Matching in Networks with Bilateral Contracts

Hatfield, John William; Kominers, Scott Duke

doi:10.2139/ssrn.1574473

Cited by 37 publications

(94 citation statements)

References 31 publications

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“…First, we define full substitutability in the language of choices, adapting and merging the Ostrovsky () same‐side substitutability and cross‐side complementarity conditions. Setting conditions on how each agent's optimal choice changes as that agent's opportunity set expands originated in the matching literature, where it is natural to consider an expansion in the set of available trades and, thus, an expansion in the set of available contracts (see Ostrovsky 2008, Westkamp 2010, Hatfield and Kominers 2012, Hatfield et al 2013, and Hatfield, Kominers, and Westkamp 2019). In choice language, we say that a choice correspondence

C_{i}

is fully substitutable if, when attention is restricted to sets of contracts for which

C_{i}

is single‐valued, whenever the set of options available to i on one side expands, i rejects a larger set of contracts on that side (same‐side substitutability) and selects a larger set of contracts on the other side (cross‐side complementarity).…”

Section: Substitutability Conceptsmentioning

confidence: 99%

“…Our third definition is essentially a reformulation of Definition , using a convenient vector notation due to Hatfield and Kominers (). For each agent i , for any set of trades

normalΨ \subseteq {normalΩ}_{i}

, define the (generalized) indicator function

e_{i} false(normalΨ false) \in {false{- 1, 0, 1 false}}^{{normalΩ}_{i}}

to be the vector with component

e_{i, ω} false(normalΨ false) = 1

for each upstream trade

ω \in {normalΨ}_{\to i}

e_{i, ω} false(normalΨ false) = - 1

for each downstream trade

ω \in {normalΨ}_{i \to}

, and

e_{i, ω} false(normalΨ false) = 0

for each trade

ω \notin normalΨ

.…”

Section: Substitutability Conceptsmentioning

confidence: 99%

“…Ostrovsky () generalized the choice‐theoretic substitutability conditions to the context of supply chain networks by introducing a pair of related assumptions, same‐side substitutability and cross‐side complementarity, which impose two constraints: First, when an agent's opportunity set on one side of the market expands, that agent does not choose any options previously rejected from that side of the market; second, when an agent's opportunity set on one side of the market expands, that agent does not reject any options previously chosen from the other side of the market. Ostrovsky () and Hatfield and Kominers () showed that under same‐side substitutability and cross‐side complementarity, a stable outcome always exists if the contractual set has a supply chain structure (see also Fleiner et al 2017).…”

Section: Introductionmentioning

confidence: 99%

“…Hatfield, Jagadeesan, and Kominers () showed that same‐side substitutability and cross‐side complementarity are together equivalent to the assumption of weak quasisubmodularity of the indirect utility function—an adaptation of submodularity to the setting without transfers Sun and Yang. () showed that the gross substitutability and complementarity condition is equivalent to submodularity of the indirect utility function.…”

Section: Introductionmentioning

confidence: 99%

“…We also extend the gross substitutes and complements condition of Sun and Yang () to settings with fully general intermediary preferences. Then, using a notation originally introduced by Hatfield and Kominers (), we give a reinterpretation of both the choice‐ and demand‐theoretic substitutability conditions, in terms of indicator functions that track the underlying goods in the economy; in the process, we show how to “fold” the general economy we consider into a Kelso and Crawford () economy in which the underlying goods are (gross) substitutes . We show moreover that all of the previously established connections between gross substitutability and more technical conditions such as submodularity and the single improvement property extend to full substitutability as well (for suitable generalizations of the technical conditions).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Full substitutability

Hatfield

Kominers

Nichifor

et al. 2019

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Various forms of substitutability are essential for establishing the existence of equilibria and other useful properties in diverse settings such as matching, auctions, and exchange economies with indivisible goods. We extend earlier models' definitions of substitutability to settings in which each agent can be both a buyer in some transactions and a seller in others, and show that all these definitions are equivalent. We then introduce a new class of substitutable preferences that allows us to model intermediaries with production capacity. We also prove that substitutability is preserved under economically important transformations such as trade endowments, mergers, and limited liability.

show abstract

C_{i}

is fully substitutable if, when attention is restricted to sets of contracts for which

C_{i}

Section: Substitutability Conceptsmentioning

confidence: 99%

“…Our third definition is essentially a reformulation of Definition , using a convenient vector notation due to Hatfield and Kominers (). For each agent i , for any set of trades

normalΨ \subseteq {normalΩ}_{i}

, define the (generalized) indicator function

e_{i} false(normalΨ false) \in {false{- 1, 0, 1 false}}^{{normalΩ}_{i}}

to be the vector with component

e_{i, ω} false(normalΨ false) = 1

for each upstream trade

ω \in {normalΨ}_{\to i}

e_{i, ω} false(normalΨ false) = - 1

for each downstream trade

ω \in {normalΨ}_{i \to}

, and

e_{i, ω} false(normalΨ false) = 0

for each trade

ω \notin normalΨ

.…”

Section: Substitutability Conceptsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Full substitutability

Hatfield

Kominers

Nichifor

et al. 2019

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Lone wolves in competitive equilibria

2020

Self Cite

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This paper develops a class of equilibrium-independent predictions of competitive equilibrium with indivisibilities. Specifically, we prove an analogue of the "Lone Wolf Theorem" of classical matching theory for the Baldwin and Klemperer (Econometrica 87(3):867-932, 2019) model of exchange economies with transferable utility, showing that any agent who does not participate in trade in some competitive equilibrium must receive her autarky payoff in every competitive equilibrium. Our results extend to approximate equilibria and to settings in which utility is only approximately transferable.

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Multi-sided assignment games on m-partite graphs

Atay

Núñez

2019

Ann Oper Res

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We consider a multi-sided assignment game with the following characteristics: (a) the agents are organized in m sectors that are connected by a graph that induces a weighted m-partite graph on the set of agents, (b) a basic coalition is formed by agents from different connected sectors, and (c) the worth of a basic coalition is the addition of the weights of all its pairs that belong to connected sectors. We provide a sufficient condition on the weights to guarantee balancedness of the related multi-sided assignment game. Moreover, when the graph on the sectors is cycle-free, we prove the game is strongly balanced and the core is fully described by means of the cores of the underlying two-sided assignment games associated with the edges of this graph. As a consequence, the complexity of the computation of an optimal matching is reduced and existence of optimal core allocations for each sector of the market is guaranteed.

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Matching in Networks with Bilateral Contracts

Cited by 37 publications

References 31 publications

Full substitutability

Full substitutability

Lone wolves in competitive equilibria

Multi-sided assignment games on m-partite graphs

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