Monotone Comparative Statics with Separable Objective Functions

A new approach for structural analysis of operations models with substitutability structures. In many operations models with substitutability structures, one often ends up with parametric optimization models that maximize submodular objective functions, and it is desirable to derive structural properties including monotone comparative statics of the optimal solutions or preservation of submodularity under the optimization operations. Yet, this task is challenging because the classical and commonly used results in lattice programming, applicable to optimization models with supermodular objective function maximization, do not apply. Using a key concept in discrete convex analysis, M♮-convexity, Chen and Li establish conditions under which the optimal solutions are nonincreasing in the parameters and the preservation property holds for parametric maximization models with submodular objectives, together with the development of several new fundamental properties of M♮-convexity. Their approach is powerful as demonstrated by applications in a classical multiproduct stochastic inventory model and a portfolio contract model.

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Matching With Complementary Contracts

Econometrica ◽

10.3982/ecta16686 ◽

2020 ◽

Vol 88 (5) ◽

pp. 1793-1827 ◽

Cited By ~ 1

Author(s):

Marzena Rostek ◽

Nathan Yoder

Keyword(s):

Social Media ◽

Comparative Statics ◽

Design Decisions ◽

Monotone Comparative Statics ◽

Patent Pools ◽

Multilateral Agreements ◽

Deferred Acceptance Algorithm ◽

Social Media Platforms ◽

Deferred Acceptance ◽

The Impact

In this paper, we show that stable outcomes exist in matching environments with complementarities, such as social media platforms or markets for patent licenses. Our results apply to both nontransferable and transferable utility settings, and allow for multilateral agreements and those with externalities. In particular, we show that stable outcomes in these settings are characterized by the largest fixed point of a monotone operator, and so can be found using an algorithm; in the nontransferable utility case, this is a one‐sided deferred acceptance algorithm, rather than a Gale–Shapley algorithm. We also give a monotone comparative statics result as well as a comparative static on the effect of bundling contracts together. These illustrate the impact of design decisions, such as increased privacy protections on social media, or the use of antitrust law to disallow patent pools, on stable outcomes.

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Preservation of Supermodularity in Parametric Optimization: Necessary and Sufficient Conditions on Constraint Structures

Operations Research ◽

10.1287/opre.2020.1992 ◽

2020 ◽

Author(s):

Xin Chen ◽

Daniel Zhuoyu Long ◽

Jin Qi

Keyword(s):

Operations Research ◽

Parametric Optimization ◽

Optimization Problems ◽

Lattice Structure ◽

Sufficient Conditions ◽

Comparative Statics ◽

Necessary And Sufficient Conditions ◽

Objective Functions ◽

New Concepts ◽

Necessary And Sufficient

The concept of supermodularity has received considerable attention in economics and operations research. It is closely related to the concept of complementarity in economics and has also proved to be an important tool for deriving monotonic comparative statics in parametric optimization problems and game theory models. However, only certain sufficient conditions (e.g., lattice structure) are identified in the literature to preserve the supermodularity. In this article, new concepts of mostly sublattice and additive mostly sublattice are introduced. With these new concepts, necessary and sufficient conditions for the constraint structures are established so that supermodularity can be preserved under various assumptions about the objective functions. Furthermore, some classes of polyhedral sets that satisfy these concepts are identified, and the results are applied to assemble-to-order systems.

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