M♮-Convexity and Its Applications in Operations

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.

Informational Herding, Optimal Experimentation, and Contrarianism

In the standard herding model, privately informed individuals sequentially see prior actions and then act. An identical action herd eventually starts and public beliefs tend to “cascade sets” where social learning stops. What behaviour is socially efficient when actions ignore informational externalities? We characterize the outcome that maximizes the discounted sum of utilities. Our four key findings are: (a) Cascade sets shrink but do not vanish, and herding should occur but less readily as greater weight is attached to posterity. (b) An optimal mechanism rewards individuals mimicked by their successor. (c) Cascades cannot start after period one under a signal logconcavity condition. (d) Given this condition, efficient behaviour is contrarian, leaning against the myopically more popular actions in every period. We make two technical contributions: As value functions with learning are not smooth, we use monotone comparative statics under uncertainty to deduce optimal dynamic behaviour.We also adapt dynamic pivot mechanisms to Bayesian learning.

Matching With Complementary Contracts

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.

A NOTE ON MONOTONE COMPARATIVE STATICS FOR MONETARY DIRECTED SEARCH MODELS

This note uses monotone methods to derive two sets of comparative statics results for monetary directed search models. First, it characterizes the impact of a higher inflation rate or a higher cost of using credit on market outcomes, regardless of the choice of matching function. Second, the seller-to-buyer ratio, output level, and money demand increase as the matching function becomes more efficient in a log-supermodular sense. I also consider an extension with endogenous search intensity and show that search intensity and trade volume always decrease in the nominal interest rate.