On the Existence of Stable Allocations in Matching Games with Infinite Contracts: Leveraging the Deferred Acceptance Algorithm

2019 ◽  
Author(s):  
James Fisher
Keyword(s):  
Deferred Acceptance Algorithm ◽  
Deferred Acceptance ◽  
Matching Games ◽  
Stable Allocations

Deferred Acceptance with Compensation Chains

2021 ◽  
Vol 69 (2) ◽  
pp. 456-468
Author(s):  
Piotr Dworczak
Keyword(s):  
Potential Function ◽  
Polynomial Time ◽  
The Other ◽  
Stable Matchings ◽  
Matching Process ◽  
Matching Market ◽  
Deferred Acceptance Algorithm ◽  
Deferred Acceptance ◽  
Proof Of Convergence ◽  
Stable Outcome

In a foundational paper, Gale and Shapley (1962) introduced the deferred acceptance algorithm that achieves a stable outcome in a two-sided matching market by letting one side of the market make proposals to the other side. What happens when both sides of the market can propose? In “Deferred Acceptance with Compensation Chains,” Dworczak answers this question by constructing an equitable version of the Gale–Shapley algorithm in which the sequence of proposers can be arbitrary. The main result of the paper shows that the extended algorithm, equipped with so-called compensation chains, is not only guaranteed to converge in polynomial time to a stable outcome, but—in contrast to the original Gale–Shapley algorithm—achieves all stable matchings (as the sequence of proposers vary). The proof of convergence uses a novel potential function. The algorithm may find applications in settings where both stability and fairness are desirable features of the matching process.

scholarly journals Heterogeneous Beliefs and School Choice Mechanisms

2020 ◽  
Vol 110 (5) ◽  
pp. 1274-1315 ◽  
Author(s):  
Adam J. Kapor ◽  
Christopher A. Neilson ◽  
Seth D. Zimmerman
Keyword(s):  
School Choice ◽  
Rational Expectations ◽  
Choice Behavior ◽  
Heterogeneous Beliefs ◽  
Welfare Costs ◽  
Subjective Beliefs ◽  
Opposite Conclusion ◽  
Deferred Acceptance Algorithm ◽  
Deferred Acceptance ◽  
Strategy Proof

This paper studies how welfare outcomes in centralized school choice depend on the assignment mechanism when participants are not fully informed. Using a survey of school choice participants in a strategic setting, we show that beliefs about admissions chances differ from rational expectations values and predict choice behavior. To quantify the welfare costs of belief errors, we estimate a model of school choice that incorporates subjective beliefs. We evaluate the equilibrium effects of switching to a strategy-proof deferred acceptance algorithm, and of improving households’ belief accuracy. We find that a switch to truthful reporting in the DA mechanism offers welfare improvements over the baseline given the belief errors we observe in the data, but that an analyst who assumed families had accurate beliefs would have reached the opposite conclusion. (JEL D83, H75, I21, I28)

scholarly journals Matching With Complementary Contracts

Econometrica ◽  
2020 ◽  
Vol 88 (5) ◽  
pp. 1793-1827 ◽  
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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