, Jagadeesan, Jankó, and Teytelboym (2017) work with
economies with finitely many firms, their proof generalizes to the large-market limit setting (Jagadeesan, 2017f).
20
When the presence of multiple forms of transfer is driven by uncertainty, market completeness
requires that all agents can costlessly trade Arrow (1953) securities corresponding to every possible
future state. Market completeness rules out complex frictions, such as variable transaction taxes
and imperfectly-insurable default risk, but permits simple frictions, such as fixed shipping costs. See
Fleiner, Jagadeesan, Jankó, and Teytelboym (2017) and Jagadeesan (2017d) for detailed discussions
of market completeness in the context of matching.
21
For example, Fox and Bajari (2013) analyze the FCC spectrum auction by assuming that the
observed outcome is pairwise stable but not necessarily stable. Thus, Fox and Bajari (2013) assume
that there is no single contract that both counterparties would like to add (possibly while dropping
some previously signed contracts), but allow the possibility that there are groups of contracts that
are jointly desirable.
7
outcomes cannot be blocked by acyclic sets of contracts, but can be blocked by sets
of contracts with cycles. Intuitively, seller-initiated blocking proposal sequences and
acyclic blocking sets do not require coordination across the whole trading network
to implement, making acyclic and seller-initiated blocking opportunities less likely
to persist than general blocks. Seller-initiated-stability may be a reasonable solution
concept in settings where it is difficult for a buyer to make requests to potential
sellers. When sales are not substitutable, I show that there are still outcomes that
cannot be blocked by a sequence of proposals of single contracts. I call such outcomes
sequentially stable. Sequential stability is a network-based strengthening of pairwise
stability22 and may be a reasonable solution concept when it is difficult for agents to
identify and propose several blocking contracts on their own. From a technical perspective, I exploit the existence of sequentially stable outcomes to prove the existence
of seller-initiated-stable and stable outcomes.
While existence results are interesting in their own right, they are also crucial to
the underlying logic of structural empirical methods (see Section IID in Fox, 2017).
These methods assume that the observed outcome is pairwise stable and hence presuppose that a (pairwise) stable outcome exists.23 Recent work by Fox (2010, 2016)
on auto part markets and Fox and Bajari (2013) on the FCC spectrum auction have
exploited results on the existence of equilibria in large markets with transferable utility to estimate demand for complementary goods. The results of this paper open up
the possibility of developing similar structural estimation methods for two-sided and
network settings without substitutability or transferable utility, imposing sequential
stability as an analogue of pairwise stability.
From a conceptual perspective, this paper relates to and connects several strands
in the general equilibrium and matching literatures. First, this paper generalizes version of previous large-market matching models, which focused on two-sided markets
and imposed structure on the set of agent types. By modeling continuously divisible
contracts, this paper encapsulates versions of classical general equilibrium models,
including models with incomplete markets. By incorporating discrete contracts in a
22
Sequential stability refines tree stability (in the sense of Ostrovsky, 2008) and trail-stability (in
the sense of Fleiner et al., 2015). See Jagadeesan (2017c) for the details regarding these relationships.
23
Initial work focused on two-sided one-to-one matching markets with transferable utility (see,
e.g., Choo and Siow, 2006), but some papers have studied one-to-one and many-to-one matching
without transfers but with substitutable preferences (Sørensen, 2007; Logan et al., 2008; Boyd et al.,
2013; Agarwal, 2015).
8
large-market setting, the model presented in this paper subsumes versions of largemarket general equilibrium models with indivisibilities. The use of the language of
matching with contracts allows this paper, unlike general equilibrium, to capture frictions. I hope that the framework developed in this paper can serve to unify general
equilibrium with matching-theoretic models of markets with frictions.
The remainder of this paper is organized as follows. Section 2 explains the results through illustrative examples. Section 3 discusses related literature. Section 4
presents the primitives of the model and defines stability. Section 5 describes several
other stability properties, both as solution concepts and as technical tools. Section 6 presents the existence results. Section 7 discusses the maximal domain results.
Section 8 concludes. Appendices A and B extend the basic model to incorporate
multilateral contracts and indifferences, respectively. The supplementary appendices
(Appendices C–F) present the omitted proofs.
2
Illustrative examples
In discrete trading networks, complementarities between inputs or between sales, as
well as cycles, preclude the existence of stable outcomes (Hatfield and Kominers,
2012). This section provides three examples that illustrate the results of the paper
by showing how aggregate continuity interacts with Hatfield and Kominers’s (2012)
negative results. The first example illustrates the main result of this paper by giving
a supply chain with complementary inputs in which continuity restores the existence
of stable outcomes. The second example shows that stable outcomes may fail to exist
when inputs are complementary and sales are complementary, highlighting the role
of substitutability in the sale-direction in my existence results. The third example
shows that, as in discrete networks (Hatfield and Kominers, 2012), cycles can obstruct
the existence of stable outcomes. The latter two examples also feature outcomes that
satisfy weaker stability properties.
2.1
Continuity helps ensure that stable outcomes exist
The first example describes a supply chain with complementarities between inputs in
which continuity restores the existence of stable outcomes.
Example 1. There are two large firms, 𝑏 and 𝑖, and a unit mass of small firms of each
9
𝑠1
𝑧1
𝑥1
𝑏O ′
𝑧2
𝑦1
?
𝑖
(
H𝑏V
6𝑏
n
𝑦
𝑥
𝑦2
𝑥2
𝑏V n
𝑦′
𝑦
𝑥
𝑠′
𝑠
𝑦′
𝑠
𝑠′
𝑠2
(b) Contracts in Example 2.
(c) Contracts in Example 3.
(a) Contracts in Example 1.
Figure 1: The trading networks described in Examples 1, 2, and 3. Contracts are
depicted by arrows pointing from sellers to buyers.
of three types, 𝑏′ , 𝑠1 , and 𝑠2 . Firms of types 𝑠1 and 𝑠2 can sell to firms of type 𝑏′
directly, and can sell to 𝑏 via the intermediary 𝑖. As shown in Figure 1(a) on page
10, there are six contracts: 𝑥1 , 𝑥2 , 𝑦1 , 𝑦2 , 𝑧1 , and 𝑧2 .
Each seller of type 𝑠1 would like to sell up to one contract and prefers to sell 𝑥1 .
Each seller of type 𝑠2 would like to sell up to one contract and prefers to sell 𝑧2 .
Each buyer of type 𝑏′ would like to buy up to one contract and prefers to buy 𝑧1 .
Intermediary 𝑖 can perfectly transform units of 𝑥𝑗 into units of 𝑦𝑗 .24 Buyer 𝑏 views
𝑦1 and 𝑦2 as perfect complements.
In this economy, there is a unique stable outcome, in which 1/2 unit of each
contract is traded. In the discrete version of this economy, where preferences are the
same but only exactly 0 or 1 units of each contract can be traded, there is no stable
outcome.
2.2
Complications of complementarities
The second example, which is adapted from Azevedo and Hatfield (2013), shows that
when there are complementary inputs and complementary sales, stable outcomes
may fail to exist (in supply chains) even under aggregate continuity. Theorems 2
and 3 generalize Example 2, showing that substitutability in the sale-direction is
necessary in a maximal domain sense for the existence of stable outcomes. However,
as Example 2 shows, there is still an outcome that is not blocked by any tree of
contracts.
24
Thus, Example 1 incorporates a simple kind of production. Permitting inputs to be complementary allows more general forms of production.
10
Example 2. There are two large firms, 𝑠 and 𝑠′ , and a unit mass of small firms of type
𝑏. As shown in Figure 1(b) on page 10, there are three contracts: 𝑥, 𝑦, and 𝑦 ′ . Seller
𝑠 views 𝑥 and 𝑦 as perfect complements. Seller 𝑠′ would like to sell as much of 𝑦 ′ as
possible. Each buyer of type 𝑏 has preference {𝑥, 𝑦 ′ } ≻𝑏 {𝑥, 𝑦} ≻𝑏 ∅.
In this economy, there is no stable outcome. Indeed, in any individually rational
outcome, the same amounts of 𝑥 and 𝑦 must be traded by construction. Thus, 𝑦 ′
cannot be traded in any individually rational outcome. If no trade occurs, then
{𝑥, 𝑦} is a blocking set; if trade occurs, then {𝑦 ′ } is a blocking set. However, the
outcome in which no trade occurs is not blocked by any tree of contracts. As will
be discussed in Example 4 in Section 5.3, the no-trade outcome is even sequentially
stable.
2.3
Setbacks from cycles
The last example, which is also adapted from Azevedo and Hatfield (2013), shows
that cycles preclude the existence of stable outcomes under aggregate continuity.
Theorem 4 generalizes Example 3, showing that acyclicity is necessary in a maximal
domain sense for the existence of stable outcomes. However, as Example 3 shows,
there is an outcome that is not blocked by any acyclic set of contracts.
Example 3. Consider the economy of Example 2, but where contract 𝑥 is redirected
(as shown in Figure 1(c) on page 10). All firms’ preferences are fully substitutable
in the modified network. However, there are still no stable outcomes, as redirecting
contracts does not affect the set of stable outcomes. On the other hand, the outcome
in which no trade occurs is not blocked by any acyclic set of contracts in the modified
network. As will be discussed in Example 5 in Section 5.4, the no-trade outcome is
even seller-initiated-stable.
3
3.1
Relationship to the literature
Matching
The two most closely-related papers on large matching markets are Azevedo and
Hatfield (2013) and Che et al. (2013). The case of this paper with no large firms
generalizes Azevedo and Hatfield’s (2013) model to trading networks. The extension
11
in Appendix A models multilateral matching with a continuum of firms, generalizing
Section 5 in Azevedo and Hatfield (2013). The case of this paper in which unit-supply
small firms match with large firms in a two-sided market recovers the discrete-type
case of Che et al.’s (2013) model25 —since Che et al. (2013) allow indifferences, the
embedding requires the extension of Appendix B. Section 6.2 and Appendices A.4
and B.5 discuss the technical details of these relationships.
Motivated by the ability of couples to enter the U.S. medical residency match
together, another strand in the literature has focused specifically on the existence
of stable outcomes in matching with couples (Kojima et al., 2013; Ashlagi et al.,
2014). These papers have studied large, finite random matching markets. Azevedo
and Hatfield’s (2013) results imply that stable outcomes exist in a large-market limit
in which all residency programs are small. The results of this paper imply that stable
outcomes exist even in the limit in which there are some large residency programs.
A separate strand in the matching literature has focused on large markets with
substitutable preferences. Those papers exploit largeness to develop tractable models
(Bodoh-Creed, 2013; Abdulkadiroğlu et al., 2015; Azevedo and Leshno, 2016), and to
study issues of incentives (Immorlica and Mahdian, 2015; Kojima and Pathak, 2009)
and core convergence (Ashlagi et al., 2017). Under substitutability, however, stable
outcomes exist even in discrete markets, while this paper focuses on settings in which
continuity is essential to the existence of stable outcomes.
The case of this paper in which there are no small firms models matching with
continuously divisible contracts. Previous papers in this strand of the matching literature have either considered specific classes of contract structures and preferences
(Fleiner, 2014; Király and Pap, 2013; Cseh et al., 2013; Cseh and Matuschke, 2017) or
assumed that utility is transferable (Hatfield and Kominers, 2015b). This paper works
with general contract structures and does not assume that utility is transferable.
3.2
General equilibrium
The key conceptual difference between this paper and classical general equilibrium
models is that this paper does not assume the existence of continuous, one-dimensional
prices. On the other hand, this paper requires that prices be discrete and bounded if
25
Since Che et al.’s (2013) setting generalizes the school choice models of Abdulkadiroğlu et al.
(2015) and Azevedo and Leshno (2016), my model nests discrete-type versions of the Abdulkadiroğlu
et al. (2015) and Azevedo and Leshno (2016) models as well.
12
they exist, but allows for multidimensional transfers.26 As discussed in the introuduction, this paper also allows frictions that are not permitted in the standard general
equilibrium approach.
The precise connection of this paper to the general equilibrium literature is as
follows. The case of this paper with no small firms nests discrete-price versions of
classical general equilibrium models (Arrow and Debreu, 1954; McKenzie, 1954, 1959).
Because preferences over contracts implicitly capture budget constraints, this paper
also naturally incorporates settings with multiple budget constraints, such as general
equilibrium models with incomplete markets (Radner, 1968, 1972; Hart, 1974, 1975;
Cass, 2006; Werner, 1985; Geanakoplos and Polemarchakis, 1986; Duffie, 1987).27
The case of this paper with no large firms captures discrete-price versions of largemarket general equilibrium models with indivisibilities (Mas-Colell, 1977; Azevedo
et al., 2013; Azevedo and Hatfield, 2013).28 The model presented in Appendix A
encapsulates discrete-price versions of club models with a continuum of firms (Ellickson et al., 1999) and multilateral matching models with continuously divisible
contracts (Hatfield and Kominers, 2015b). The existence results presented in this
paper are matching-theoretic analogues of existence results in the general equilibrium
literature.
The extension presented in Appendix B allows preferences to be incomplete. Thus,
this paper proves matching-theoretic analogues of results on existence of general equilibrium with incomplete preferences (Schmeidler, 1969; Mas-Colell, 1974; Shafer and
Sonnenschein, 1975; Yamazaki, 1978; Aliprantis and Brown, 1983). To my knowledge, this is the first paper in matching theory to consider incomplete preferences
and intransitive indifferences.
26
Like Radner (1972) and unlike other papers (e.g., Cass, 2006; Werner, 1985; Geanakoplos and
Polemarchakis, 1986; Duffie, 1987), I impose exogenous upper bounds on trade.
27
Matching naturally captures both settings in which assets pay off in commodities (Radner, 1972;
Hart, 1975; Geanakoplos and Polemarchakis, 1986) and settings in which assets pay off in the units
of account (Hart, 1974; Cass, 2006; Werner, 1985; Duffie, 1987).
28
See also Section 6 in Azevedo and Hatfield (2013).
13
4
4.1
Model
Firms
There is a finite set 𝐹large of “large” firms. Intuitively, each member of 𝐹large can
be party to a positive proportion of contracts in the market (as the market grows).
There is also a finite set 𝐹small of “small” firm types. Intuitively, small firms each
form a bounded number of contracts (as the market grows).
Formally, there is continuum of each type of small firm, but only one instance of
each large firm. For each 𝑓 ∈ 𝐹small , let 𝜁 𝑓 ∈ R≥0 be the mass of firms of type 𝑓 that
are present. Let 𝐹 = 𝐹large ∪ 𝐹small denote the set of firm types.
4.2
Contracts
Let 𝑋 be a finite set of contracts. Each 𝑥 ∈ 𝑋 has an associated buyer type b(𝑥) ∈ 𝐹
and seller type s(𝑥) ∈ 𝐹 . I always assume that b(𝑥) ̸= s(𝑥) for all 𝑥 ∈ 𝑋.
As in the literature, contracts (conceptually) specify what is being traded and who
is trading, as well as prices and other contract terms (Roth, 1984b, 1985; Hatfield
and Milgrom, 2005; Ostrovsky, 2008; Klaus and Walzl, 2009; Hatfield and Kominers,
2012, 2017). Since I do not impose any extra structure on the set of contracts, prices
can be multidimensional. Thus, contracts can specify complex payments, capturing
(partial) financing by trade credit and payments in several currencies. More generally,
contracts can specify payments that incorporate several forms of transfer, which is a
relevant feature in settings with incomplete financial markets (see Section 1).
Each small firm of type 𝑓 trades a set of contracts 𝑌 ⊆ 𝑋𝑓 .29 Thus, in the
𝑋
aggregate, a small firm type 𝑓 trades a mass 𝜇 ∈ R≥0𝑓 of contracts. Large firms trade
masses of continuously divisible contracts among one another and discrete contracts
𝑋
with masses of small firms, so that large firms each trade a mass 𝜇 ∈ R≥0𝑓 of contracts.
More formally, I summarize aggregate trade in the economy by a mass of contracts
in the following fashion. When b(𝑥), s(𝑥) ∈ 𝐹small , a mass 𝑚 of contract 𝑥 represents
the trade of 𝑥 between mass 𝑚 of firms of type b(𝑥) and mass 𝑚 of firms of type
s(𝑥). When b(𝑥) ∈ 𝐹small and s(𝑥) ∈ 𝐹large , a mass 𝑚 of contract 𝑥 represents
29
I implicitly require each small firm to trade at most one unit of each contract. This assumption
plays no role in the existence results, but is made for sake of consistency with the matching literature.
14
the trade of 𝑥 between large firm s(𝑥) and mass 𝑚 of small firms of type b(𝑥).30
When b(𝑥), s(𝑥) ∈ 𝐹large , a mass 𝑚 of contract 𝑥 represents the trade of 𝑚 units of
continuously divisible contract 𝑥 between large firms b(𝑥) and s(𝑥).
For each 𝑥 ∈ 𝑋, there is an exogenous upper bound M𝑥 on the amount of 𝑥 that
can be traded.31,32 Thus, an allocation, which specifies how much of each contract is
traded, is an element of
X=
[0, M𝑥 ] .
×
𝑥∈𝑋
I always assume that M𝑥 ≥ 𝜁 𝑓 for all 𝑓 ∈ {b(𝑥), s(𝑥)} ∩ 𝐹small , so that it is feasible for
all small firms of a given type to trade all possible contracts. However, preferences
and the mass vector 𝜁 generally restrict trade in equilibrium.
Given firm types 𝑓, 𝑓 ′ ∈ 𝐹 and a set of contracts 𝑌 ⊆ 𝑋, let
𝑌𝑓 → = {𝑥 ∈ 𝑌 | s(𝑥) = 𝑓 }
𝑌→𝑓 ′ = {𝑥 ∈ 𝑌 | b(𝑥) = 𝑓 ′ }
𝑌𝑓 = 𝑌𝑓 → ∪ 𝑌→𝑓
𝑌𝑓 →𝑓 ′ = 𝑌𝑓 → ∩ 𝑌→𝑓 ′
denote the sets of contracts in 𝑌 that are sold by 𝑓 (resp., bought by 𝑓 ′ , involve 𝑓 ,
are sold by 𝑓 to 𝑓 ′ ). Given an allocation 𝜇 ∈ X and a firm 𝑓 ∈ 𝐹, let
(︀
)︀
𝜇𝑓 → = 𝜇𝑋𝑓 → , 0𝑋r𝑋𝑓 →
(︀
)︀
𝜇→𝑓 = 𝜇𝑋→𝑓 , 0𝑋r𝑋→𝑓
(︀
)︀
𝜇𝑓 = 𝜇𝑋𝑓 , 0𝑋r𝑋𝑓
denote the masses of contracts that are bought by type 𝑓 (resp., are sold by 𝑓 , involve
30
The homogeneity of the continuum of firms of type b(𝑥) ensures that s(𝑥) is indifferent as to
exactly which firms of type b(𝑥) trade with s(𝑥).
31
The bounds M𝑥 are analogous to the bound 𝐿 in Radner (1972) and the bounds 𝑟𝜔max in Hatfield
and Kominers (2015b). Hart (1974, 1975) has illustrated the role of such bounds in deriving general
results on the existence of equilibrium in incomplete markets. For particular asset structures, the
bounds on the trade of contracts can be removed (see, e.g., Cass, 2006; Werner, 1985; Geanakoplos
and Polemarchakis, 1986; Duffie, 1987). Restricting the asset structure corresponds to imposing
conditions on the set of contracts and firms’ choice correspondences in the matching model studied
in this paper. This paper focuses on general contract structures and choice functions, necessitating
the imposition of exogenous bounds on the quantity of trade.
32
In Examples 1–3 in Section 2, I set M𝑧 = 1 for all contracts 𝑧.
15
×
𝑓 ) under 𝜇. Let X𝑓 = 𝑥∈𝑋𝑓 [0, M𝑥 ] denote the set of bundles of contracts that can
be traded by firm 𝑓 (if 𝑓 ∈ 𝐹large ) or in the aggregate by firms of type 𝑓 (if 𝑓 ∈ 𝐹small ).
4.3
Outcomes
An allocation does not determine an outcome as one has to specify which firms of
each small firm type trade. Let 𝑓 be a small firm type. A distribution for 𝑓 specifies
the masses of firms of type 𝑓 that trade each possible set of contracts 𝑌 ⊆ 𝑋𝑓 .
[︀
]︀𝒫(𝑋𝑓 )
Definition 1. Let 𝑓 ∈ 𝐹small . An distribution for 𝑓 is a vector 𝐷𝑓 ∈ 0, 𝜁 𝑓
satisfying
∑︁ 𝑓
𝐷𝑌 = 𝜁 𝑓 .
𝑌 ⊆𝑋𝑓
Allocation A(𝐷𝑓 ) is defined to be the total mass of contracts that firms of type 𝑓
sign in distribution 𝐷𝑓 .
Definition 2. A distribution 𝐷𝑓 for 𝑓 induces allocation A(𝐷𝑓 ) ∈ X𝑓 by
∑︁
A(𝐷𝑓 )𝑥 =
𝐷𝑌𝑓
𝑥∈𝑌 ⊆𝑋𝑓
for 𝑥 ∈ 𝑋.
An outcome specifies how much of each contract is traded (an allocation) and how
the mass of contracts traded by each small firm type is distributed among the firms
of that type (a distribution for 𝑓 for each small type 𝑓 ). Distributions do not have
to be specified for large firms since there is only one instance of each large firm.
^
Definition 3. An outcome consists(︁of an
)︁ allocation 𝜇 ∈ X and, for each 𝑓 ∈ 𝐹small ,
^
^
a distribution 𝐷𝑓 for 𝑓^ satisfying A 𝐷𝑓 = 𝜇𝑓^.
4.4
Preferences
Large firms 𝑓 have choice functions defined over masses of contracts involving 𝑓 . More
formally, each large firm 𝑓 ∈ 𝐹large has a choice function 𝐶 𝑓 : X𝑓 → X𝑓 , assumed to
satisfy 𝐶 𝑓 (𝜇) ≤ 𝜇 for all 𝜇 ∈ X𝑓 .
16
Small firm types 𝑓 trade sets of contracts, and therefore have preferences defined
over sets of contracts involving 𝑓 . Formally, each type 𝑓 ∈ 𝐹small has a complete,
strict preference ≻𝑓 over 𝒫(𝑋𝑓 ). Define choice function 𝑐𝑓 : 𝒫(𝑋𝑓 ) → 𝒫(𝑋𝑓 ) by33
𝑐𝑓 (𝑌 ) = max 𝒫(𝑌 ).
≻𝑓
4.5
Stability
Stability requires individual rationality and the absence of a block (Roth, 1984b;
Blair, 1988; Hatfield and Milgrom, 2005; Echenique and Oviedo, 2006; Klaus and
Walzl, 2009; Hatfield and Kominers, 2012, 2017). As in Roth (1984a), individual
rationality requires that no firm wants to drop any contracts assigned to it.
(︂ (︁ )︁
)︂
𝑓^
Definition 4. An outcome 𝒪 = 𝜇, 𝐷
is individually rational if:
𝑓^∈𝐹small
∙ 𝜇𝑓 ∈ 𝐶 𝑓 (𝜇𝑓 ) for all 𝑓 ∈ 𝐹large ; and
∙ 𝑐𝑓 (𝑌 ) = 𝑌 for all 𝑓 ∈ 𝐹small and 𝑌 ⊆ 𝑋𝑓 with 𝐷𝑌𝑓 > 0.
Consider an outcome 𝒪 and a mass 𝛽 ∈ R𝑋
≥0 of contracts. Let 𝑍(𝛽) be the set of
contracts that appear in 𝛽 with positive mass. Intuitively, I say that 𝛽 blocks 𝒪 if:
∙ Every large firm 𝑓 wants mass 𝛽𝑓 when given access to the contracts it already
signs (possibly while dropping masses of contracts that are not in 𝑍(𝛽)).
∙ For every small firm type 𝑓 , the mass 𝛽𝑓 can be distributed to the firms of type
𝑓 such that every firm wants all of the blocking contracts assigned to it when
given access to the contracts that it already signs (possibly while dropping some
of the previously-signed contracts).
The motivation for this definition is that it is the limit of the definition of blocks in
discrete matching (Hatfield and Kominers, 2012, 2017) as the number of small agents
grows large and contracts between large agents become continuously divisible.34 In
33
I could instead take the choice functions 𝑐𝑓 as the primitives and assume that the functions 𝑐𝑓
satisfies the irrelevance of rejected contracts condition (Aygün and Sönmez, 2012, 2013). For ease of
notation in the proofs, I instead assume that small firms’ choice functions satisfy the strong axiom
of revealed preferences. A companion note (Jagadeesan, 2017e) presents a detailed analysis of the
role of choice functions in this setting, showing that the irrelevance of rejected contracts condition
for 𝑐𝑓 is the key to ensuring that aggregate demand is continuous. See also Footnote 42.
34
See Galichon et al. (2016) for a formal result in this vein, in a different setting.
17
two-sided markets with no large firms, my definition of stability agrees with Azevedo
and Hatfield (2013).
)︂
(︂ (︁ )︁
𝑋
𝑓^
Definition 5. A non-zero mass 𝛽 ∈ R≥0 blocks an outcome 𝒪 = 𝜇, 𝐷
𝑓^∈𝐹small
if 𝛽 + 𝜇 ≤ M and:
∙ for all 𝑓 ∈ 𝐹large , we have 𝐶 𝑓 (𝛽𝑓 + 𝜇𝑓 )𝑍(𝛽)𝑓 ≥ (𝛽 + 𝜇)𝑍(𝛽)𝑓 ; and
∙ for all 𝑓 ∈ 𝐹small , there exist 𝑌 1 , . . . , 𝑌 𝑘 ⊆ 𝑋𝑓 , 𝑍 1 , . . . , 𝑍 𝑘 ⊆ 𝑍(𝛽)𝑓 , and
𝛾 1 , . . . , 𝛾 𝑘 ∈ R≥0 such that 𝑌 𝑗 ∩ 𝑍 𝑗 = ∅ and 𝑐𝑓 (𝑌 𝑗 ∪ 𝑍 𝑗 ) ⊇ 𝑍 𝑗 for all 𝑗,
𝑘
∑︁
𝛾
𝑗
(︀
)︀
1{𝑌 𝑗 } , 0𝒫(𝑋𝑓 )r{𝑌 𝑗 } ≤ 𝐷
𝑓
and
𝑗=1
𝑘
∑︁
)︀
(︀
𝛾 𝑗 1𝑍 𝑗 , 0𝑋𝑓 r𝑍 𝑗 = 𝛽𝑓 .
𝑗=1
Here, we write 𝑍(𝛽) = {𝑥 ∈ 𝑋 | 𝛽𝑥 > 0}. An outcome is stable if it is individually
rational and is not blocked by any mass of contracts.
Remark 1. In the second part of Definition 5, 𝛾 𝑗 is the mass of small firms that are
assigned 𝑌 𝑗 in 𝒪 and 𝑍 𝑗 in the block.
5
Other stability properties
This section describes several stability properties in the model presented in Section 4,
including “strong stability” and several weakened stability conditions. The definitions
introduced in this section rely on blocking sets of contracts instead of blocking masses
of contracts, which makes them technically simpler.
In addition to yielding existence results under weaker conditions, the weakened
stability properties are technically useful in proving the existence of stable outcomes. To prove the existence of strongly tree stable (Corollary 1), seller-initiatedstable (Corollary 2), strongly acyclically stable (Corollary 3), and strongly stable
(Corollary 4) outcomes, I relate sequential stability to the other stability conditions.
More precisely, I show that any sequentially stable outcome is strongly tree stable
(Lemma 2), any sequentially stable outcome is seller-initiated-stable if all firms’ preferences are substitutable in the sale-direction (Lemma 4), and no seller-initiatedstable outcome can be blocked by an acyclic set of contracts (Lemma 3). In Section 6, I show the existence of sequentially stable outcomes (Theorem 1), thereby
proving the other existence results (Corollaries 1–4) as well.
18
Section 5.1 defines blocking sets and strong stability. Section 5.2 defines strong
tree stability and strong acyclic stability. Section 5.3 defines sequential stability and
relates sequential stability to strong stability and strong tree stability. Section 5.4
defines seller-initiated-stability and substitutability in the sale-direction and relates
seller-initiated-stability to strong acyclic stability and sequential stability.
5.1
Blocking sets and strong stability
Stability rules out deviations where all firms want to take the full mass of blocking
contracts. To define strong stability, I relax “wanting to take 𝛽” to “wanting to take
some of each contract in 𝑍(𝛽) when given access to some of an appropriate subset of
𝑍(𝛽).” I formalize the latter concept as the rationality of 𝑍(𝛽).35 Intuitively, a set of
contracts 𝑍 ⊆ 𝑋𝑓 is rational for firm type 𝑓 at an outcome given 𝑊 ⊆ 𝑋 when for
all contracts 𝑧 ∈ 𝑍:
∙ if 𝑓 is a large firm, then 𝑓 demands more 𝑧 when given access to additional
units of contracts of an appropriately chosen subset of 𝑊 ∪ 𝑍 (in addition to
the previously-signed contracts);
∙ if 𝑓 is a small firm type, then a positive mass of firms of type 𝑓 demand 𝑧 when
given access to an appropriately chosen subset of 𝑊 ∪ 𝑍 (in addition to the
previously-signed contracts).
(︂ (︁ )︁
)︂
𝑓^
Definition 6. Let 𝑓 ∈ 𝐹 be a firm type and let 𝒪 = 𝜇, 𝐷
be an
𝑓^∈𝐹small
outcome. A set 𝑍 ⊆ 𝑋 is rational for 𝑓 at 𝒪 given 𝑊 ⊆ 𝑋 if 𝑍 ⊆ 𝑋𝑓 and:
Case 1: 𝑓 ∈ 𝐹large . For all 𝑧 ∈ 𝑍, there exists 𝜇𝑓 ≤ 𝛽 ∈ X𝑓 such that 𝛽𝑋𝑓 r𝑊𝑓 r𝑍 =
𝜇𝑋𝑓 r𝑊𝑓 r𝑍 and 𝐶 𝑓 (𝛽)𝑧 > 𝜇𝑧 .
Case 2: 𝑓 ∈ 𝐹small . For all 𝑧 ∈ 𝑍, there exists 𝑊 ′ ⊆ 𝑊 ∪ 𝑍 and 𝑌 ⊆ 𝑋𝑓 r {𝑧}
such that 𝐷𝑌𝑓 > 0 and 𝑧 ∈ 𝑐𝑓 (𝑊 ′ ∪ 𝑌 ).
If 𝑍 is rational for 𝑓 at 𝒪 given ∅, then we say that 𝑍 is rational for 𝑓 at 𝒪.
A set of contracts is blocking if it is a rational deviation for all firm types. The
corresponding stability property is called strong stability.
35
The terminology “rationality” in this setting is due to Fleiner et al. (2015). My definition of
rationality is less restrictive than that of Fleiner et al. (2015).
19
Definition 7. A non-empty set 𝑍 ⊆ 𝑋 of contracts blocks an outcome 𝒪 if 𝑍𝑓 is
rational for 𝑓 at 𝒪 for all 𝑓 ∈ 𝐹 . An outcome is strongly stable if it is individually
rational and is not blocked by any set of contracts.
Firms need not agree on quantities of contracts for a set to be blocking. The
following proposition formally shows that blocking masses give rise to blocking sets,
so that strong stability indeed strengthens stability. Thus, I focus on proving the
existence of strongly stable outcomes in the rest of the paper.
Proposition 1. If a mass 𝛽 ∈ R𝑋
≥0 blocks an outcome 𝒪, then 𝑍(𝛽) blocks 𝒪. In
particular, strongly stable outcomes are stable.
5.2
Strong tree stability and strong acyclic stability
Ostrovsky (2008) introduced a stability property called tree stability that refines path
stability.36 Tree stability weakens stability by allowing blocks that are not trees. Since
I rule out blocking sets that are trees (as opposed to merely blocking masses that form
trees), I call the analogous stability property in my setting “strong tree stability.”
Definition 8. A set of contracts 𝑍 ⊆ 𝑋 is a tree if there do not exist distinct contracts
𝑧1 , . . . , 𝑧𝑛 ∈ 𝑍 and distinct firms 𝑓1 , . . . , 𝑓𝑛 ∈ 𝐹 such that {𝑏(𝑧𝑖 ), s(𝑧𝑖 )} = {𝑓𝑖 , 𝑓𝑖+1 } for
all 1 ≤ 𝑖 ≤ 𝑛, where 𝑓𝑛+1 = 𝑓1 . An outcome is strongly tree stable if it is individually
rational and is not blocked by any non-empty tree of contracts.
The last stability property, which is new to this paper, interpolates between strong
tree stability and strong stability. This stability property, strong acyclic stability,
rules out blocking sets that are directed acyclic. Acyclicity makes makes blocking
sets easier implement because acyclic blocking sets flow from terminal sellers (within
the block) to terminal buyers (within the block). Blocking sets that contain several
contracts between a pair of agents could be acyclic but are never trees. This discussion
motivates the consideration of strongly acyclically stable outcomes.
Definition 9. A set of contracts 𝑍 ⊆ 𝑋 is acyclic if if there do not exist contracts
𝑧1 , . . . , 𝑧𝑛 ∈ 𝑍 and firms 𝑓1 , . . . , 𝑓𝑛 ∈ 𝐹 such that b(𝑧𝑖 ) = 𝑓𝑖 = s(𝑧𝑖−1 ) for all 𝑖 =
36
Ostrovsky (2008) introduced path stability in acyclic networks under the name chain stability.
Chain stability and path stability agree in acyclic networks. Fleiner et al. (2015) coined the term
“path stability” to distinguish path stability from chain stability in general trading networks.
20
1, . . . , 𝑛, where 𝑧0 = 𝑧𝑛 . An outcome is strongly acyclically stable if it is individually
rational and is not blocked by any acyclic set of contracts.
Lemma 1. If 𝑋 is acyclic, then every strongly acyclically stable outcome is strongly
stable.
Proof. If 𝑋 is acyclic, every set of contracts 𝑍 ⊆ 𝑋 is acyclic.
5.3
Proposal sequences and sequential stability
Fleiner et al. (2015) have defined blocking conditions based on sequences of contracts
instead of unordered sets of contracts. This section introduces blocking conditions
based on sequences of sets of contracts. The basic concept is that of proposal sequences, which play the role of blocking sets. Intuitively, a proposal sequence consists
of a sequence of firms 𝑓𝑖 and sets of contracts 𝑍𝑖 such that all contracts in 𝑍𝑖 involve 𝑓𝑖 . A proposal sequence can be interpreted as a sequence of proposals of sets of
contracts, where 𝑓𝑖 proposes 𝑍𝑖 at the 𝑖th stage.
Definition 10. A proposal sequence is a sequence of sets of contracts and firms
((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑛 , 𝑓𝑛 )) ∈ (𝒫(𝑋) × 𝐹 )𝑛
satisfying 𝑍𝑖 ⊆ 𝑋𝑓𝑖 for all 1 ≤ 𝑖 ≤ 𝑛.
A proposal sequence blocks an outcome if both of the following conditions are
satisfied.
∙ For all 𝑖, the firm type 𝑓𝑖 wants to propose all the contracts in 𝑍𝑖 when given ac𝑓𝑖
cess to the previously signed contracts and (some of) the set 𝑍≤(𝑖−1)
of contracts
that have been proposed to 𝑓𝑖 before stage 𝑖.
∙ Some firm wants some of the contracts that have been proposed to it by the
end.
Blocking is formally defined in terms of rational deviations to capture the preceding
intuition.
Definition 11. A proposal sequence ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑛 , 𝑓𝑛 )) blocks an outcome 𝒪 if
𝑓𝑖
∙ for all 1 ≤ 𝑖 ≤ 𝑛, the set 𝑍𝑖 is rational for 𝑓𝑖 at 𝒪 given 𝑍≤(𝑖−1)
; and
21
𝑓
𝑓
∙ there exists 𝑓 ∈ 𝐹 and 𝑧 ∈ 𝑍≤𝑛
such that {𝑧} is rational for 𝑓 at 𝒪 given 𝑍≤𝑛
.
Here, we write
⎛
𝑓
𝑍≤𝑖
=⎝
⎞
⋃︁
𝑗≤𝑖|𝑓𝑗 ̸=𝑓
𝑍𝑗 ⎠ .
𝑓
A proposal sequence is rooted if only one contract is proposed at each stage.
Rooted proposal sequences are the simplest proposal sequences. Sequential stability
rules out the existence of any rooted blocking proposal sequence.
Definition 12. A proposal sequence ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑛 , 𝑓𝑛 )) is rooted if |𝑍𝑖 | = 1 for
all 1 ≤ 𝑖 ≤ 𝑛. An outcome is sequentially stable if it is individually rational is not
blocked by any rooted proposal sequence.
Example 4. Recall that, in the economies studied in Examples 2 and 3, there are two
large firms, 𝑠 and 𝑠′ , and one small firm type 𝑏. As shown in Figures 1(b) and 1(c)
on page 10, there are three contracts, 𝑥, 𝑦, and 𝑦 ′ . Contracts 𝑥 and 𝑦 are traded by
𝑏 and 𝑠 and contract 𝑦 ′ is traded by 𝑏 and 𝑠′ .
Examples 2 and 3 observed that the no-trade outcome is strongly tree stable in
these economies. The no-trade outcome is even sequentially stable in these examples.
Suppose that outcome 𝒪 is blocked by a tree of contracts. Fixing a root in the
tree, one can build a rooted proposal sequence from the blocking tree so that all
proposals flow toward the root in the tree. This rooted proposal sequence blocks 𝒪.
Thus, sequential stability implies strong tree stability.
Lemma 2. Every sequentially stable outcome is strongly tree stable.
Proof. See Appendix F.
In particular, sequential stability refines path stability and pairwise stability. Sequential stability also strengthens Fleiner et al.’s (2015) trail-stability property.37 As
will be shown in the next section, sequential stability has a convenient relationship
with strong acyclic stability, in that sequential stability and substitutability in the
sale-direction together imply strong acyclic stability. Thus, sequential stability provides a conceptually natural and technically useful strengthening of path stability,
pairwise stability, and trail-stability.
37
See also Jagadeesan (2017c) for a discussion of sequential stability in discrete trading networks.
22
While Lemma 2 is technically useful, it also illustrates a sense in which trees are
easier to implement than other blocking sets. To be precise, Lemma 2 shows that any
potential blocking tree can be decentralized to a blocking rooted proposal sequence,
and hence does not require coordination across the network to implement. Section 5.4
gives a related interpretation of acyclic blocking sets.
On the other hand, any blocking proposal sequence gives rise to a blocking set
by considering the set of contracts in the proposal sequence that the non-proposing
counterparty desires. Thus, stability implies sequential stability, thereby reconciling
the approaches to stability via blocking sets and blocking proposal sequences.
Proposition 2. Every strongly stable outcome is sequentially stable.
Proof. See Appendix F.
5.4
Seller-initiated-stability
I say that a proposal sequence is seller-initiated if every proposed contract is proposed
by its seller. Ruling out seller-initiated blocking proposal sequences defines a stability
property. This stability property is natural when it is difficult for buyers to identify
willing sellers but might be easy for sellers to find potential buyers.
Definition 13. A proposal sequence ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑛 , 𝑓𝑛 )) is seller-initiated if 𝑍𝑖 ⊆
𝑋𝑓𝑖 → for all 1 ≤ 𝑖 ≤ 𝑛. An outcome is seller-initiated-stable if it is individually
rational and is not blocked by any seller-initiated proposal sequence.
Example 5. Recall that, in the economy studied in Example 3, there are two large
firms, 𝑠 and 𝑠′ , and one small firm type 𝑏. As shown in Figure 1(c) on page 10, there
are three contracts, 𝑥, 𝑦, and 𝑦 ′ , and contracts 𝑥 and 𝑦 form a cycle.
Example 3 observed that the no-trade outcome is strongly tree stable in this
economy. The no-trade outcome is even seller-initiated-stable in this economy.
Seller-initiated-stability is technically useful due to its relationships with strong
acyclic stability and sequential stability. Any acyclic set 𝑍 defines a seller-initiated
proposal sequence by letting proposals flow from firms who don’t buy any contracts
in 𝑍 to firms who don’t sell any contracts in 𝑍. If 𝑍 blocks an outcome, then the
corresponding seller-initiated proposal sequence blocks the outcome as well. Thus,
seller-initiated-stability implies strong acyclic stability.
23
Lemma 3. Every seller-initiated-stable outcome is strongly acyclically stable.
Proof. See Appendix F.
In addition to being technically useful, Lemma 3 illustrates a sense in which acyclic
blocking sets are easier to implement than blocking sets with cycles. To precise,
Lemma 3 shows that any acyclic potential blocking set can be decentralized to a
blocking seller-initiated proposal sequence, and hence does not require coordination
across the network to implement.
When sales are substitutable, any seller-initiated blocking proposal sequence can
be simplified into a rooted proposal sequence by making each seller propose the blocking contracts one by one—this property is the technical use of seller-initiated-stability and substitutability in the sale-direction. Formally, substitutability in the saledirection is an analogue of one half of Ostrovsky’s (2008) same-side substitutability condition. However, unlike same-side substitutability, substitutability in the saledirection allows complementarities between inputs; unlike full substitutability, substitutability in the sale-direction does not require inputs and sales to be complementary
to one another. In two-sided markets, substitutability in the sale-direction requires
that sellers have substitutable choice functions but places no restriction on buyers’
choice functions.38,39 For large firms, I naturally extend the condition that sales are
substitutable to the continuous setting.
Definition 14.
∙ For 𝑓 ∈ 𝐹small , choice function 𝑐𝑓 is substitutable in the saledirection 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 in the sale-direction if for
all 𝜇 ≤ 𝜇′ ∈ X𝑓 and all 𝑥 ̸= 𝑦 ∈ 𝑋𝑓 → with 𝜇𝑋𝑓 r{𝑥} = 𝜇′𝑋𝑓 r{𝑥} , we have
𝐶 𝑓 (𝜇′ )𝑦 ≤ 𝐶 𝑓 (𝜇)𝑦 .
Lemma 4. If 𝐶 𝑓 is substitutable in the sale-direction for all 𝑓 ∈ 𝐹large and 𝑐𝑓 is
substitutable in the sale-direction for all 𝑓 ∈ 𝐹small , then every sequentially stable
outcome is seller-initiated-stable.
38
In particular, the assumption in Azevedo and Hatfield’s (2013) existence result is equivalent in
their setting to the assumption that all firms’ preferences are substitutable in the sale-direction.
39
See Definition 17 in Appendix A.3 for a formal definition of substitutability in the continuous
setting, motivated by Hatfield and Milgrom’s (2005) definition.
24
Proof. See Appendix F.
As will be seen in Section 6.1, Lemma 4 plays a key role in proving the existence
of seller-initiated-stable, strongly acyclically stable, and strongly stable outcomes. I
do not know of an analogue of Lemma 4 for stability properties defined in terms of
blocking sets or blocking masses.
6
Existence results
Section 6.1 states the existence results. Section 6.2 discusses the relationship of the
existence results to Azevedo and Hatfield (2013) and Che et al. (2013). Section 6.3
sketches the proof of the existence of sequentially stable outcomes (Theorem 1).
6.1
Existence results
In order to ensure that stable outcomes exist, I need to impose Che et al.’s (2013) continuous analogue of the irrelevance of rejected contracts condition.40 The irrelevance
of rejected contracts condition is automatic if 𝐶 𝑓 is rationalizable by a preference.41
Definition 15 (Che et al., 2013). Choice function 𝐶 𝑓 satisfies the irrelevance of
rejected contracts condition if 𝐶 𝑓 (𝜇) = 𝐶 𝑓 (𝜇′ ) whenever 𝜇, 𝜇′ ∈ X𝑓 satisfy 𝐶 𝑓 (𝜇) ≤
𝜇′ ≤ 𝜇.
The first result asserts that continuity and the irrelevance of rejected contracts
condition (for large firms’ choice functions) together ensure that a sequentially stable
outcome exists.
Theorem 1. If 𝐶 𝑓 is continuous and satisfies the irrelevance of rejected contracts
condition for all 𝑓 ∈ 𝐹large , then a sequentially stable outcome exists.
Proof. See Section 6.3.
Because every sequentially stable outcome is strongly tree stable (Lemma 2),
strongly tree stable outcomes also exist under the hypotheses of Theorem 1.
40
Aygün and Sönmez (2012, 2013) have shown that the irrelevance of rejected contracts condition
is crucial to the existence of stable outcomes in matching with contracts.
41
In settings without indifferences, the irrelevance of rejected contracts condition is equivalent to
the weak axiom of revealed preferences.
25
Corollary 1. Under the hypotheses of Theorem 1, a strongly tree stable outcome
exists.
Proof. Follows from Theorem 1 and Lemma 2.
Under substitutability in the sale-direction, sequential stability implies seller-initiated-stability (Lemma 4). Thus, seller-initiated-stable outcomes exist under substitutability in the sale-direction (in conjunction with the hypotheses of Theorem 1).
Corollary 2. Under the hypotheses of Theorem 1, if furthermore 𝐶 𝑓 is substitutable
in the sale-direction for all 𝑓 ∈ 𝐹large and 𝑐𝑓 is substitutable in the sale-direction for
all 𝑓 ∈ 𝐹small , then a seller-initiated-stable outcome exists.
Proof. Follows from Theorem 1 and Lemma 4.
Finally, seller-initiated-stable outcomes are strongly acyclically stable (Lemma 3),
and, in acyclic networks, strongly acyclically stable outcomes are strongly stable
(Lemma 1). Thus, strongly acyclically stable outcomes exist under substitutability
in the sale-direction (in conjunction with the hypotheses of Theorem 1); when the
network is also acyclic, strongly stable outcomes also exist.
Corollary 3. Under the hypotheses of Corollary 2, a strongly acyclically stable outcome exists.
Proof. Follows from Corollary 2 and Lemma 3.
Corollary 4. Suppose that:
∙ 𝐶 𝑓 is continuous and satisfies the irrelevance of rejected contracts condition for
all 𝑓 ∈ 𝐹large ;
∙ 𝐶 𝑓 is substitutable in the sale-direction for all 𝑓 ∈ 𝐹large ;
∙ 𝑐𝑓 is substitutable in the sale-direction for all 𝑓 ∈ 𝐹small ; and
∙ the trading network is acyclic.
Then, a strongly stable outcome exists.
Proof. Follows from Corollary 3 and Lemma 1.
26
6.2
Relationship to Azevedo and Hatfield (2013) and Che
et al. (2013)
Corollary 4 generalizes recent results of Azevedo and Hatfield (2013) and Che et al.
(2013) on the existence of stable outcomes in large two-sided matching markets. A
matching market is two-sided if 𝐹 = 𝐵 ∪ 𝑆 with 𝐵 ∩ 𝑆 = ∅ and (b(𝑥), s(𝑥)) ∈ 𝐵 × 𝑆
for all 𝑥 ∈ 𝑋. Such markets are clearly acyclic. In these markets, 𝐶 𝑏 (resp. 𝑐𝑏 )
is automatically substitutable in the sale-direction for all 𝑏 ∈ 𝐵 ∩ 𝐹large (resp., 𝑏 ∈
𝐵 ∩ 𝐹small ).
The case of the two-sided model in which 𝐹large = ∅ recovers the model of Azevedo
and Hatfield (2013). Corollary 4 strengthens and generalizes Theorem 1 in Azevedo
and Hatfield (2013). On the other hand, the case of the two-sided model in which
𝐵 = 𝐹large , 𝑆 = 𝐹small , and sellers each sign at most one contract recovers the case
of the model of Section 3 in Che et al. (2013) in which the space of seller/worker
types is a finite set. Corollary 4 strengthens and generalizes the finite-type case of
Theorem 2 in Che et al. (2013). Appendices A.4 and B.5 discuss similar relationships
in multilateral matching and in settings with indifferences, respectively.
6.3
Proof of Theorem 1
The proof of Theorem 1 is conceptually similar to the proof of Theorem 1 in Azevedo
and Hatfield (2013). The proof proceeds in three steps. I first define the aggregate
demand of small firm types following Azevedo and Hatfield (2013). I then use the
aggregate demand functions of small firm types and the choice functions of large firms
to define a Gale-Shapley operator. I conclude by showing that the Gale-Shapley has
fixed points and that fixed points give rise to sequentially stable outcomes.
The aggregate demand of each small firm type is defined as follows. Fix a small
firm type 𝑓 ∈ 𝐹small . Let 𝑌 𝐾 , . . . , 𝑌 0 be the acceptable sets of contracts for 𝑓 in order
of preference—formally, let ≻𝑓 be given by
𝑌 𝐾 ≻𝑓 · · · ≻𝑓 𝑌 1 ≻𝑓 𝑌 0 = ∅.
27
Given 𝜇 ∈ X𝑓 , define ℎ𝑓𝑌 𝑘 inductively for 𝑘 = 𝐾, 𝐾 − 1, . . . , 0 by
{︃
{︃
ℎ𝑓𝑌 𝑘 (𝜇) = min 𝜁 𝑓 −
∑︁
𝑗>𝑘
ℎ𝑓𝑌 𝑗 (𝜇), min
𝑥∈𝑌 𝑘
𝜇𝑥 −
}︃}︃
∑︁
𝜒𝑌 𝑗 (𝑥)ℎ𝑓𝑌 𝑗 (𝜇)
.
(1)
𝑗>𝑘
Set ℎ𝑓𝑌 (𝜇) = 0 for all 𝑌 ⊆ 𝑋𝑓 with ∅ ≻𝑓 𝑌 , so that ℎ𝑓 (𝜇) is a distribution for 𝑓 .
Intuitively, ℎ𝑓𝑌 (𝜇) is the mass of firms of type 𝑓 that are assigned 𝑌 when mass 𝜇
of contracts is available and the distribution of sets of contracts is maximized in the
lexicographic order (with respect to ≻𝑓 ). Define the aggregate demand 𝐶 𝑓 : X𝑓 → X𝑓
(︀
)︀
by 𝐶 𝑓 (𝜇) = A ℎ𝑓 (𝜇) . Note that 𝐶 𝑓 is single-valued, continuous,42 and satisfies
𝐶 𝑓 (𝜇) ≤ 𝜇 for all 𝜇 ∈ X𝑓 by construction.
My generalized Gale-Shapley operator Φ : X 4 → X 4 is a continuous analogue
of one of the Gale-Shapley operators in the literature (Fleiner, 2003; Hatfield and
Milgrom, 2005; Hatfield and Kominers, 2012; Che et al., 2013; Fleiner et al., 2015;
Fleiner, Jagadeesan, Jankó, and Teytelboym, 2017), but applied in an auxiliary twosided market where firms are matched to contracts.43 The use of a novel Gale-Shapley
operator ensures that fixed-points yield equilibria even when large firms have “lumpy”
preferences (in the sense of Che et al., 2013), a property that I exploit in Appendix B.44
Formally, let
⎛
M − 𝜎 + 𝜌 ∧ 𝜎, M − 𝜌 + 𝜌 ∧ 𝜎, M − 𝜆 +
⎜
⎜
Φ(𝜅, 𝜆, 𝜌, 𝜎) = ⎜
∑︁
⎝M − 𝜅 +
𝐶 𝑓 (𝜅→𝑓 + 𝜆𝑓 → )→𝑓
∑︁
𝑓 ∈𝐹
𝐶 𝑓 (𝜅→𝑓 + 𝜆𝑓 → )𝑓 → ,
⎞
⎟
⎟
⎟
⎠
𝑓 ∈𝐹
for all (𝜅, 𝜆, 𝜌, 𝜎) ∈ X 4 , where − ∧ − denotes componentwise minimum. Intuitively,
𝜅 is the mass of contracts available to buyers and 𝜆 denotes the mass of contracts
available to sellers. The operator Φ is a t^atonnement procedure, and 𝜌 and 𝜎 are
42
A companion note (Jagadeesan, 2017e) shows that the irrelevance of rejected condition for 𝑐𝑓 is
the key condition to ensure that 𝐶 𝑓 is continuous. See also Footnote 33.
43
Other papers have used a different Gale-Shapley operator to ensure that there is a one-to-one
correspondence between fixed points and stable outcomes (see, e.g., Adachi, 2000, 2017; Echenique
and Oviedo, 2004, 2006; Ostrovsky, 2008; Azevedo and Hatfield, 2013; Hatfield and Kominers, 2017).
However, this alternative operator behaves poorly when there are indifferences (Jagadeesan, 2017b,c).
44
Lumpiness is ruled out by continuity and the irrelevance of rejected contracts condition (see
Example 2 in Che et al., 2013). However, lumpiness is allowed in settings with indifferences, such
as in Appendix B.
28
auxiliary allocations that are used to deal with lumpy demand.45
Third, I show Φ has a fixed point, which yields a sequentially stable outcome. It
turns out that 𝜌 ∧ 𝜎 is the analogue of 𝑋 𝐵 ∩ 𝑋 𝑆 in the usual correspondence between
fixed points and stable outcomes (Adachi, 2000, 2017; Fleiner, 2003; Hatfield and
Milgrom, 2005; Ostrovsky, 2008; Hatfield and Kominers, 2012, 2017; Fleiner et al.,
2015). I distribute contracts in 𝜌 ∧ 𝜎 among small firms according to the functions
ℎ𝑓 —formally, define
(︂
Ψ(𝜌 ∧ 𝜎) =
(︁
𝑓^
𝜌 ∧ 𝜎, ℎ
(︁
)︂
)︁)︁
(𝜌 ∧ 𝜎)𝑓^
𝑓^∈𝐹small
.
When (𝜅, 𝜆) is a fixed-point of the Gale-Shapley operator, Ψ(𝜌 ∧ 𝜎) is a sequentially
stable outcome, as the following proposition shows.
Proposition 3. If 𝐶 𝑓 satisfies the irrelevance of rejected contracts condition for
all 𝑓 ∈ 𝐹large and (𝜅, 𝜆, 𝜌, 𝜎) = Φ(𝜅, 𝜆, 𝜌, 𝜎), then Ψ (𝜌 ∧ 𝜎) is a sequentially stable
outcome.
Proof. See Appendix D.
Proof of Theorem 1. We clearly have M − 𝜎 ≤ M − 𝜎 + 𝜌 ∧ 𝜎 ≤ M for all 𝜌, 𝜎 ∈ X .
Because 0 ≤ 𝐶 𝑓 (𝜅→𝑓 + 𝜆𝑓 → ) ≤ 𝜅→𝑓 + 𝜆𝑓 → for all 𝑓 ∈ 𝐹, we have
M−𝜆≤M−𝜆+
∑︁
𝐶 𝑓 (𝜅→𝑓 + 𝜆𝑓 → )𝑓 → ≤ M
𝑓 ∈𝐹
for all 𝜅, 𝜆 ∈ X . It follows that Φ : X 4 → X 4 .
Because 𝐶 𝑓 is continuous for all 𝑓 ∈ 𝐹 (by construction and assumption), Φ is
continuous. As X is a compact, convex subset of R𝑋 , Brouwer’s fixed point theorem
guarantees that Φ has a fixed point (𝜅, 𝜆, 𝜌, 𝜎). By Proposition 3, Ψ (𝜌 ∧ 𝜎) is a
sequentially stable outcome.
7
Maximal domain results
This section shows that the conditions of Corollaries 2 and 4 define maximal domains
for the existence of seller-initiated-stable outcomes and stable outcomes, respectively.
45
In the auxiliary economy in which firms are matched to contracts, 𝜌 and 𝜎 are the masses
available to the contract-side of the economy.
29
Intuitively, blocking masses and seller-initiated proposal sequences are rich enough
to detect any complementarities between sales in the preferences of small firms. It
follows that the substitutability in the sale-direction of the preferences of small firms
is necessary in a maximal domain sense for seller-initiated-stable or stable outcomes
to exist in large trading networks.46
Theorem 2. Let f ∈ 𝐹small , and suppose that |𝐹 | ≥ 4. Suppose furthermore that
|𝑋𝑓 ∩ 𝑋𝑓 ′ | ≥ 1 for all 𝑓 ̸= 𝑓 ′ ∈ 𝐹, and that M𝑥 > 0 for all 𝑥 ∈ 𝑋. If 𝑐f is not
substitutable in the sale-direction, then there exist preferences for types 𝑓 ∈ 𝐹 r {f }
and a non-empty open set 𝑈 ⊆ R𝐹>0small such that:
(1) 𝐶 𝑓 is substitutable in the sale-direction, continuous, and satisfies the irrelevance
of rejected contracts condition for all 𝑓 ∈ 𝐹large ;
(2) 𝑐𝑓 is substitutable in the sale-direction for all 𝑓 ∈ 𝐹small r {f };
(3) 𝜁 𝑓 ≤ M𝑥 for all 𝜁 ∈ 𝑈, 𝑥 ∈ 𝑋, and 𝑓 ∈ {b(𝑥), s(𝑥)}; and
(4) the economy does not have a seller-initiated-stable outcome or a stable outcome
for any 𝜁 ∈ 𝑈 .
Proof. See Appendix E.
A version of Theorem 2 is also true for large firms. As there are only finitely
many possible preferences for small firms, small firms’ preferences are not always rich
enough to witness all possible complementarities between sales in the preferences of
large firms. Thus, I assume that all firms are large in order to formulate the analogue
of Theorem 2 for large firms. This assumption can be relaxed considerably, but
without providing further intuition and at the expense of complicating the statement
of the theorem.
Theorem 3. Let f ∈ 𝐹 , and suppose that 𝐹small = ∅ and |𝐹 | ≥ 4. Suppose furthermore that |𝑋𝑓 ∩ 𝑋𝑓 ′ | ≥ 1 for all 𝑓 ̸= 𝑓 ′ ∈ 𝐹, and that M𝑥 > 0 for all 𝑥 ∈ 𝑋. If 𝐶 f
satisfies the irrelevance of rejected contracts condition but is not substitutable in the
sale-direction, then there exist preferences for firms 𝑓 ∈ 𝐹 r {f } such that:
46
On the other hand, it is an open problem to determine a maximal domain for the existence
of strongly acyclically stable outcomes in large markets. A companion paper (Jagadeesan, 2017a)
proves a maximal domain result for a stability concept that interpolates between strong tree stability
and strong stability.
30
(1) 𝐶 𝑓 is substitutable in the sale-direction, continuous, and satisfies the irrelevance
of rejected contracts condition for all 𝑓 ∈ 𝐹 r {f }; and
(2) the economy does not have a seller-initiated-stable outcome or a stable outcome.
As in discrete matching (Hatfield and Kominers, 2012), acyclicity is necessary in a
maximal domain sense for the existence of stable outcomes. Intuitively, in Example 3,
cycles can hide complementarities in both directions, which preclude the existence of
stable outcomes (Theorems 2 and 3).
Theorem 4. Suppose that there exist firms f1 , f2 , . . . , f𝑘 and f ′ such that:
∙ 𝑋f𝑖 →f𝑖+1 ̸= ∅ for all 1 ≤ 𝑖 ≤ 𝑘, where f𝑘+1 = f1 ; and
∙ 𝑋f1 ∩ 𝑋f ′ ̸= ∅.
If M𝑥 > 0 for all 𝑥 ∈ 𝑋, then there exist preferences for all types 𝑓 ∈ 𝐹 such that:
(1) 𝐶 𝑓 is substitutable in the sale-direction,47 continuous, and satisfies the irrelevance of rejected contracts condition for all 𝑓 ∈ 𝐹large ;
(2) 𝑐𝑓 is substitutable in the sale-direction for all 𝑓 ∈ 𝐹small ; and
(3) the economy does not have a stable outcome for any 𝜁 ∈ R𝐹>0small .
Proof. See Appendix F.
Remark 2. Theorems 2–4 imply that substitutability in the sale-direction and acyclicity are necessary in a maximal domain sense for strongly stable outcomes to exist.
In contrast, substitutability in the sale-direction is not necessary for the existence of stable outcomes in two-sided matching.48 Indeed, in two-sided markets with
complementarities on both sides, contracts can sometimes be redirected to obtain an
acyclic trading network in which sales are substitutable. For example, if one seller
has complementary preferences over contracts with several different buyers, contracts
can be redirected to make all firms’ preferences substitutable in the sale-direction,
without creating cycles. Proposition 4 formalizes this intuition.
47
In fact, the preferences of firm types other than f can be taken to be fully substitutable, in the
sense that sales and inputs are substitutable for each other and complementary to one another, in
Theorem 4, as in Theorem 6 in Hatfield and Kominers (2012).
48
Note that the hypotheses of Theorems 2 and 3 cannot be satisfied in two-sided markets. Indeed,
the hypotheses require that there are at least four firms and that every pair of firms can trade (in
some direction), which is incompatible with two-sidedness.
31
Proposition 4. Suppose that 𝐹 = 𝐵 ∪𝑆 with 𝐵 ∩𝑆 = ∅ and that (b(𝑥), s(𝑥)) ∈ 𝐵 ×𝑆
for all 𝑥 ∈ 𝑋. Let s ∈ 𝑆 be arbitrary. Suppose furthermore that 𝐶 𝑓 is substitutable
in the sale-direction, continuous, and satisfies the irrelevance of rejected contracts
condition for all 𝑓 ∈ 𝐹large r {s }; and 𝑐𝑓 is substitutable in the sale-direction for all
𝑓 ∈ 𝐹small r {s }. If there exists Z ⊆ 𝑋s such that |b(Z)| = |Z| and either
(a) s ∈ 𝐹large , the choice function 𝐶 s is continuous and satisfies the irrelevance of
rejected contracts condition, and 𝐶 s (𝜇)𝑋s rZ = 0 for all 𝜇 ∈ Xs ; or
(b) s ∈ 𝐹small and ∅ ≻s 𝑌 for all 𝑌 ⊆ 𝑋s with 𝑌 ̸⊆ Z,
then a strongly stable outcome exists.
Proof. See Appendix F.
In trading networks satisfying the conditions of Theorem 2, changes of direction
always create cycles, which preclude the existence of stable outcomes (by Theorem 4).
8
Conclusion
This paper developed a model of trading networks in which a large number of firms or
goods are present. Aggregate continuity helps restore the existence of stable outcomes.
I introduced stability properties based on sequential blocking conditions to obtain
existence results in trading networks with complementarities and cycles. My model
captures complex frictions that are ruled out by the standard general equilibrium
approach. Previous matching models of networks, while also capturing frictions,
suffered from the non-existence of equilibrium in the presence of complementarities,
preventing matching from being applied to analyze complex, real-world markets.
This paper opens several avenues for future research. First, one could develop
structural empirical methods that assume that the observed outcome is sequentially
stable, underpinned by the existence results of this paper. Second, the real-world
validity of the novel solution concepts, sequential stability and seller-initiated-stability, could be investigated. Third, the problem of finding computationally efficient
algorithms to find or approximate equilibria in my model could be explored.49
49
The proof of Theorem 1 relies on Brouwer’s Fixed-Point Theorem and hence does not yield an
efficient algorithm to compute or approximate equilibria.
32
A
Multilateral matching
This section extends the existence results to multilateral matching. Appendix A.1
adapts the model to allow for multilateral contracts. Appendix A.2 shows that
strongly tree stable outcomes exist, and Appendix A.3 shows that stable outcomes
exist if all firms’ preferences are substitutable. Appendix A.4 relates the existence
results to Azevedo and Hatfield (2013).
A.1
Model
I model multilateral contracts 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 contract-specific—formally, I assume 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 bundles of roles.
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, blocking sets, and stable
outcomes are exactly as in Section 5.
33
A.2
Tree stability in multilateral matching
As in settings with bilateral contracts, continuity and convexity alone ensure the
existence of strongly tree stable outcomes in large multilateral matching markets. In
order to define strong 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 16. 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 5. If 𝐶 𝑓 is continuous and satisfies the irrelevance of rejected contracts
condition for all 𝑓 ∈ 𝐹large , then a strongly tree stable outcome exists.
Proof. See Appendix F.
To prove Corollary 5, 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 5.
A.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 17.
∙ (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}).
34
∙ For 𝑓 ∈ 𝐹large , choice function 𝐶 𝑓 is substitutable if for all 𝜇 ≤ 𝜇′ ∈ X𝑓 and
𝑥 ̸= 𝑦 ∈ 𝑋𝑓 with 𝜇𝑋𝑓 r{𝑥} = 𝜇′𝑋𝑓 r{𝑥} , we have 𝐶 𝑓 (𝜇′ )𝑦 ≤ 𝐶 𝑓 (𝜇)𝑦 .50
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 6. Suppose that:
∙ 𝐶 𝑓 is continuous and satisfies the irrelevance of rejected contracts condition for
all 𝑓 ∈ 𝐹large ;
∙ 𝐶 𝑓 is substitutable for all 𝑓 ∈ 𝐹large ; and
∙ 𝑐𝑓 is substitutable for all 𝑓 ∈ 𝐹small .
Then, a strongly stable outcome exists.
Proof. See Appendix F.
To prove Corollary 6, I consider with the auxiliary economy described in Appendix A.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 6.
Corollary 6 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.
A.4
Relationship to Azevedo and Hatfield (2013)
Corollary 6 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 6 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 6.
50
Che et al. (2013) have defined substitutability in settings with continuously divisible contracts
in terms of the monotonicity of the rejection function (see also Fleiner, 2003). My definition of
substitutability is stronger than theirs.
35
B
Incorporating indifferences
This section extends the results to settings with indifferences, including intransitive indifferences. Firms, contracts, and outcomes are exactly as in Section 4. Appendix B.1
presents the model of preferences with indifferences. Appendix B.2 defines strong stability. Appendix B.3 describes an analogue of substitutability in the sale-direction.
Appendix B.4 presents the existence results, and Appendix B.5 relates the extended
model to Che et al. (2013).
B.1
Preferences and individual rationality
Each large firm 𝑓 ∈ 𝐹large has a choice correspondence 𝐶 𝑓 : X𝑓 ⇒ X𝑓 , assumed to
satisfy 𝜈 ≤ 𝜇 for all 𝜈 ∈ 𝐶 𝑓 (𝜇) and 𝜇 ∈ X𝑓 .
Each small firm type 𝑓 ∈ 𝐹small has a strict, potentially incomplete preference ≻𝑓
over 𝒫(𝑋𝑓 ). That is, ≻𝑓 is a partial order on 𝒫(𝑋𝑓 ). For example, ≻𝑓 could be the
acyclic part of a weak order on 𝒫(𝑋𝑓 ). Define choice correspondence 𝑐𝑓 : 𝒫(𝑋𝑓 ) ⇒
𝒫(𝑋𝑓 ) by setting
𝑐𝑓 (𝑌 ) = max 𝒫(𝑌 ),
≻𝑓
to be the set of maximal elements of 𝒫(𝑌 ) with respect to (the restriction of) the
partial order ≻𝑓 .
Intuitively, individual rationality requires that no firm strictly prefers dropping
some of the contracts assigned to it. In discrete matching with contracts settings 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 the setting of large
markets.
(︂ (︁ )︁
)︂
′
𝑓^
Definition 4 . An outcome 𝒪 = 𝜇, 𝐷
is individually rational if:
𝑓^∈𝐹small
∙ 𝜇𝑓 ∈ 𝐶 𝑓 (𝜇𝑓 ) for all 𝑓 ∈ 𝐹large ; and
∙ 𝑌 ∈ 𝑐𝑓 (𝑌 ) for all 𝑓 ∈ 𝐹small and 𝑌 ⊆ 𝑋𝑓 with 𝐷𝑌𝑓 > 0.
B.2
Blocking sets and strong stability
Hatfield et al. (2013) have given a definition of blocking sets and stability in settings
with indifferences. This section adapts Hatfield et al.’s (2013) definitions to large
36
markets, and defines seller-initiated-stability in the extended model.
First, I adapt the definition of rational deviations given in Section 5.1 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 6.51
)︂
(︂ (︁ )︁
′
𝑓^
be an
Definition 6 . Let 𝑓 ∈ 𝐹 be a firm type and let 𝒪 = 𝜇, 𝐷
𝑓^∈𝐹small
outcome. A set 𝑍 ⊆ 𝑋 is rational for 𝑓 at 𝒪 given 𝑊 ⊆ 𝑋 if 𝑍 ⊆ 𝑋𝑓 and
Case 1: 𝑓 ∈ 𝐹large . For all 𝑧 ∈ 𝑍, there exists 𝜇𝑓 ≤ 𝛽 ∈ X𝑓 such that 𝛽|𝑋𝑓 r𝑊𝑓 r𝑍 =
𝜇|𝑋𝑓 r𝑊𝑓 r𝑍 and 𝜈𝑧 > 𝜇𝑧 for all 𝜈 ∈ 𝐶 𝑓 (𝛽).
Case 2: 𝑓 ∈ 𝐹small . For all 𝑧 ∈ 𝑍, there exist 𝑊 ′ ⊆ 𝑊𝑓 ∪ 𝑍 and 𝑌 ⊆ 𝑋𝑓 r 𝑍
such that 𝐷𝑌𝑓 > 0 and 𝑧 ∈ 𝑌 ′ for all 𝑌 ′ ∈ 𝑐𝑓 (𝑊 ′ ∪ 𝑌 ).
Lemmata 2 and 3 persist in the extended model.
Lemma 2′ . Every sequentially stable outcome is strongly tree stable.
Proof. See Appendix F.
Lemma 3′ . Every seller-initiated-stable outcome is strongly acyclically stable.
Proof. See Appendix F.
B.3
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. (2015) and Fleiner, Jagadeesan, Jankó, and
Teytelboym (2017) have studied substitutability in settings with continuous prices
and indifferences. One of their substitutability conditions, choice-language expansion full substitutability (CEFS), focuses on retaining substitutability as the set of
available contracts grows. Expansion-substitutability in the sale-direction, which is a
51
Che et al. (2013) has 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 exists
𝑓
𝑌 ∈ 𝐶 (𝐴 ∪ 𝑍) with 𝑍 ⊆ 𝑌 . In Appendix B.5, I show that, under the existence assumptions made
by Che et al. (2013), my notion of sstrong stability is strictly stronger than their notion of stability.
37
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).52
Definition 14′ .
∙ For 𝑓 ∈ 𝐹small , choice correspondence 𝑐𝑓 is expansion-substitutable in the sale-direction if for all 𝑌 ⊆ 𝑋𝑓 and 𝑥 ̸= 𝑦 ∈ 𝑋𝑓 → such that there
exists 𝑊 ∈ 𝑐𝑓 (𝑌 ∪ {𝑦}) with 𝑦 ∈
/ 𝑊, there exists 𝑊 ′ ∈ 𝑐𝑓 (𝑌 ∪ {𝑥, 𝑦}) with
𝑦∈
/ 𝑊 ′.
∙ For 𝑓 ∈ 𝐹large , choice correspondence 𝐶 𝑓 is expansion-substitutable in the saledirection if for all 𝜇 ≤ 𝜇′ ∈ X𝑓 , all 𝑥 ̸= 𝑦 ∈ 𝑋𝑓 → with 𝜇𝑋𝑓 r{𝑥} = 𝜇′𝑋𝑓 r{𝑥} , and
all 𝜈 ∈ 𝐶 𝑓 (𝜇), there exists 𝜈 ′ ∈ 𝐶 𝑓 (𝜇′ ) with 𝜈𝑦′ ≤ 𝜈𝑦 .
Expansion-substitutability in the sale-direction is technically analogous to substitutability in the sale-direction, in that expansion-substitutability in the sale-direction
and sequential stability together imply seller-initiated-stability.
Lemma 4′ . If 𝐶 𝑓 is expansion-substitutable in the sale-direction for all 𝑓 ∈ 𝐹large
and 𝑐𝑓 is expansion-substitutable in the sale-direction for all 𝑓 ∈ 𝐹small , then every
sequentially stable outcome is seller-initiated-stable.
Proof. See Appendix F.
B.4
Existence results
The existence of equilibrium in matching markets always relies on some form of the
irrelevance of rejected contracts condition (Aygün and Sönmez, 2012, 2013). The
following irrelevance of rejected contracts condition, adapted to settings with indifferences, requires that adding contracts in a way that does not strictly improve a firm
can only shrink the choice set. The condition is a form of Sen’s 𝛼. Che et al.’s (2013)
revealed preference property also imposes Sen’s 𝛽 by requiring that adding contracts
in a way that does not strictly improve a firm cannot change the choice set. Thus,
while Che et al.’s (2013) condition conceptually rules out incomplete preferences and
intransitive indifferences, my irrelevance of rejected contracts condition allows both.
52
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 (2017c) shows that IIFS is the key condition to
ensure existence results in discrete trading networks with indifferences.
38
Definition 15′ . 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).53
The following result is analogous to Theorem 1.
Theorem 1′ . If 𝐶 𝑓 is upper hemi-continuous, non-empty compact convex-valued,
and satisfies the irrelevance of rejected contracts condition for all 𝑓 ∈ 𝐹large , then a
sequentially stable outcome exists.
Proof. See Appendix D.
Applying Lemmata 2′ –4′ yields analogues of Corollaries 1–4.
Corollary 1′ . Under the hypotheses of Theorem 1′ , a strongly tree stable outcome
exists.
Proof. Follows from Theorem 1′ and Lemma 2′ .
Corollary 2′ . Under the hypotheses of Theorem 1′ , if furthermore 𝐶 𝑓 is expansionsubstitutable in the sale-direction for all 𝑓 ∈ 𝐹large and 𝑐𝑓 is expansion-substitutable
in the sale-direction for all 𝑓 ∈ 𝐹small , then a seller-initiated-stable outcome exists.
Proof. Follows from Theorem 1′ and Lemma 4′ .
Corollary 3′ . Under the hypotheses of Corollary 2′ , a strongly acyclically stable outcome exists.
Proof. Follows from Corollary 2′ and Lemma 3′ .
Corollary 4′ . Suppose that:
∙ 𝐶 𝑓 is upper hemi-continuous, non-empty compact convex-valued, and satisfies
the irrelevance of rejected contracts condition for all 𝑓 ∈ 𝐹large ;
∙ 𝐶 𝑓 is expansion-substitutable in the sale-direction for all 𝑓 ∈ 𝐹large ;
∙ 𝑐𝑓 is expansion-substitutable in the sale-direction for all 𝑓 ∈ 𝐹small ; and
∙ the trading network is acyclic.
53
It is straightforward to verify that 𝑐𝑓 satisfies the following analogous irrelevance of rejected
contracts condition—𝑊 ∈ 𝑐𝑓 (𝑌 ) whenever 𝑊 ∈ 𝑐𝑓 (𝑌 ′ ) and 𝑊 ⊆ 𝑌 ⊆ 𝑌.
39
Then, a strongly stable outcome exists.
Proof. Follows from Corollary 3′ and Lemma 1.54
B.5
Relationship to Che et al. (2013)
Corollary 4′ generalizes the cases of Theorems 4 and S5 in Che et al. (2013) when
the set of seller/worker types Θ is finite. Indeed, the finite-type case of the Che et al.
(2013) model is a case of the extended model 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 4′ guarantees that a strongly
stable outcome exists.
To be precise, suppose that 𝑋 = 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(𝑠,𝑏) = 𝜁 𝑠 for all (𝑠, 𝑏) ∈ 𝑋.
These assumptions yield the case of the model of Section 6 in Che et al. (2013) in
which the seller/workertype space Θ is finite.
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
∙
∑︀
{𝑥}≻s(𝑥) {𝑥′ }
𝑠
𝐷{𝑥
′ } ≥ 𝛽𝑥 for all 𝑥 ∈ 𝑋 and
∙ 𝛽 ∈ 𝐶 𝑏 (𝛽 ∨ 𝜇𝑏 ) and 𝜇𝑏 ∈
/ 𝐶 𝑏 (𝛽 ∨ 𝜇𝑏 ) ,
where − ∨ − denotes componentwise maximum. However, under the assumptions of
Che et al.’s (2013) existence results, the definition of strong stability considered in
this paper is strictly stronger than the definition of stability by Che et al. (2013).
The following example shows that, even under the assumptions of Che et al.’s (2013)
existence results, stability in the sense of Definition 5 in Che et al. (2013) does not
imply strong stability (in the sense of this paper).
Example 6 (Stability in the sense of Definition 5 in Che et al. (2013) does not imply
strong stability). Let 𝐹small = {𝑠}, let 𝐹large = {𝑏, 𝑏′ }, and let 𝜁 𝑠 = 2. Let 𝑋 = {𝑏, 𝑏′ }
54
Lemma 1 asserts that strong acyclic stability implies stability in acyclic networks. Although
Lemma 1 is stated formally for the basic model, identical logic shows that the result is true in the
extended model.
40
with s(𝑥) = 𝑠 and b(𝑥) = 𝑥 for all 𝑥 ∈ 𝑋. Define ≻𝑠 by 𝑏 ≻𝑠 𝑏′ ≻𝑠 ∅. Define
′
𝐶 𝑏 : [0, 2] → [0, 2] by 𝐶 𝑏 (𝜇) = {min{𝜇, 1}} and 𝐶 𝑏 : [0, 2] → [0, 2] by
⎧
⎨[0, 𝜇] if 𝜇 ≤ 1
′
𝐶 𝑏 (𝜇) =
⎩{𝜇} if 𝜇 > 1.
′
The choice correspondences 𝐶 𝑏 and 𝐶 𝑏 are clearly upper hemi-continuous and maximize quasi-concave utility functions.
(︀
(︀
)︀)︀
Consider the outcome 𝒪 = (1𝑏 , 0𝑏′ ) , 1{𝑏} , 1∅ , 0𝒫(𝑋)r{∅,{𝑏}} where 1 unit of 𝑠 to
𝑏 and 1 unit of 𝑠 remains unmatched. The set {𝑏′ } blocks 𝒪, but 𝒪 is stable in the
sense of Definition 5 in Che et al. (2013).
The following proposition shows that, assuming that 𝐶 𝑓 is non-empty compactvalued and satisfies Che et al.’s (2013) revealed preference property, strong 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 5. 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 ∅ ≻𝑓 𝑌 whenever 𝑌 ⊆ 𝑋𝑓 satisfies |𝑌 | > 1. If 𝐶 𝑓 is non-empty compactvalued and satisfies Che et al.’s (2013) revealed preference property for all 𝑓 ∈ 𝐹large ,
every outcome that is strongly stable is also stable in the sense of Definition 5 in Che
et al. (2013).
41
Supplementary appendix
C
A property of rational deviations
This appendix proves a useful technical property of rational deviations, which applies
in both the basic model (of Section 4) and the extended model (of Appendix B). This
property will be used in several of the subsequent proofs.
Lemma 5. Let 𝑓 ∈ 𝐹 be a firm type, let 𝒪 be an outcome, and let 𝑍 ⊆ 𝑍 ′ ⊆ 𝑋𝑓
and 𝑊, 𝑊 ′ ⊆ 𝑋 be sets of contracts. If 𝑍 ′ is rational for 𝑓 at 𝒪 given 𝑊 ′ and
𝑊 ⊇ 𝑊 ′ ∪ 𝑍 ′ r 𝑍, then 𝑍 is rational for 𝑓 at 𝒪 given 𝑊 . In particular, if 𝑍 ′ is
rational for 𝑓 at 𝒪 and 𝑊 ⊇ 𝑍 ′ r 𝑍, then 𝑍 is rational for 𝑓 at 𝒪 given 𝑊 .
Proof. The second assertion clearly
from
)︂ the first, and so we prove only the
(︂ (︁ follows
)︁
^
. As 𝑊 ⊇ 𝑊 ′ ∪ 𝑍 ′ r 𝑍, we have
first assertion. Write 𝒪 = 𝜇, 𝐷𝑓
′
′
𝑓^∈𝐹small
𝑊 ∪ 𝑍 ⊇ 𝑊 ∪ 𝑍 . We divide into cases based on whether 𝑓 is a large firm or a small
firm type to complete the proof.
Case 1: 𝑓 ∈ 𝐹large . Let 𝑧 ∈ 𝑍 be arbitrary. Because 𝑍 ′ is rational for 𝑓 at 𝒪
given 𝑊 ′ , there exists 𝜇𝑓 ≤ 𝛽 ∈ X𝑓 such that 𝛽𝑋𝑓 r𝑊 ′ r𝑍 ′ = 𝜇𝑋𝑓 r𝑊 ′ r𝑍 ′ and 𝜈𝑧 > 𝜇𝑧
for all 𝜈 ∈ 𝐶 𝑓 (𝛽). As 𝑊 ∪ 𝑍 ⊇ 𝑊 ′ ∪ 𝑍 ′ , we have 𝛽𝑋𝑓 r𝑊 r𝑍 = 𝜇𝑋𝑓 r𝑊 r𝑍 . Since
𝑧 ∈ 𝑍 was arbitrary, it follows that 𝑍 is rational for 𝑓 at 𝒪 given 𝑊 .
Case 2: 𝑓 ∈ 𝐹small . Let 𝑧 ∈ 𝑍 be arbitrary. Because 𝑍 ′ is rational for 𝑓 at 𝒪
given 𝑊 ′ , there exists 𝑇 ⊆ 𝑊 ′ ∪ 𝑍 ′ and 𝑌 ⊆ 𝑋𝑓 r {𝑧} such that 𝐷𝑌𝑓 > 0 and
𝑧 ∈ 𝑌 ′ for all 𝑌 ′ ∈ 𝑐𝑓 (𝑇 ∪ 𝑌 ). As 𝑊 ∪ 𝑍 ⊇ 𝑊 ′ ∪ 𝑍 ′ , we have 𝑇 ⊆ 𝑊 ∪ 𝑍. Since
𝑧 ∈ 𝑍 was arbitrary, it follows that 𝑍 is rational for 𝑓 at 𝒪 given 𝑊 .
The cases clearly exhaust all possibilities, completing the proof of the lemma.
D
Proofs of Proposition 3 and Theorem 1′
This appendix works with the extended model of Appendix B. Appendix D.1 constructs and proves basic properties of the aggregate demand functions 𝐶 𝑓 for 𝑓 ∈
𝐹small . Appendix D.2 extends the Gale-Shapley operator Φ to this setting and proves
basic properties of the fixed points of Φ. Appendix D.3 formulates and proves a
general version of Proposition 3. Appendix D.4 concludes the proof of Theorem 1′ .
42
D.1
Aggregate demand of small firm types
̂︀ 𝑓 be a complete, strict preference that
Let 𝑓 ∈ 𝐹small be a small firm type. Let ≻
̂︀ 𝑓 as
refines ≻𝑓 . Write ≻
̂︀ 𝑓 · · · ≻
̂︀ 𝑓 𝑌 0 = ∅.
𝑌𝐾 ≻
Define ℎ𝑓 and 𝐶 𝑓 according to (1) in Section 6.3. The first claim asserts that no small
firm can want any contract of which there is excess supply.
Claim D.1. Let 𝑓 ∈ 𝐹small , let 𝜇 ∈ X𝑓 , and let 𝑌 ⊆ 𝑋𝑓 . Let
𝑊 = {𝑥 ∈ 𝑋𝑓 | 𝐶 𝑓 (𝜇)𝑥 < 𝜇𝑥 }.
If 𝑌 ∈
/ 𝑐𝑓 (𝑊 ∪ 𝑌 ), then ℎ𝑓𝑌 (𝜇) = 0.
Proof. Let 𝑌 𝑘 ⊆ 𝑐𝑓 (𝑊 ∪ 𝑌 ) be such that 𝑌 𝑘 ≻𝑓 𝑌. For all 𝑥 ∈ 𝑊, we have
∑︁
𝜒𝑌 𝑗 (𝑥)ℎ𝑓𝑌 𝑗 (𝜇) ≤ 𝐶 𝑓 (𝜇)𝑥 < 𝜇𝑥 .
𝑗≥𝑘
Thus, by the definition of ℎ𝑓 , either
∑︁
ℎ𝑓𝑌 𝑗 (𝜇) = 𝜁 𝑓
𝑗≥𝑘
or there exists 𝑥 ∈ 𝑌 such that
∑︁
𝜒𝑌 𝑗 (𝑥)ℎ𝑓𝑌 𝑗 (𝜇) = 𝜇𝑥 .
𝑗≥𝑘
In either case, it is straightforward to verify that ℎ𝑓𝑌 (𝜇) = 0, since 𝑌 𝑘 ≻𝑓 𝑌.
The second claim asserts that the aggregate demand functions satisfy the irrelevance of rejected contracts condition.
Claim D.2. Let 𝑓 ∈ 𝐹small and let 𝜇 ≤ 𝜇′ ∈ X𝑓 . If 𝐶 𝑓 (𝜇′ ) ≤ 𝜇, then ℎ𝑓 (𝜇) = ℎ𝑓 (𝜇′ )
and hence 𝐶 𝑓 (𝜇′ ) = 𝐶 𝑓 (𝜇).
Proof. It is straightforward to prove that ℎ𝑓𝑌 𝑘 (𝜇) = ℎ𝑓𝑌 𝑘 (𝜇′ ) for 𝑘 = 𝐾, 𝐾 − 1, . . . , 0
by descending strong induction on 𝑘.
43
D.2
The Gale-Shapley operator
I begin by extending the generalized Gale-Shapley operator defined in Section 6.3 to
settings with indifferences. Define Φ : X 4 ⇒ X 4 by
⎧⎛
⎫
⎞
⎪
⎪
⎨ M − 𝜎 + 𝜌 ∧ 𝜎, M − 𝜌 + 𝜌 ∧ 𝜎,
⎬
⎜
⎟ 𝑓
∑︁
∑︁
𝑓
Φ(𝜅, 𝜆, 𝜌, 𝜎) = ⎝ M − 𝜆 +
.
𝑓 ⎠ | 𝜐 ∈ 𝐶 (𝜅→𝑓 + 𝜆𝑓 → )
𝜐𝑓𝑓 → , M − 𝜅 +
𝜐→𝑓
⎪
⎪
⎩
⎭
𝑓 ∈𝐹
𝑓 ∈𝐹
The remainder of this section is devoted to proving two technical properties of
fixed points of Φ. The first claim shows that fixed points of Φ yield individually
rational outcomes under Ψ.
Claim D.3. Suppose that 𝐶 𝑓 satisfies the irrelevance of rejected contracts condition
for all 𝑓 ∈ 𝐹large . Let (𝜅, 𝜆, 𝜌, 𝜎) ∈ X 4 . If (𝜅, 𝜆, 𝜌, 𝜎) ∈ Φ(𝜅, 𝜆, 𝜌, 𝜎), then:
(a) 𝜌 ∧ 𝜎 = 𝜅 + 𝜎 − M = 𝜆 + 𝜌 − M;
(b) (𝜌 ∧ 𝜎)𝑓 ∈ 𝐶 𝑓 (𝜅→𝑓 + 𝜆𝑓 → ) for all 𝑓 ∈ 𝐹 ; and
(c) Ψ (𝜌 ∧ 𝜎) is an individually rational outcome.
Proof. First, we prove Part (a). Because (𝜅, 𝜆, 𝜌, 𝜎) ∈ Φ(𝜅, 𝜆, 𝜌, 𝜎), we have
M − 𝜎 + 𝜌 ∧ 𝜎 = 𝜅.
Rearranging yields that 𝜌 ∧ 𝜎 = 𝜅 + 𝜎 − M. By symmetry, we have 𝜌 ∧ 𝜎 = 𝜆 + 𝜌 − M.
We now prove Part (b). Let 𝑓 ∈ 𝐹 be a firm type. The definition of Φ ensures
that there exists 𝜐 𝑓 ∈ 𝐶 𝑓 (𝜅→𝑓 + 𝜆𝑓 → ) such that
𝜌𝑓 → = M𝑓 → − 𝜆𝑓 → + 𝜐𝑓𝑓 →
𝑓
𝜎→𝑓 = M→𝑓 − 𝜅→𝑓 + 𝜐→𝑓
.
Summing the two above equations yields that
𝜐 𝑓 = 𝜅→𝑓 + 𝜆𝑓 → + 𝜌𝑓 → + 𝜎→𝑓 − M𝑓 .
In light of Part (a), it follows that 𝜐 𝑓 = (𝜌 ∧ 𝜎)𝑓 .
44
It remains to prove Part (b). Claim D.2 and the previous paragraph yield that
)︁
(︁
𝐶 𝑓 (𝜌 ∧ 𝜎)𝑓 = 𝐶 𝑓 (𝜅→𝑓 + 𝜆𝑓 → ) = (𝜌 ∧ 𝜎)𝑓
(︁
𝑓
(︁
)︁)︁
= (𝜌 ∧ 𝜎)𝑓 for all 𝑓 ∈ 𝐹small ,
for all 𝑓 ∈ 𝐹small . It follows that A ℎ (𝜌 ∧ 𝜎)𝑓
which implies that Ψ (𝜌 ∧ 𝜎) is an outcome.
(︁
)︁
It follows from Claim D.1 that, for all 𝑓 ∈ 𝐹small , we have ℎ𝑓𝑌 (𝜌 ∧ 𝜎)𝑓 = 0
whenever 𝑌 ∈
/ 𝑐𝑓 (𝑌 ). The irrelevance of rejected contracts condition and Part (b)
yield that (𝜌 ∧ 𝜎)𝑓 ∈ 𝐶 𝑓 (𝜅→𝑓 + 𝜆𝑓 → ) for all 𝑓 ∈ 𝐹large . Thus, Ψ (𝜌 ∧ 𝜎) is individually
rational.
The second claim serves as the inductive step of an argument to show that, at
fixed points of Φ, the full mass (according to M) of any proposed contract must be
available to non-proposing counterparty.
Claim D.4. Let 𝑓 ∈ 𝐹 be a firm type, let 𝑊 ⊆ 𝑋 be a set of contracts, and let 𝑧 ∈ 𝑋𝑓
be a contract. Let (𝜅, 𝜆, 𝜌, 𝜎) ∈ Φ(𝜅, 𝜆, 𝜌, 𝜎) and suppose that {𝑧} is rational for 𝑓 at
Ψ(𝜅, 𝜆) given 𝑊 . If 𝐶 𝑓 satisfies the irrelevance of rejected contracts condition for all
𝑓 ∈ 𝐹large and (𝜅→𝑓 + 𝜆𝑓 → )𝑊 = M𝑊 , then
(𝜅→𝑓 + 𝜆𝑓 → )𝑧 < M𝑧 = (𝜆→𝑓 + 𝜅𝑓 → )𝑧 .
Proof. Without loss of generality, assume that 𝑧 ∈ 𝑋→𝑓 . We divide into cases based
on whether 𝑓 is a large firm or a small firm type to prove that 𝜅𝑧 < M𝑧 .
Case 1: 𝑓 ∈ 𝐹large . Because {𝑧} is rational for 𝑓 at Ψ(𝜅, 𝜆) given 𝑊 , there exists
(𝜌 ∧ 𝜎)𝑓 ≤ 𝛽 ∈ X𝑓 such that 𝛽𝑋𝑓 r𝑊 r{𝑧} = (𝜌 ∧ 𝜎)𝑋𝑓 r𝑊 r{𝑧} and 𝜈𝑧 > (𝜌 ∧ 𝜎)𝑧
for all 𝜈 ∈ 𝐶 𝑓 (𝛽)𝑧 . (︁By the irrelevance of rejected
)︁ contracts condition, it follows
𝑓
that (𝜌 ∧ 𝜎)𝑓 ∈
/ 𝐶 (𝜌 ∧ 𝜎)𝑋𝑓 r𝑊 r{𝑧} , M𝑊𝑓 ∪{𝑧} . Due to Claim D.3(b) and the
assumption that (𝜅→𝑓 + 𝜆𝑓 → )𝑊 = M𝑊 , we must have 𝜅𝑧 < M𝑧 .
Case 2: 𝑓 ∈ 𝐹small . Because {𝑧} is rational for 𝑓(︁ at Ψ(𝜅, )︁
𝜆) given 𝑊 , there exist
𝑓
′
𝑊 ⊆ 𝑊 ∪ {𝑧} and 𝑌 ⊆ 𝑋𝑓 r {𝑧} such that ℎ𝑌 (𝜌 ∧ 𝜎)𝑓 > 0 and 𝑧 ∈ 𝑍 for all
𝑍 ∈ 𝑐𝑓 (𝑊 ′ ∪ 𝑌 ). By shrinking 𝑊 ′ to 𝑊 ′ r 𝑌, we can assume that 𝑊 ′ and 𝑌 are
disjoint.
45
)︁
(︁
Because 𝑌 ∩ 𝑊 ′ = ∅ and ℎ𝑓𝑌 (𝜌 ∧ 𝜎)𝑓 > 0, we have
𝐶
𝑓
(︁
(𝜌 ∧ 𝜎)𝑓
)︁
𝑥
< 𝜁 𝑓 ≤ M𝑥
for all 𝑥 ∈ 𝑊 ′ . By Claims D.2 and D.3(b), it follows that
𝐶 𝑓 (𝜅→𝑓 + 𝜆𝑓 → )𝑥 < 𝜁 𝑓 ≤ M𝑥
for all 𝑥 ∈ 𝑊 ′ . The contrapositive of Claim D.1 and the assumption that M𝑊 =
(𝜅→𝑓 + 𝜆𝑓 → )𝑊 together yield that
M𝑧 > 𝐶 𝑓 (𝜅→𝑓 + 𝜆𝑓 → )𝑧 = 𝜅𝑧 .
The cases clearly exhaust all possibilities, and so we have proved that 𝜅𝑧 < M𝑧 .
By Claim D.3(a), it follows that
(𝜌 ∧ 𝜎)𝑧 = 𝜅𝑧 + 𝜎𝑧 − M𝑧 < 𝜎𝑧 .
Thus, we have 𝜌𝑧 = (𝜌 ∧ 𝜎)𝑧 . Because (𝜅, 𝜆, 𝜌, 𝜎) ∈ Φ(𝜅, 𝜆, 𝜌, 𝜎), we must have
𝜆𝑧 = M𝑧 − 𝜌𝑧 + (𝜌 ∧ 𝜎)𝑧 = M𝑧 ,
as desired.
D.3
Completion of the proof of Proposition 3
The following result generalizes Proposition 3 to settings with indifferences.
Proposition 3′ . If 𝐶 𝑓 satisfies the irrelevance of rejected contracts condition for
all 𝑓 ∈ 𝐹large and (𝜅, 𝜆, 𝜌, 𝜎) ∈ Φ(𝜅, 𝜆, 𝜌, 𝜎), then Ψ(𝜌 ∧ 𝜎) is a sequentially stable
outcome.
Proof. Claim D.3(c) guarantees that Ψ (𝜅, 𝜆) is an individually rational outcome. It
remains to prove that Ψ (𝜅, 𝜆) is not blocked by any rooted proposal sequence.
Let ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑛 , 𝑓𝑛 )) be a rooted proposal sequence, and suppose that for
𝑓𝑖
all 1 ≤ 𝑖 ≤ 𝑛, the set 𝑍𝑖 is rational for 𝑓𝑖 at 𝒪 given 𝑍≤(𝑖−1)
. For 1 ≤ 𝑖 ≤ 𝑛, let
46
𝑍𝑖 = {𝑧𝑖 } and let {𝑓𝑖′ } = {b(𝑧𝑖 ), s(𝑧𝑖 )} r {𝑓𝑖 }. Note that
}︀
{︀
𝑓
𝑍≤𝑖
= 𝑧𝑗 | 𝑗 ≤ 𝑖 and 𝑓𝑗′ = 𝑓
(2)
for all 𝑓 ∈ 𝐹 and 0 ≤ 𝑖 ≤ 𝑛. Note furthermore that
(︀
𝜅→𝑓𝑖′ + 𝜆𝑓𝑖′ →
)︀
𝑧𝑖
= (𝜆→𝑓𝑖 + 𝜅𝑓𝑖 → )𝑧𝑖
(3)
for all 1 ≤ 𝑖 ≤ 𝑛.
(︀
)︀
We claim that 𝜅→𝑓𝑘′ + 𝜆𝑓𝑘′ → 𝑧 = M𝑧𝑘 for all 1 ≤ 𝑘 ≤ 𝑛. To prove this claim, we
𝑘
proceed
by
strong
induction
on
𝑘.
The base case of 𝑘 ≤ 0 is obvious. Assume that
(︁
)︁
𝜅→𝑓𝑗′ + 𝜆𝑓𝑗′ →
= M𝑧𝑗 for all 𝑗 ≤ 𝑖 − 1. The inductive hypothesis and (2) yield that
𝑧𝑗
(𝜅→𝑓𝑖 + 𝜆𝑓𝑖 → )𝑍 𝑓𝑖
≤(𝑖−1)
= M𝑍 𝑓𝑖
≤(𝑖−1)
. Claim D.4 guarantees that (𝜆→𝑓𝑖 + 𝜅𝑓𝑖 → )𝑧𝑖 = M𝑧𝑖 .
By (3), we have
)︀
(︀
𝜅→𝑓𝑖′ + 𝜆𝑓𝑖′ → 𝑧 = (𝜆→𝑓𝑖 + 𝜅𝑓𝑖 → )𝑧𝑖 = M𝑧𝑖 ,
𝑖
completing the inductive argument.
By (2), it follows that (𝜅→𝑓 + 𝜆𝑓 → )𝑍 𝑓 = M𝑍 𝑓 for all 𝑓 ∈ 𝐹. Claim D.4 guaran≤𝑛
≤𝑛
𝑓
𝑓
tees that {𝑧} is not rational for 𝑓 at Ψ(𝜅, 𝜆) given 𝑍≤𝑛
for any 𝑓 ∈ 𝐹 and 𝑧 ∈ 𝑍≤𝑛
.
Thus, ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑛 , 𝑓𝑛 )) does not block Ψ(𝜅, 𝜆). Since ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑛 , 𝑓𝑛 ))
was an arbitrary rooted proposal sequence, we have proved that Ψ(𝜅, 𝜆) is sequentially stable.
D.4
Completion of the proof of Theorem 1′
We clearly have M − 𝜎 ≤ M − 𝜎 + 𝜌 ∧ 𝜎 ≤ M for all 𝜌, 𝜎 ∈ X . Because 0 ≤ 𝜐 𝑓 ≤
𝜅→𝑓 + 𝜆𝑓 → when 𝜐 𝑓 ∈ 𝐶 𝑓 (𝜅→𝑓 + 𝜆𝑓 → ) , we have M − 𝜆 ≤ 𝜌′ ≤ M for all 𝜅, 𝜆 ∈ X and
(𝜅′ , 𝜆′ , 𝜌′ , 𝜎 ′ ) ∈ Φ(𝜅, 𝜆, 𝜌, 𝜎). It follows that Φ(𝜅, 𝜆, 𝜌, 𝜎) ⊆ X 4 for all (𝜅, 𝜆, 𝜌, 𝜎) ∈ X 4 .
As we saw in Section 6.3, 𝐶 𝑓 is single-valued and continuous for all 𝑓 ∈ 𝐹small .
It follows that Φ is upper hemi-continuous and non-empty compact-convex valued.
Kakutani’s Fixed Point Theorem guarantees that Φ has a fixed point (𝜅, 𝜆, 𝜌, 𝜎).
Proposition 3′ guarantees that Ψ(𝜌 ∧ 𝜎) is a sequentially stable outcome.
47
b =I U b ′
x
y
y
bO
f
z
x
x ′ =z
f
f
(a) Case of b = b ′ .
f
x′
/ b′
(b) Case of b ̸= b ′ .
Figure 2: 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 ′ }.
E
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 2 on page 48.
Before proving Theorems 2 and 3, I define when a mass is a rational deviation
and relate such masses to Definition 6. This result will be useful in streamlining the
arguments for Theorems 2 and 3.
E.1
Preliminaries on rational deviations and stability
Rationally deviating masses are defined by analogy with Definition 6. The key is that
𝛽 ∈ R𝑋
≥0 blocks an outcome 𝒪 if and only if 𝛽𝑓 is rational for 𝑓 at 𝒪 for all 𝑓 ∈ 𝐹 .
(︂ (︁ )︁
)︂
𝑓^
Definition 18. Let 𝑓 ∈ 𝐹 be a firm type and let 𝒪 = 𝜇, 𝐷
be an
𝑓^∈𝐹small
outcome. A mass 𝛽 ∈ R
𝑋𝑓
is rational for 𝑓 at 𝒪 if 𝛽 + 𝜇 ≤ M and:
Case 1: 𝑓 ∈ 𝐹large . We have 𝐶 𝑓 (𝛽𝑓 + 𝜇𝑓 ) ≥ (𝛽 + 𝜇)𝑍(𝛽)𝑓 .
Case 2: 𝑓 ∈ 𝐹small . There exist 𝑌 1 , . . . , 𝑌 𝑘 ⊆ 𝑋𝑓 , 𝑍 1 , . . . , 𝑍 𝑘 ⊆ 𝑍(𝛽), and
𝛾 1 , . . . , 𝛾 𝑘 ∈ R≥0 such that 𝑌 𝑗 ∩ 𝑍 𝑗 = ∅ and 𝑐𝑓 (𝑌 𝑗 ∪ 𝑍 𝑗 ) ⊇ 𝑍 𝑗 for all 𝑗,
𝑘
∑︁
(︀
)︀
𝛾 𝑗 1{𝑌 𝑗 } , 0𝒫(𝑋𝑓 )r{𝑌 𝑗 } ≤ 𝐷𝑓
𝑗=1
and
𝑘
∑︁
(︀
)︀
𝛾 𝑗 1𝑍 𝑗 , 0𝑋𝑓 r𝑍 𝑗 = 𝛽.
𝑗=1
The following lemma refines Proposition 1 to the level of rational deviations.
48
𝑋
Lemma 6. Let 𝑓 ∈ 𝐹 and let 𝒪 be an outcome. If mass 𝛽 ∈ R≥0𝑓 is rational for 𝑓
at 𝒪, then set 𝑍(𝛽) is rational for 𝑓 at 𝒪.
Proof. We divide into cases based on whether 𝑓 is a large firm or a small firm type
to prove the lemma.
′
Case 1: 𝑓 ∈ 𝐹large . Let 𝛽 ′ = 𝛽 + 𝜇𝑓 . Note that 𝜇𝑓 ≤ 𝛽 ′ and 𝜇𝑋𝑓 r𝑍(𝛽) = 𝛽𝑋
.
𝑓 r𝑍(𝛽)
Since 𝛽 is rational for 𝑓 at 𝒪, we have
𝐶 𝑓 (𝛽 ′ )𝑧 = 𝐶 𝑓 (𝛽 + 𝜇)𝑧 = 𝜇𝑧 + 𝛽𝑧 > 𝜇𝑧
for all 𝑧 ∈ 𝑍(𝛽). Thus, 𝑍(𝛽) is rational for 𝑓 at 𝒪.
Case 2: 𝑓 ∈ 𝐹small . Let 𝑧 ∈ 𝑍(𝛽). Let (𝑌 𝑗 , 𝑍 𝑗 , 𝛾 𝑗 )𝑗 be as in Definition 18. By
definition, there exists 𝑗 with 𝛾 𝑗 > 0 and 𝑧 ∈ 𝑍 𝑗 . Note that 𝐷𝑌𝑓 𝑗 ≥ 𝛾 𝑗 > 0,
𝑍 𝑗 ⊆ 𝑍(𝛽), and 𝑧 ∈ 𝑍 𝑗 ⊆ 𝑐𝑓 (𝑌 𝑗 ∪ 𝑍 𝑗 ) by construction. Since 𝑧 ∈ 𝑍(𝛽) was
arbitrary, it follows that 𝑍(𝛽) is rational for 𝑓 at 𝒪.
The cases clearly exhaust all possibilities, completing the proof of the claim.
E.2
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 . For 𝜇 ∈ X𝑓 , let 𝐶 𝑓 (𝜇) = 𝜇Z′𝑓 , 0𝑋𝑓 rZ′𝑓 .
Case 2: 𝑓 ∈ 𝐹small . Let ∅ ≻𝑓 𝑌 if 𝑌 ̸⊆ Z′𝑓 and 𝑌 ≻𝑓 𝑌 ′ if 𝑌 ′ ⊆ 𝑌 ⊆ Z′𝑓 .
Arbitrarily extend ≻𝑓 to a complete, strict preference.
49
Note that the preferences defined above are substitutable, hence in particular substitutable in the sale-direction.
We divide into cases based on whether b = b ′ to define a contract z ∈ 𝑋b ∩ 𝑋b ′
as in Figure 2 on page 48.
Case 1: b = b ′ . Let z = x ′ (as in Figure 2(a) on page 48).
Case 2: b ̸= b ′ . Let z ∈ 𝑋b ∩ 𝑋b ′ be arbitrary (as in Figure 2(b) on page 48).
{︀
}︀
Let 𝐾 = min Mx , Mx ′ , My , Mz . Define the preference of b as follows.
Case 1: b ∈ 𝐹large . For 𝜇 ∈ Xb , let
(︃
b
𝐶 (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 . Let ∅ ≻b 𝑌 if 𝑌 ̸⊆ Zb ∪ {x , y , z }, {y , z } ⊆ 𝑌 , or |𝑌 ∩
{x , y , z }| = 1, and let 𝑌 ≻b ∅ if 𝑌 ̸= ∅ and ∅ ̸≻b 𝑌. Also, let 𝑌 ≻b 𝑌 ′ if 𝑌, 𝑌 ′ ≻b ∅
and 𝑌 ′ ⊆ 𝑌 ∪ {z }. Arbitrarily extend ≻b to a complete, strict preference.
Note that the preference of b exhibits complementarities only between x and y , and
between x and z . Since b = b(x ), it follows that the preference 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
b′
(︁
)︁
𝐶 (𝜇) = 𝜇Zb ′ , min {𝜇x ′ , 𝜇z , 𝐾}{x ′ ,z } , 0𝑋b ′ rZb ′ r{x ′ ,z } .
Subcase 2.2: b ′ ∈ 𝐹small . Let ∅ ≻b ′ 𝑌 if 𝑌 ̸⊆ Zb ′ ∪ {x ′ , z }, or |𝑌 ∩ {x ′ , z }| = 1,
and let 𝑌 ≻b ′ ∅ if 𝑌 ̸= ∅ and ∅ ̸≻b ′ 𝑌. Let 𝑌 ≻b ′ 𝑌 ′ if 𝑌, 𝑌 ′ ≻b ′ ∅ and
𝑌 ′ ⊆ 𝑌. Arbitrarily extend ≻b ′ to a complete, strict preference.
50
Note that the preference of 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. In this case, the preference of b ′ is even fully substitutable.
Define an open set 𝑈 ⊆ R𝐹>0small by
{︃
𝑈=
𝜁 ∈ R𝐹>0small
𝜁 𝑓 < M𝑥 for all 𝑥 ∈ 𝑋 and 𝑓 ∈ 𝐹
|
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 E.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 ′ , yz } be arbitrary, and
let 𝑓 ∈ {b(𝑥), s(𝑥)} r {f } be arbitrary—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|
51
which is positive due to the definition of 𝑈 .
Claim E.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. It follows
that 𝜇𝑓 + 𝛽 ≤ M. Because 𝒪 is individually rational, Claim E.1(b) guarantees that
𝜇x ′ = 𝜇z . As 𝒪 is an outcome, we have 𝜇x , 𝜇x ′ ≤ 𝜁 f . 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
𝛾 𝑗 = 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
∑︁
𝛾
𝑗
(︀
)︀
1{𝑌 𝑗 } , 0𝒫(𝑋𝑓 )r{𝑌 𝑗 } ≤ 𝐷
𝑗=1
𝑓
and
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 𝛾 𝑗 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 }. Since 𝜁 𝑓 − 𝜇𝑥 ≥ |𝑋| · 2|𝑋| · 𝜖 there exists 𝑌 𝑘 ⊆ 𝑋𝑓 r {𝑥} with
52
/ 𝑌 𝑘.
𝐷𝑌𝑓 𝑘 ≥ |𝑋| · 𝛽𝑥 . Since 𝒪 is individually rational, we must have z ∈
Moreover, we have 𝑍 𝑘 ⊆ 𝑐𝑓 (𝑌 𝑘 ⊆ 𝑍 𝑘 ). By construction, we have
𝑘
∑︁
𝛾
𝑗
(︀
)︀
1{𝑌 𝑗 } , 0𝒫(𝑋𝑓 )r{𝑌 𝑗 } ≤ 𝐷
𝑓
and
𝑘
∑︁
(︀
)︀
1𝑍 𝑗 , 0𝑋𝑓 r𝑍 𝑗 = 𝛽.
𝑗=1
𝑗=1
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^
Claim E.3. Suppose that 𝜁 ∈ 𝑈 . If 𝒪 = 𝜇, 𝐷
is a stable or seller𝑓^∈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 E.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 E.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 E.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 𝑍𝑖 = 𝑍𝑓𝑖 → . Let 𝑍𝑘+1 = 𝑍 r 𝑘𝑖=1 𝑍𝑖 and let 𝑓𝑘+1 = 𝑓, so that
((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑘+1 , 𝑓𝑘+1 )) is a seller-initiated proposal sequence.
We claim that ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑘+1 , 𝑓𝑘+1 )) blocks 𝒪. For all 1 ≤ 𝑖 ≤ 𝑘, the
f
set 𝑍𝑖 is rational for 𝑓 at 𝒪 by Claim E.2 and Lemma 6. The set 𝑍f = 𝑍≤𝑘 ∪
𝑍𝑘+1 is rational for f at 𝒪 by Lemma 6. If 𝑍f → = ∅, then we have shown that
((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑘+1 , 𝑓𝑘+1 )) blocks 𝒪 (by Lemma 5). If 𝑧 ∈ 𝑍f → , then Claim E.2
and Lemma 6 guarantee that {𝑧} is rational for b(𝑧) at 𝒪. By Lemma 5, the
53
proposal sequence ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑘+1 , 𝑓𝑘+1 )) blocks 𝒪, so that 𝒪 is not sellerinitiated-stable either.
(︀
)︀
Case 2: x ∈
/ 𝑌. Let 𝑍 = 𝑐𝑓 (Z ∪ {x , x ′ }) r 𝑌 ∪ {z }, and note that {x , x ′ } ⊆ 𝑍.
Let 𝛽 = (𝜉𝑍 , 0𝑋r𝑍 ).
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 E.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
𝑍𝑘+2
⎧
⎨{z } if b ̸= b ′
=
⎩∅
if b = b ′ ,
and let 𝑓𝑘+2 = s(z ), so that ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑘+2 , 𝑓𝑘+2 )) is a seller-initiated proposal
sequence.
f
We claim that ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑘+2 , 𝑓𝑘+2 )) blocks 𝒪. The set 𝑍f = 𝑍𝑘+1 ∪𝑍<(𝑘+1)
is rational for f at 𝒪 by Lemma 6. For all 1 ≤ 𝑖 ≤ 𝑘, the set 𝑍𝑖 is rational for 𝑓 at
𝒪 by Claim E.2 and Lemma 6. The set {z } = 𝑍𝑘+2 is rational for 𝑓𝑘+2 at 𝒪 given
𝑓𝑘+2
b(z )
𝑍≤(𝑘+1)
and rational for b(z ) at 𝒪 given 𝑍≤(𝑘+2) by Claim E.2 and Lemmata 5
and 6 Thus, ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑘+1 , 𝑓𝑘+1 )) blocks 𝒪, so that 𝒪 is not seller-initiatedstable either.
The cases clearly exhaust all possibilities, and thus we have proved the claim.
(︂ (︁ )︁
)︂
𝑓^
Claim E.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.
54
Case 1: b ∈ 𝐹large . Since 𝛿 ≤ 𝜇z , we have
𝐶 b (𝜇b + 𝛽)y = 𝜇y + 𝛽y
by construction. Thus, 𝛽 is rational for b at 𝒪.
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 E.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 . Since 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 ({y }, s(y )) blocks 𝒪 by Lemma 6 and the discussion of the previous two
paragraphs.
Claims E.3 and E.4 together imply that, if 𝜁 ∈ 𝑈, then 𝜇z < 𝜇x ′ in any stable
or seller-initiated-stable outcome. But Claim E.1(b) guarantees that 𝜇z = 𝜇x ′ in any
individually rational outcome. Thus, no stable or seller-initiated-stable outcome can
exist when 𝜁 ∈ 𝑈.
E.3
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 (as in the statement of Theorem 3), and then complete
the proof.
55
Note that since 𝐹small = ∅, outcomes are uniquely determined by their associated allocations. By abuse of notation, I identify outcomes with their associated
allocations.
Defining the preferences of firms other than f Because 𝐶 f is not substitutable
in the sale-direction, there exist 𝜈 ≤ 𝜂 ∈ Xf and x , x ′ ∈ 𝑋f → 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 2(a) on page 48).
Case 2: b ̸= b ′ . Let z ∈ 𝑋b ∩ 𝑋b ′ be arbitrary (as in Figure 2(b) on page 48).
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 (𝜈)
⎟
𝐶 b (𝜇) = ⎜
min
min
{𝜇
,
𝜂
}
−
min
𝜇
,
𝐶
,
𝜇
,
x
x
x
x
y
⎠
⎝
y
{︀
{︀
}︀
}︀
min max min {𝜇x , 𝜂x } − 𝐶 f (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.
56
Case 2: b ̸= b ′ . For 𝜇 ∈ Xb ′ , let
)︁
(︁
′
𝐶 b (𝜇) = (𝜇 ∧ 𝜈b ′ )𝑋b ′ r{x ′ ,z } , min {𝜇x ′ , 𝜇z , 𝜂x ′ }{x ′ ,z } .
′
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.
In this case, the preference of b ′ is even fully substitutable.
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 E.1.
Claim E.5. If 𝜇 ∈ X is an individually rational outcome, then:
(a) 𝜇𝑋r{y ,z }∪{x ′ } ≤ 𝜂𝑋r{y ,z }∪{x ′ } ;
(b) 𝜇x ′ = 𝜇z ; and
(c) 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 E.2.
𝑋
Claim E.6. If 𝜇 ∈ X is an individually rational outcome, 𝑓 ̸= f , and 𝛽 ∈ R≥0𝑓
satisfies
∙ 𝛽f + 𝜇f ≤ 𝜈;
57
∙ 𝛽x + 𝜇x ≤ 𝐶 f (𝜈)x ;
∙ 𝛽z + 𝜇z ≤ 𝜈x ′ ;
∙ 𝑍(𝛽) ⊆ 𝑋f ∪ {z }; 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 E.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 E.3 and E.4, respectively.
Claim E.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 ′ . In light of Claim E.5(a), it follows that 𝜇𝑋r{y ,z }∪{x ′ } ≤ 𝜈𝑋r{y ,z }∪{x ′ } . In
particular, we have 𝜇f ≤ 𝜈. Note also that 𝜇z ≤ 𝜈x ′ by Claim E.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, whence
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 E.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 ), so that ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑘+2 , 𝑓𝑘+2 )) is a
seller-initiated proposal sequence.
58
We claim that ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑘+2 , 𝑓𝑘+2 )) blocks 𝜇. For all 1 ≤ 𝑖 ≤ 𝑘, the set
f
𝑍𝑖 is rational for 𝑓 at 𝜇 by Claim E.6 and Lemma 6. The set 𝑍f = 𝑍≤𝑘 ∪ 𝑍𝑘+1
is rational for f at 𝜇 by Lemma 6. The set 𝑍𝑘+2 is rational for 𝑓𝑘+2 at 𝜇 given
𝑓𝑘+2
by Claim E.6 and Lemmata 5 and 6. We divide into cases to prove that
𝑍≤(𝑘+1)
there exists 1 ≤ 𝑗 ≤ 𝑘 + 2 and 𝑧 ∈ 𝑍𝑗 such that {𝑧} is rational for b(𝑧) at 𝜇 given
b(𝑧)
𝑍≤(𝑘+2) .
Subcase 1.1: 𝑍f → = ∅. We can take any 𝑧 ∈ 𝑍f (by Lemma 5).
Subcase 1.2: x ′ ∈ 𝑍f → . Note that z ∈ 𝑍𝑘+2 . It follows from Claim E.6 and
b(z )
Lemmata 5 and 6 that {z } is rational for b(z ) at 𝜇 given 𝑍≤(𝑘+2) .
Subcase 1.3: 𝑍f → ̸= ∅ and x ′ ∈
/ 𝑍f → . If 𝑧 ∈ 𝑍f → , then Claim E.6 and Lemma 6
guarantee that {𝑧} is rational for b(𝑧) at 𝜇, so that {𝑧} is rational for b(𝑧)
b(𝑧)
at 𝜇 given 𝑍≤(𝑘+2) by Lemma 5.
In all cases, we have shown that there exists 1 ≤ 𝑗 ≤ 𝑘 + 2 and 𝑧 ∈ 𝑍𝑗 such that
b(𝑧)
{𝑧} is rational for b(𝑧) at 𝜇 given 𝑍≤(𝑘+2) . Since the cases exhaust all possibilities,
((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑘+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 E.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 ′
so that ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑘+2 , 𝑓𝑘+2 )) is a seller-initiated proposal sequence.
59
f
We claim that ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑘+2 , 𝑓𝑘+2 )) blocks 𝜇. The set 𝑍f = 𝑍𝑘+1 ∪ 𝑍≤𝑘
is rational for f at 𝜇 by construction. For all 1 ≤ 𝑖 ≤ 𝑘, the set 𝑍𝑖 is rational
for 𝑓 at 𝜇 by Claim E.6 and Lemma 6. The set 𝑍𝑘+2 is rational for 𝑓𝑘+2 at
𝑓𝑘+2
b(z )
and rational for b(z ) at 𝜇 given 𝑍≤(𝑘+2) by Lemma 5. Thus,
𝜇 given 𝑍≤(𝑘+1)
((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑘+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.
Claim E.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 , 0𝑋ry .
We claim that 𝛽 blocks 𝜇. Because 𝜇 is individually rational, we have
𝐶 b (𝜇b + 𝛽)y = 𝜇y + 𝛽y
by construction and Claim E.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
({y }, s(y )) blocks 𝒪 by Lemma 6 and the discussion of the previous paragraph.
Claims E.7 and E.8 together imply that 𝜇x ′ > 𝜈x ′ ≥ 𝜇z in any stable or sellerinitiated-stable outcome. But Claim E.5(b) guarantees that 𝜇x ′ = 𝜇z in any individually rational outcome 𝜇. Thus, no stable or seller-initiated-stable outcome can
exist.
F
F.1
Other proofs omitted from the text
Proof of Proposition 1
Suppose that mass 𝛽 ∈ R𝑋
≥0 blocks outcome 𝒪. By Definition 18, 𝛽𝑓 is rational for
𝑓 at 𝒪 for all 𝑓 ∈ 𝐹 . Lemma 6 guarantees that 𝑍(𝛽)𝑓 is rational for 𝑓 at 𝒪 for all
𝑓 ∈ 𝐹 . Thus, 𝑍(𝛽) blocks 𝒪.
60
F.2
Proof of Lemmata 2 and 2′
We prove the contrapositive. Suppose that outcome 𝒪 is not strongly tree stable.
If 𝒪 is not individually rational, then it is not sequentially stable. Thus, we can
assume that some tree 𝑍 blocks 𝒪. As 𝑍 is a tree, there exists an ordering of firms
𝐹 = {𝑓1 , . . . , 𝑓𝑛 } such that
⃒
⃒
⃒
⃒
⋃︁
⃒
⃒
𝑍𝑓𝑗 ⃒ ≤ 1
⃒𝑍𝑓𝑖 r
⃒
⃒
𝑗<𝑖
⋃︀
for all 1 ≤ 𝑖 ≤ 𝑛. Let 𝑍𝑖 = 𝑍𝑓𝑖 r 𝑗<𝑖 𝑍𝑓𝑗 , so that ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑛 , 𝑓𝑛 )) is a rooted
proposal sequence. Because the firms 𝑓1 , . . . , 𝑓𝑛 are pairwise distinct, we have
(︃
𝑓𝑖
𝑍≤(𝑖−1)
=
)︃
⋃︁
𝑍𝑗
𝑗<𝑖
= 𝑍𝑓𝑖 r 𝑍𝑖
𝑓
for all 1 ≤ 𝑖 ≤ 𝑛.
We claim ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑛 , 𝑓𝑛 )) blocks 𝒪. Because 𝑍 blocks 𝒪, the set 𝑍𝑓𝑖 is
rational for 𝑓𝑖 at 𝒪 for all 1 ≤ 𝑖 ≤ 𝑛. Lemma 5 guarantees 𝑍𝑖 is rational for 𝑓𝑖 at
𝑓𝑖
𝒪 given 𝑍≤(𝑖−1)
for all 1 ≤ 𝑖 ≤ 𝑛. Let 𝑘 be such that 𝑍𝑘 = ∅ and 𝑍𝑓𝑘 ̸= ∅—such
𝑓𝑘
a 𝑘 exists because 𝑍 is a non-empty tree. Note that 𝑍≤𝑛
= 𝑍𝑓𝑘 because the firms
𝑓1 , . . . , 𝑓𝑛 are pairwise distinct. Let 𝑧 ∈ 𝑍𝑓𝑘 be arbitrary. Lemma 5 guarantees that
𝑓𝑘
{𝑧} is rational for 𝑓𝑘 at 𝒪 given 𝑍≤𝑛
. Thus, ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑛 , 𝑓𝑛 )) blocks 𝒪, which
implies that 𝒪 is not sequentially stable.
F.3
Proof of Proposition 2
We prove the contrapositive. Suppose that outcome 𝒪 is not sequentially stable. If
𝒪 is not individually rational, then it is not stable. Thus, we can assume that some
rooted proposal sequence ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑛 , 𝑓𝑛 )) blocks 𝒪.
Write 𝑍𝑖 = {𝑧𝑖 } and let {𝑓𝑖′ } = {b(𝑧𝑖 ), s(𝑧𝑖 )} r {𝑓𝑖 } for 1 ≤ 𝑖 ≤ 𝑛. Let
{︁
}︁
𝑓
𝑍 = 𝑧𝑖 | {𝑧𝑖 } is rational for 𝑓𝑖′ at 𝒪 given 𝑍≤𝑛
.
Note that 𝑍 is non-empty because ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑛 , 𝑓𝑛 )) blocks 𝒪. We will prove
that 𝑍 blocks 𝒪.
𝑓
Claim F.1. Let 𝑓 ∈ 𝐹 and let 𝑧 ∈ 𝑍. If {𝑧} is rational for 𝑓 at 𝒪 given 𝑍≤𝑛
, then
61
{𝑧} is rational for 𝑓 at 𝒪 given 𝑍.
Proof. We divide into cases based on whether 𝑓 is a large firm or a small firm type
to prove that there exists 𝑧 ∈ 𝑊 ⊆ 𝑍 such that 𝑊 is rational for 𝑓 at 𝒪.
Case 1: 𝑓 ∈ 𝐹large . By the definition of rational deviations, there exists 𝜇𝑓 ≤
𝛽 ∈ X𝑓 such that 𝛽𝑋𝑓 r𝑍 𝑓 r{𝑧} = 𝜇𝑋𝑓 r𝑍 𝑓 r{𝑧} and 𝐶 𝑓 (𝛽)𝑧 > 𝜇𝑧 . Let 𝑊 = {𝑥 ∈
≤𝑛
≤𝑛
𝑋 | 𝐶 𝑓 (𝛽)𝑥 > 𝜇𝑥 }, and note that 𝑧 ∈ 𝑊. By the irrelevance of rejected contracts
condition, we can assume that 𝛽𝑋𝑓 r𝑊 = 𝜇𝑋𝑓 r𝑊 , so that 𝑊 is rational for 𝑓 at 𝒪.
𝑓
Case 2: 𝑓 ∈ 𝐹small . By the definition of rational deviations, there exists 𝑊 ⊆ 𝑍≤𝑛
and 𝑌 ⊆ 𝑋𝑓 r {𝑧} such that 𝐷𝑌𝑓 > 0 and 𝑧 ∈ 𝑐𝑓 (𝑊 ∪ 𝑌 ). By the irrelevance of
rejected contracts condition, we can assume that 𝑊 ⊆ 𝑐𝑓 (𝑊 ∪ 𝑌 ) r 𝑌, so that 𝑊
is rational for 𝑓 at 𝒪.
The cases clearly exhaust all possibilities. Lemma 5 guarantees that {𝑤} is rational
𝑓
for 𝑓 at 𝒪 given 𝑍≤𝑛
for all 𝑤 ∈ 𝑊, so that 𝑊 ⊆ 𝑍. In light of Lemma 5, it follows
that {𝑧} is rational for 𝑓 at 𝒪 given 𝑍, as desired.
𝑓
First, we prove that {𝑧𝑖 } is rational for 𝑓 at 𝒪 given 𝑍≤𝑛
for all 𝑧𝑖 ∈ 𝑍 and
′
′
𝑓 = 𝑓𝑖 , 𝑓𝑖 . This is true by assumption for 𝑓 = 𝑓𝑖 . Note that {𝑧𝑖 } is rational for 𝑓𝑖 at
𝑓𝑖
𝑓𝑖
by Lemma 5. Thus,
𝒪 given 𝑍≤(𝑖−1)
, so that {𝑧𝑖 } is rational for 𝑓𝑖 at 𝒪 given 𝑍≤𝑛
𝑓
{𝑧𝑖 } is rational for 𝑓 at 𝒪 given 𝑍≤𝑛 for all 𝑧𝑖 ∈ 𝑍 and 𝑓 = 𝑓𝑖 , 𝑓𝑖′ .
Claim F.1 guarantees {𝑧𝑖 } is rational for 𝑓 at 𝒪 given 𝑍 for all 𝑧𝑖 ∈ 𝑍 and
𝑓 = 𝑓𝑖 , 𝑓𝑖′ . It follows from Lemma 5 that 𝑍 blocks 𝒪, so that 𝒪 is not stable.
F.4
Proof of Lemmata 3 and 3′
We prove the contrapositive. Suppose that outcome 𝒪 is not strongly acyclically
stable. If 𝒪 is not individually rational, then it is not seller-initiated-stable. Thus,
we can assume that some acyclic set 𝑍 ⊆ 𝑋 blocks 𝒪. As 𝑍 is acyclic, there exists
an ordering of firms 𝐹 = {𝑓1 , . . . , 𝑓𝑛 } such that 𝑍𝑓𝑖 →𝑓𝑗 = ∅ whenever 𝑖 > 𝑗. Define a
seller-initiated proposal sequence ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑛 , 𝑓𝑛 )) by 𝑍𝑖 = 𝑍𝑓𝑖 → for 1 ≤ 𝑖 ≤ 𝑛.
Because the firms 𝑓1 , . . . , 𝑓𝑛 are pairwise distinct, we have
(︃
𝑓𝑖
𝑍≤(𝑖−1)
=
)︃
⋃︁
𝑍𝑗
𝑗<𝑖
= 𝑍 𝑓𝑖 r 𝑍 𝑖
𝑓
62
for all 1 ≤ 𝑖 ≤ 𝑛.
We claim that ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑛 , 𝑓𝑛 )) blocks 𝒪. Because 𝑍𝑓𝑖 is rational for 𝑓𝑖 at
𝑓𝑖
𝒪 for all 𝑖, Lemma 5 guarantees that the set 𝑍𝑖 is rational for 𝑓𝑖 at 𝒪 given 𝑍≤(𝑖−1)
.
Let 𝑘 be such that 𝑍𝑘 = ∅ and 𝑍𝑓𝑘 ̸= ∅—such a 𝑘 exists because 𝑍 is non-empty and
𝑓𝑘
= 𝑍𝑓𝑘 because the firms 𝑓1 , . . . , 𝑓𝑛 are pairwise distinct. Let
acyclic. Note that 𝑍≤𝑛
𝑓𝑘
.
𝑧 ∈ 𝑍𝑓𝑘 be arbitrary. Lemma 5 guarantees that {𝑧} is rational for 𝑓𝑘 at 𝒪 given 𝑍≤𝑛
Thus, ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑛 , 𝑓𝑛 )) blocks 𝒪, which implies that 𝒪 is not seller-initiatedstable.
F.5
Proof of Lemmata 4 and 4′
The following claim shows that, under substitutability in the sale-direction, if a set
𝑍 ′ of sales is rational, then any subset of 𝑍 ′ is rational.
Claim F.2. Let 𝑓 ∈ 𝐹 be a firm type, let 𝒪 be an outcome, and let 𝑍 ⊆ 𝑍 ′ ⊆ 𝑋𝑓 →
and 𝑊 ⊆ 𝑋 be sets of contracts. Suppose that either 𝑓 ∈ 𝐹large and 𝐶 𝑓 is expansionsubstitutable in the sale-direction or 𝑓 ∈ 𝐹small and 𝑐𝑓 is expansion-substitutable in
the sale-direction. If 𝑍 ′ is rational for 𝑓 at 𝒪 given 𝑊 , then 𝑍 is rational for 𝑓 at
𝒪 given 𝑊 .
(︂ (︁ )︁
)︂
𝑓^
Proof. Write 𝒪 = 𝜇, 𝐷
. We divide into cases based on whether 𝑓 is a
𝑓^∈𝐹small
large firm or a small firm type to complete the proof.
Case 1: 𝑓 ∈ 𝐹large . Let 𝑧 ∈ 𝑍 be arbitrary. Because 𝑍 ′ is rational for 𝑓 at 𝒪
′
given 𝑊 ′ , there exists 𝜇𝑓 ≤ 𝛽 ′ ∈ X𝑓 such that 𝛽𝑋
′ = 𝜇𝑋𝑓 r𝑊 r𝑍 ′ and 𝜈𝑧 > 𝜇𝑧
𝑓 r𝑊 r𝑍
𝑓
′
′
for all 𝜈 ∈ 𝐶 (𝛽 ). Let 𝛽 = 𝛽𝑊𝑓 ∪𝑍𝑓 + 𝜇𝑋𝑓 r𝑊 r𝑍 , so that 𝜇𝑓 ≤ 𝛽 ≤ 𝛽 ′ .
′
′
′
Because 𝛽𝑋
′ = 𝜇𝑋𝑓 r𝑊 r𝑍 ′ and 𝑍 ⊆ 𝑋𝑓 → , we have 𝛽→𝑓 = 𝛽→𝑓 . Note also
𝑓 r𝑊 r𝑍
that 𝛽𝑧 = 𝛽𝑧′ . It follows from the expansion-substitutability in the sale-direction
of 𝐶 𝑓 that 𝜈𝑧 > 𝜇𝑧 for all 𝜈 ∈ 𝐶 𝑓 (𝛽). Since 𝑧 ∈ 𝑍 was arbitrary, it follows that 𝑍
is rational for 𝑓 at 𝒪 given 𝑊 .
Case 2: 𝑓 ∈ 𝐹small . Let 𝑧 ∈ 𝑍 ′ be arbitrary. Because 𝑍 ′ is rational for 𝑓 at 𝒪
given 𝑊 ′ , there exists 𝑇 ′ ⊆ 𝑊 ∪ 𝑍 ′ and 𝑌 ⊆ 𝑋𝑓 r {𝑧} such that 𝐷𝑌𝑓 > 0 and
𝑧 ∈ 𝑌 ′ for all 𝑌 ′ ∈ 𝑐𝑓 (𝑇 ′ ∪ 𝑌 ). Let 𝑇 = 𝑇 ′ ∩ (𝑊 ∪ 𝑍).
′
Because 𝑍 ′ ⊆ 𝑋𝑓 → , we have 𝑇→𝑓 = 𝑇→𝑓
. It follows from the expansion-sub𝑓
stitutability in the sale-direction of 𝑐 that 𝑧 ∈ 𝑌 ′ for all 𝑌 ′ ∈ 𝑐𝑓 (𝑇 ∪ 𝑌 ) . Since
𝑧 ∈ 𝑍 was arbitrary, 𝑍 is rational for 𝑓 at 𝒪 given 𝑊 .
63
The cases clearly exhaust all possibilities, which completes the proof of the lemma.
We prove the contrapositive of Lemma 4′ . Suppose that outcome 𝒪 is not sellerinitiated-stable. If 𝒪 is not individually rational, then it is not sequentially stable.
Thus, we can assume that a seller-initiated proposal sequence ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑛 , 𝑓𝑛 ))
blocks 𝒪.
}︀
{︀
∑︀
𝑖
. Define a
Let 𝑚𝑖 = |𝑍𝑖 | and let 𝑀𝑖 = 𝑖𝑗=1 𝑚𝑖 . For each 𝑖, let 𝑍𝑖 = 𝑧1𝑖 , . . . , 𝑧𝑚
𝑖
(︀ ′ ′
(︀ ′
)︀)︀
′
rooted proposal sequence (𝑍1 , 𝑓1 ) , . . . , 𝑍𝑀𝑛 , 𝑓𝑀𝑛 by
(𝑍𝑗′ , 𝑓𝑗′ )
(︁{︁
}︁
)︁
𝑖(𝑗)
=
𝑧𝑗−𝑀𝑖(𝑗)−1 , 𝑓𝑖(𝑗)
for 1 ≤ 𝑗 ≤ 𝑀𝑛 , where 1 ≤ 𝑖(𝑗) ≤ 𝑛 is the unique integer 𝑖 satisfying 𝑀𝑖−1 < 𝑗 ≤ 𝑀𝑖 .
𝑓
Note that 𝑍≤𝑛
= (𝑍 ′ )𝑓≤𝑀𝑛 for all 𝑓 ∈ 𝐹.
)︀)︀
(︀
(︀ ′
′
,
𝑓
blocks 𝒪. As ((𝑍1 , 𝑓1 ) , . . . , (𝑍𝑛 , 𝑓𝑛 ))
We claim that (𝑍1′ , 𝑓1′ ) , . . . , 𝑍𝑀
𝑀
𝑛
𝑛
blocks 𝒪 and 𝑍𝑖 ⊆ 𝑍𝑓𝑖 → , Claim F.2 guarantees that 𝑍𝑗′ is rational for 𝑓𝑗′ at 𝒪 given
𝑓′
𝑓
𝑗
for all 1 ≤ 𝑗 ≤ 𝑀𝑛 . Because 𝑍≤(𝑖(𝑗)−1)
= (𝑍 ′ )𝑓≤𝑀𝑖(𝑗)−1 ⊆ (𝑍 ′ )𝑓≤𝑗 for all 𝑓 ∈ 𝐹
𝑍≤(𝑖(𝑗)−1)
𝑓
and 1 ≤ 𝑗 ≤ 𝑀𝑛 , Lemma 5 implies that 𝑍𝑗′ is rational for 𝑓𝑗′ at 𝒪 given (𝑍 ′ )≤𝑗𝑗 for
𝑓
all 1 ≤ 𝑗 ≤ 𝑀𝑛 . Let 𝑓 ∈ 𝐹 and 𝑧 ∈ 𝑍≤𝑛
be such that {𝑧} is rational for 𝑓 at 𝒪
𝑓
given 𝑍≤𝑛
. Then, 𝑧 ∈ (𝑍 ′ )𝑓≤𝑀𝑛 and {𝑧} is rational for 𝑓 at 𝒪 given (𝑍 ′ )𝑓≤𝑀𝑛 . Thus,
(︀ ′ ′
(︀ ′
)︀)︀
′
(𝑍1 , 𝑓1 ) , . . . , 𝑍𝑀
, 𝑓𝑀
blocks 𝒪, which implies that 𝒪 is not sequentially stable.
𝑛
𝑛
F.6
Proof of Theorem 4
The proof is similar to the proof of Theorem 5 in Hatfield and Kominers (2012).
Let 𝐹 ′ = {f1 , . . . , f𝑛 , f ′ }. Let x𝑖 ∈ 𝑋f𝑖 →f𝑖+1 be arbitrary for 𝑖 = 1, . . . , 𝑛, and
let y ∈ 𝑋f1 ∩ 𝑋f ′ be arbitrary. By redirecting all the contracts in the economy if
necessary, we can assume that y ∈ 𝑋f1 →f ′ .55
Define the preferences of firm types 𝑓 ∈ 𝐹 as follows.
Case 1: 𝑓 ∈
/ 𝐹 ′ . If 𝑓 ∈ 𝐹large , then let 𝐶 𝑓 (𝜇) = 0 for all 𝜇 ∈ X𝑓 . If 𝑓 ∈ 𝐹small ,
then let ∅ ≻𝑓 𝑌 for all ∅ =
̸ 𝑌 ⊆ 𝑋𝑓 .
55
Although substitutability in the sale-direction is not in general preserved by redirecting all
contracts, the preferences that we construct will be substitutable in the sale-direction will remain
substitutable in the sale-direction upon redirecting all contracts. Indeed, as discussed in Footnote 47
in Section 7, the preferences of all firms can be taken to be fully substitutable, hence in particular
substitutable in both directions.
64
Case 2: 𝑓 = f1 . If f1 ∈ 𝐹large , then let 𝐶 f1 be given by
(︃
f1
𝐶 (𝜇) =
)︃
{︀
}︀
{︀
}︀
min 𝜇x1 , 𝜇x𝑛 + 𝜇y x , min max{𝜇x1 − 𝜇y , 0}, 𝜇x𝑛 x ,
𝑛
1
{︀
}︀
min 𝜇x1 , 𝜇y y , 0𝑋f1 r{x1 ,x𝑛 ,y }
for all 𝜇 ∈ Xf𝑗 . If f1 ∈ 𝐹small , then let ≻f1 be given by
≻f1 : {x1 , y } ≻f1 {x1 , x𝑛 } ≻f1 ∅.
Case 3: 𝑓 = f𝑗 with 2 ≤ 𝑗 ≤ 𝑛. If f𝑗 ∈ 𝐹large , then let 𝐶 f𝑗 be given by
)︁
(︁
{︀
}︀
𝐶 f𝑗 (𝜇) = min 𝜇x𝑗−1 , 𝜇x𝑗 {x𝑗−1 ,x𝑗 } , 0𝑋f𝑗 r{x𝑗−1 ,x𝑗 }
for all 𝜇 ∈ Xf𝑗 . If f𝑗 ∈ 𝐹small , then let ≻f𝑗 be given by ≻f𝑗 : {x𝑗−1 , x𝑗 } ≻f𝑗 ∅.
′
Case 4: 𝑓 = f ′ . If f ′ ∈ 𝐹large , then let 𝐶 f be given by
)︁
(︁
′
𝐶 f (𝜇) = 𝜇y , 0𝑋f ′ r{y }
for all 𝜇 ∈ Xf ′ . If f ′ ∈ 𝐹small , then let ≻f ′ be given by ≻f ′ : {y } ≻f ′ ∅.
Note that 𝐶 𝑓 (resp. 𝑐𝑓 ) is substitutable in the sale-direction for all 𝑓 ∈ 𝐹large (resp.
𝑓 ∈ 𝐹small ) by construction.56
The following claim shows that in any individually rational outcome, the same
quantity of each of the contracts x𝑖 must be traded and y cannot be traded.
(︂ (︁ )︁
)︂
𝑓^
Claim F.3. If 𝒪 = 𝜇, 𝐷
is an individually rational outcome, then
𝑓^∈𝐹small
𝜇x1 = 𝜇x2 = · · · = 𝜇x𝑛 and 𝜇y = 0.
Proof. To prove the first part of the claim, it suffices to show that 𝜇x𝑗−1 = 𝜇x𝑗 for all
2 ≤ 𝑗 ≤ 𝑛. This assertion is immediate from the individual rationality of 𝒪 for f𝑗 .
To prove the second part of the claim, note that 𝜇x1 = 𝜇x𝑛 + 𝜇y due to the
individual rationality of 𝒪 for f1 . Thus, the second part of the claim follows from the
first part.
56
In fact, 𝐶 𝑓 (resp. 𝑐𝑓 ) is even fully substitutable for all 𝑓 ∈ 𝐹large (resp. 𝑓 ∈ 𝐹small ) by
construction.
65
(︂
Let 𝒪 =
(︁
𝜇, 𝐷
)︂
)︁
𝑓^
𝑓^∈𝐹small
be any individually rational outcome. Let
{︃
}︃
𝜖 = min min M𝑥 , inf 𝐷𝑌𝑓
𝑥∈𝑋
𝑓
𝐷𝑌
>0
.
Note that 𝜇y = 0 by Claim F.3. To prove that 𝒪 is blocked, we divide into cases
based on whether 𝜇x1 = 0.
Case 1: 𝜇x𝑛 = 0. By Claim F.3, we have x𝑖 = 0 for all 1 ≤ 𝑖 ≤ 𝑛. It is straight(︀
)︀
forward to verify that 𝜖{x1 ,...,x𝑛 } , 0𝑋r{x1 ,...,x𝑛 } blocks 𝒪.
(︀
)︀
Case 2: 𝜇x𝑛 > 0. It is straightforward to verify that 𝜖y , 0𝑋r{y } blocks 𝒪.
The cases are clearly exhaustive, and thus we have proved that 𝒪 is not stable. Since
𝒪 was an arbitrary individually rational outcome, the economy does not have a stable
outcome.
F.7
Proof of Proposition 4
Note that we can assume that 𝑋s = Z without loss of generality—indeed, s always
rejects all contracts outside Z. Define b′ , s′ : 𝑋 → 𝐹 by
⎧
⎨(b(𝑥), s(𝑥)) if s(𝑥) ̸= s
.
(b′ (𝑥), s′ (𝑥)) =
⎩(s(𝑥), b(𝑥)) if s(𝑥) = s .
For the remainder of the proof, we write
𝑊𝑓 → = {𝑥 ∈ 𝑊 | s′ (𝑥) = 𝑓 }
for all 𝑊 ⊆ 𝑋 by abuse of notation, and attach analogous meanings to 𝑊→𝑓 , 𝜇𝑓 → ,
and 𝜇→𝑓 . The following claim shows that, under the assumptions of Proposition 4,
all firms’ preferences are substitutable in the sale-direction in the modified economy.
Claim F.4. For all 𝑓 ∈ 𝐹large , the choice function 𝐶 𝑓 is substitutable in the sale-direction. For all 𝑓 ∈ 𝐹small , the choice function 𝑐𝑓 is substitutable in the sale-direction.
Proof. We divide into cases based on whether 𝑓 is in {s }, 𝑆 r {s }, or 𝐵 to prove the
claim.
66
Case 1: 𝑓 = s . Note that 𝑋s → = ∅ by construction. Thus, the claim is vacuously
true for 𝑓 = s .
Case 2: 𝑓 ∈ 𝑆 r {s }. The substitutability of 𝐶 𝑓 (resp. 𝑐𝑓 ) implies the substitutability in the sale-direction of 𝐶 𝑓 (resp. 𝑐𝑓 ) if 𝑓 ∈ 𝐹large (resp. 𝑓 ∈ 𝐹small ).
Case 3: 𝑓 ∈ 𝐵. Because |b(Z)| = |Z|, we have |𝑋𝑓 → | ≤ 1 by construction, so
that the claim is vacuously true.
The cases clearly exhaust all possibilities, and thus we have proved the claim.
The modified economy is acyclic by construction. By Corollary 4 and Claim F.4,
the modified economy has a strongly stable outcome 𝒪. But 𝒪 is a strongly stable
outcome in the original network because changing the directions of contracts does not
affect the set of strongly stable outcomes.
F.8
Proofs of Corollaries 5 and 6
To prove Corollaries 5 and 6, 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 define preference ≻𝑥 by
≻𝑥 : r(𝑥) ≻𝑥 ∅.
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 F.5. Let 𝜇 , 𝐷
be an individually rational outcome in the auxil′
𝑓^∈𝐹small
iary economy and let 𝑥 ∈ 𝑋. We have 𝜇′r = 𝜇′r ′ for all r , r ′ ∈ a(𝑥).
Proof. Note that r(𝑥) and ∅ are the only individually rational sets of contracts for 𝑥
in the auxiliary economy.
(︂ (︁ )︁
)︂
′
𝑓^
Given an individually rational outcome 𝜇 , 𝐷
in the auxiliary econ′
𝑓^∈𝐹small
omy, let
(︂(︂
𝜏
′
(︁
𝜇, 𝐷
)︂)︂
)︁
𝑓^
′
𝑓^∈𝐹small
(︂ (︁ )︁
^
= 𝜇, 𝐷𝑓
𝑓^∈𝐹small
67
)︂
,
where 𝜇𝑥 = 𝜇′r for all r ∈ a(𝑥). The above formula yields a well-defined outcome by
Claim F.5. The following claim shows that blocks of 𝜏 (𝒪′ ) induce blocks of 𝒪′ .
Claim F.6. Let 𝒪′ be an individually rational outcome in the auxiliary economy.
Then, 𝒪 = 𝜏 (𝒪′ ) is an individually rational outcome. If 𝑍 ⊆ 𝑋 blocks 𝒪, then r(𝑍)
blocks 𝒪′ .
Proof. We have already shown that 𝒪 is an outcome (by Claim F.5). The individual rationality of 𝒪 is immediate from the definitions of the choice functions in the
auxiliary economy.
Suppose that 𝑍 ⊆ 𝑋 blocks 𝜏 (𝒪′ ) . Let 𝑍 ′ = r(𝑍). It is clear that 𝑍𝑓′ is rational
for 𝑓 at 𝒪 for all 𝑓 ∈ 𝐹 . Note that 𝑍𝑥′ = ∅ for 𝑥 ∈ 𝑋 r 𝑍, so that in particular 𝑍𝑥′ is
rational for 𝑥 at 𝒪 for all 𝑥 ∈ 𝑋 r 𝑍. It remains to prove that 𝑍𝑧′ is rational for 𝑥 at
𝒪 for all 𝑧 ∈ 𝑍.
Let 𝑧 ∈ 𝑍 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 < M𝑧 ,
(︁ )︁
^
where 𝒪 = 𝜇, 𝐷𝑓
.
𝑓^∈𝐹small
Case 1: 𝑓 ∈ 𝐹large . Because 𝑍𝑓 is rational for 𝑓 at 𝒪, there exists 𝜇𝑓 ≤ 𝛽 ∈ X𝑓
such that 𝛽𝑋𝑓 r𝑍𝑓 = 𝜇𝑋𝑓 r𝑍𝑓 and 𝐶 𝑓 (𝛽)𝑧 > 𝜇𝑧 . Because 𝐶 𝑓 (𝛽) ∈ X𝑓 , we have
M𝑧 ≥ 𝐶 𝑓 (𝛽)r > 𝜇𝑧 .
Case 2: 𝑓 ∈ 𝐹small . Because 𝑍𝑓 is rational for 𝑓 at 𝒪, there exists 𝑊 ′ ⊆ 𝑍 and
𝑌 ⊆ 𝑋𝑓 r {𝑧} such that 𝐷𝑌𝑓 > 0 and r ∈ 𝑐𝑓 (𝑊 ′ ∪ 𝑌 ). Because 𝒪 is an outcome,
it follows that 𝜇𝑧 < 𝜁 𝑓 ≤ M𝑧 .
The cases clearly exhaust all possibilities, and thus we(︂have proved that)︂ 𝜇𝑧 < M𝑧 .
(︁ )︁
^
It follows that 𝜇′r < 𝜁 𝑧 for all r ∈ a(𝑧), where 𝒪′ = 𝜇′ , 𝐷𝑓
. Because
′
𝑓^∈𝐹small
𝑋𝑧′ = 𝑍𝑧′ and ∅ are the only individually rational sets for 𝑧 in the auxiliary economy,
we have (𝐷)𝑧∅ > 0, so that 𝑍𝑧′ is rational for 𝑧 at 𝒪′ (in the auxiliary economy). Since
𝑧 ∈ 𝑍 was arbitrary, it follows that 𝑍 ′ blocks 𝒪′ .
Note that r(𝑍) is a tree in the auxiliary economy whenever 𝑍 is a tree in the
original economy by Definitions 8 and 16. Thus, Corollaries 5 and 6 follow from
Corollaries 1 and 4 and Claim F.6.
68
F.9
Proof of Proposition 5
As in Che et al. (2013), let − ∨ − denote componentwise
maximum.
)︂
(︂ (︁ )︁
𝑓^
be an outcome. If 𝒪 is
We prove the contrapositive. Let 𝒪 = 𝜇, 𝐷
𝑓^∈𝐹small
not individually rational, then it is not stable. Thus, we can assume that there exists
∑︀
𝑏 ∈ 𝐹large , 𝛽 ∈ X𝑏 such that 𝛽 ∈ 𝐶 𝑏 (𝛽 ∨ 𝜇𝑏 ), 𝜇𝑏 ∈
/ 𝐶 𝑏 (𝛽 ∨ 𝜇𝑏 ), and {(𝑠,𝑏)}≻𝑌 𝐷𝑌𝑠 ≥ 𝛽
for all 𝑠 ∈ 𝐹small .
Because 𝐶 𝑏 (𝛽 ∨ 𝜇𝑏 ) is non-empty and compact and the partial order on X is
continuous, there exists 𝛽 ′ ∈ 𝐶 𝑏 (𝛽 ∨ 𝜇𝑏 ) such that (𝛽 ′ − 𝜇) ∨ 0 is minimal in
{(𝛾 − 𝜇) ∨ 0 | 𝛾 ∈ 𝐶 𝑏 (𝛽 ∨ 𝜇𝑏 )}
by Theorem 1 in Ward (1954). As 𝜇𝑏 ∈
/ 𝐶 𝑏 (𝛽 ∨ 𝜇𝑏 ), we have 𝜇𝑏 ∈
/ 𝐶 𝑏 (𝛽 ′ ∨ 𝜇𝑏 ) (by
the revealed preference property), so that 𝛽 ′ ̸≤ 𝜇𝑏 . Let 𝑍 = {𝑥 | 𝛽𝑥′ > 𝜇𝑥 }, which is
non-empty because 𝛽 ′ ̸≤ 𝜇. By the revealed preference property, we have 𝜈𝑧 ≥ 𝛽𝑧′ for
all 𝜈 ∈ 𝐶 𝑏 (𝛽 ′ ∨ 𝜇𝑏 ) and 𝑧 ∈ 𝑍. It follows that 𝜈𝑧 > 𝜇𝑧 for all 𝜈 ∈ 𝐶 𝑏 (𝛽 ′ ∨ 𝜇𝑏 ) and
∑︀
𝑧 ∈ 𝑍. Because {(𝑠,𝑏)}≻𝑌 𝐷𝑌𝑠 ≥ 𝛽 for all 𝑠 ∈ 𝐹small , the set 𝑍 is rational for 𝑠 at 𝒪
for all 𝑠 ∈ 𝐹small . Thus, 𝑍 blocks 𝒪, which implies that 𝒪 is not strongly stable.
69
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