Virtual Demand and Stable Mechanisms

Abstract

We study conditions for the existence of stable, strategy-proof mechanisms in a many-to-one matching model with salaries. Workers and firms want to match and agree on the terms of their match. Firms demand different sets of workers at different salaries. Workers have preferences over different firm-salary combinations. Workers' preferences are monotone in salaries. We show that for this model, a descending auction mechanism is the only candidate for a stable mechanism that is strategy-proof for workers. Moreover, we identify a maximal domain of demand functions for firms, such that the mechanism is stable and strategy-proof. In the special case, where demand functions are generated by quasi-linear profit functions, our domain reduces to the domain of demand functions under which workers are gross substitutes for firms. We provide two versions of the results for the quasi-linear case. One for a discrete model, where salaries are restricted to discrete units and one for a continuous model, where salaries can take on arbitrary positive numbers. More generally, we show that several celebrated results (the existence of a worker-optimal stable allocation, the rural hospitals theorem, the strategyproofness of the worker-optimal stable mechanism) generalize from the discrete to the continuous model.

Trading Networks With General Preferences

“…The construction is by an approximation argument when the existence in the continuum is obtained from the existence of an equilibrium in a discrete markets with smaller and smaller price increments. Different versions of a (group-)strategy-proofness result for a many-to-one matching model with continuous transfers have been established by Hatfield et al (2014); Schlegel (2016); Jagadeesan et al (2018).…”

Equivalent choice functions and stable mechanisms

“…Our original working paper (Schlegel, 2016) contained a version of our maximal domain result for a model of matching with contracts where also colleges' choice functions are monotone in contract-terms. Technically the two maximal domain results are independent.…”

“…After, our original working paper (Schlegel, 2016), Hatfield et al (2020) released a new version of their paper, where the authors extend their analysis to the case of restricted preference domains for the applicant side and show that their analysis of observable substitutes and the observable law of aggregate demand extends to restricted preference domains if the notion of "observability" is adjusted to the domain restriction. Their additional work allows us to shorten the proof of our maximal domain result, since we can now make use of their Theorem 5.…”

Equivalent Choice Functions and Stable Mechanisms

“…However, Abizada and Dur (2017) also analyze pairwise-stable outcomes and this part of their analysis does not have a counter-part in our paper. Our original working paper (Schlegel, 2016) contained a version of our maximal domain result for a model of matching with contracts where also colleges' choice functions are monotone in contract-terms. Technically the two maximal domain results are independent.…”