Comparative Statics for Size-Dependent Discounts in Matching Markets

David  Delacretaz; Scott Duke Kominers; Alexandru   Nichifor

Article | Journal of Mathematical Economics | Forthcoming

Comparative Statics for Size-Dependent Discounts in Matching Markets

by David Delacretaz, Scott Duke Kominers and Alexandru Nichifor

Abstract

We prove a natural comparative static for many-to-many matching markets in which agents’ choice functions exhibit size-dependent discounts: reducing the extent to which some agent discounts additional partners leads to improved outcomes for the agents on the other side of the market, and worsened outcomes for the agents on the same side of the market. Our argument draws upon recently developed methods bringing tools from choice theory into matching.

Keywords: size-dependent discounts; path-independence; respect for improvements; Market Design; Mathematical Methods;

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Delacretaz, David, Scott Duke Kominers, and Alexandru Nichifor. "Comparative Statics for Size-Dependent Discounts in Matching Markets." Journal of Mathematical Economics (forthcoming).

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, showing that when utility is perfectly transferable, any agent who does not participate in trade in one 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.

Hatfield and Kominers (2012) introduced a model of matching in networks with bilateral contracts and showed that stable outcomes exist in supply chains when firms' preferences over contracts are fully substitutable. Hatfield and Kominers (2012) also asserted that in their setting, full substitutability is equivalent to the assumption that all indirect utility representations of each firm's preferences are quasisubmodular; we show here that this claimed equivalence result does not hold in general. We show instead that full substitutability is equivalent to weak quasisubmodularity of all indirect utility representations.

Home delivery has shifted from a luxury service aimed at young urban professionals to a core part of many businesses, which is used by almost everyone. That upheaval has strained capacity of many delivery services and changed how they relate to the suppliers that they depend on. Specifically, it has meant a less immediate focus on customer acquisition and retention, and more investment in metrics and technology for supporting retailer operations. As delivery becomes a substantial fraction of stores’ sales, marketplaces can help coach suppliers on more efficient fulfillment; identify and share best practices around staffing, store layout, and delivery processes; and provide technology solutions to help small businesses optimize their operations around delivery, such as cloud-based inventory logs and modern point-of-sale systems.

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Comparative Statics for Size-Dependent Discounts in Matching Markets

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