Matching theory in the September issue of Games and Economic Behavior

 The September 2021 issue of Games and Economic Behavior (Volume 129, Pages 1-590) has five papers on matching theory.

In the order in which they appear:

An improved bound to manipulation in large stable matches  by Gustavo Saraiva

https://doi.org/10.1016/j.geb.2021.05.005Get rights and content

Abstract: This paper builds on Kojima and Pathak (2009)'s result of vanishing manipulability in large stable mechanisms. We show that convergence toward truth-telling in stable mechanisms can be achieved much faster if colleges' preferences are independently drawn from an uniform distribution. Another novelty from our results is that they can be applied to competitive environments in which virtually all vacancies end up being filled. So this paper adds evidence to the fact that, though stable matching mechanisms are not entirely strategy-proof, in practice, when the number of participants in the market is sufficiently large, they can be treated as being effectively strategy-proof.

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How lotteries in school choice help to level the playing field by Christian Basteck, Bettina Klaus, Dorothea Kübler

https://doi.org/10.1016/j.geb.2021.05.010Get rights and content

Abstract: School authorities in the UK and the US advocate the use of lotteries to desegregate schools. We study a school choice mechanism employed in Berlin where a lottery quota is embedded in the immediate acceptance (IA) mechanism, and compare it to the deferred acceptance mechanism (DA) with a lottery quota. In both mechanisms, some seats are allocated based on academic achievement (e.g., grades), while seats in the lottery quota are allocated randomly. We find that, in theory, a lottery quota strengthens truth-telling in DA by eliminating non-truth-telling equilibria. Furthermore, the equilibrium outcome is stable for DA with a lottery but not for IA with a lottery. These predictions are borne out in the experiment. Moreover, the lottery quota leads to more diverse school populations in the experiment, as predicted. Students with the lowest grades profit more from the introduction of the lottery under IA than under DA.

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Substitutes and stability for many-to-many matching with contracts  by Keisuke Bando, Toshiyuki Hirai, Jun Zhang

https://doi.org/10.1016/j.geb.2021.07.002Get rights and content

Abstract:We examine the roles of (slightly weakened versions of) the observable substitutability condition and the observable substitutability across doctors condition of Hatfield et al. (2021) in many-to-many matching with contracts. We modify the standard cumulative offer algorithm to find stable outcomes and prove new results on the existence of stable outcomes. It is remarkable that size monotonicity at the offer-proposing side is essential for the existence result under observable substitutability across doctors.

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Slot-specific priorities with capacity transfers  by Michelle Avataneo and BertanTurhan

https://doi.org/10.1016/j.geb.2021.07.005

Abstract: In many real-world matching applications, there are restrictions for institutions either on priorities of their slots or on the transferability of unfilled slots over others (or both). Motivated by the need in such real-life matching problems, this paper formulates a family of practical choice rules, slot-specific priorities with capacity transfers (SSPwCT). These rules invoke both slot-specific priorities structure and transferability of vacant slots. We show that the cumulative offer mechanism (COM) is stable, strategy-proof and respects improvements with regards to SSPwCT choice rules. Transferring the capacity of one more unfilled slot, while all else is constant, leads to strategy-proof Pareto improvement of the COM. Following Kominers' (2020) formulation, we also provide comparative static results for expansion of branch capacity and addition of new contracts in the SSPwCT framework. Our results have implications for resource allocation problems with diversity considerations.

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Stability in sequential matching with incomplete information by Fanqi Shi

https://doi.org/10.1016/j.geb.2021.07.001Get rights and content

Abstract: I study a two-period matching model where one side of the market (e.g. workers) have an option to invest and delay matching in the first period. Investment increases each agent's matching surplus in the second period, by a magnitude of the worker's investment ability in the match pair. Assuming each worker's investment ability is her private information that unfolds in the second period, I define a notion of sequential stability, and show that the set of sequentially stable outcomes is a superset of the complete information stable outcomes. Moreover, with transferable utility, as long as the cost of delay coincides on the same side of the market, efficient investment occurs in any sequentially stable outcome. When every agent shares the same cost of delay, efficient investment also occurs in any sequentially stable outcome with non-transferable utility. My analysis suggests that efficient investment is a robust prediction in sequential matching markets.