A cumulative offer process for supply chain networks
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A cumulative offer process for supply chain networks Juan F. Fung1
· Chia-Ling Hsu2
Received: 12 August 2020 / Accepted: 9 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We introduce a version of the Cumulative Offer Algorithm of Hatfield and Milgrom (Am Econ Rev 95(4):913–935, 2005) modified for the supply chain networks setting of Ostrovsky (Am Econ Rev 98(3):897–923, 2008). The algorithm provides an alternative proof for existence of a chain stable network. Moreover, we show that when choice functions satisfy same-side substitutes and cross-side complements, and the Law of Aggregate Demand and Supply, then contracts satisfy Irrelevance of Rejected Contracts condition, a condition implicitly assumed in Ostrovsky (2008). Keywords Supply chain networks · Matching with contracts · Market design JEL Classification C78 · D63 · D85 · L14
1 Introduction The matching with contracts model of Hatfield and Milgrom (2005) is one of the most important contributions to the literature on two-sided matching. Hatfield and Milgrom (2005) provide a general and elegant framework that subsumes many of the previously separate specialized matching models—both with and without endogenously determined salaries, as well as more general relationships specified by a bilateral contract for each match. Moreover, it has led to various important applications in settings traditionally outside the scope of two-sided matching. The supply chain networks model of Ostrovsky (2008) is a significant extension beyond the standard “two-sided” matching framework that considers matching on a network. Ostrovsky (2008) presents a novel approach to an industrial organization
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Chia-Ling Hsu [email protected] Juan F. Fung [email protected]
1
Department of Economics, University of Illinois at Urbana-Champaign, Urbana, IL, USA
2
School of Economics, Southwestern University of Finance and Economics, Chengdu, China
123
J. F. Fung, C.-L. Hsu
topic, using the tools of matching, and the result is a mathematically elegant and intuitive model of supply chains. He extends the two-sided matching notion of a pairwise stable allocation to that of a chain stable network for the supply chain setting. Hatfield and Milgrom (2005) propose two methods to show the existence of a stable allocation in the two-sided, many-to-one matching with contracts setting: the generalized Gale-Shapely algorithm, which utilizes a fixed-point theorem, and the cumulative offer process, in which agents on one side propose and agents on the other side choose contracts from from a set of accumulated offers. The fixed-point approach in Hatfield and Milgrom (2005) generalizes Adachi (2000)’s use of Tarski’s fixed-point theorem to prove existence of stable matchings in the setting of one-to-one matching. Echenique and Oviedo (2004, 2006) extend Adachi (2000)’s fixed-point approach to the settings of many-to-one and many-to-many matching, respectively. Ostrovsky (2008)’s approach extends the fixed-point approach to the supply chain matching setting
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