Choice and Matching

2017 ◽  
Vol 9 (3) ◽  
pp. 126-147 ◽  
Author(s):  
Christopher P. Chambers ◽  
M. Bumin Yenmez
Keyword(s):  
The Other ◽  
Matching Theory ◽  
Matching Markets ◽  
Powerful Technique ◽  
Static Result ◽  
Deferred Acceptance Algorithm ◽  
Deferred Acceptance

We study path-independent choice rules applied to a matching context. We use a classic representation of these choice rules to introduce a powerful technique for matching theory. Using this technique, we provide a deferred acceptance algorithm for many-to-many matching markets with contracts and study its properties. Next, we obtain a compelling comparative static result: if one agent's choice expands, the remaining agents on her side of the market are made worse off, while agents on the other side of the market are made better off. Finally, we establish several results related to path-independent choice rules. (JEL C78, D11, D71, D86)

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 Stability in Matching Markets with Complex Constraints

2021 ◽  
Author(s):  
Hai Nguyen ◽  
Thành Nguyen ◽  
Alexander Teytelboym
Keyword(s):  
Refugee Resettlement ◽  
The Novel ◽  
Matching Markets ◽  
Constraint Violation ◽  
Constraint Matrix ◽  
Knapsack Constraints ◽  
Multidimensional Knapsack ◽  
Practical Constraint ◽  
Deferred Acceptance Algorithm ◽  
Deferred Acceptance

We develop a model of many-to-one matching markets in which agents with multiunit demand aim to maximize a cardinal linear objective subject to multidimensional knapsack constraints. The choice functions of agents with multiunit demand are therefore not substitutable. As a result, pairwise stable matchings may not exist and even when they do, may be highly inefficient. We provide an algorithm that finds a group-stable matching that approximately satisfies all the multidimensional knapsack constraints. The novel ingredient in our algorithm is a combination of matching with contracts and Scarf’s Lemma. We show that the degree of the constraint violation under our algorithm is proportional to the sparsity of the constraint matrix. The algorithm, therefore, provides practical constraint violation bounds for applications in contexts, such as refugee resettlement, day care allocation, and college admissions with diversity requirements. Simulations using refugee resettlement data show that our approach produces outcomes that are not only more stable, but also more efficient than the outcomes of the Deferred Acceptance algorithm. Moreover, simulations suggest that in practice, constraint violations under our algorithm would be even smaller than the theoretical bounds. This paper was accepted by Gabriel Weintraub, revenue management and market analytics.

scholarly journals Two-Sided Matching with Endogenous Preferences

2015 ◽  
Vol 7 (3) ◽  
pp. 241-258 ◽  
Author(s):  
Yair Antler
Keyword(s):  
Stable Matching ◽  
The Other ◽  
Matching Theory ◽  
Endogenous Preferences ◽  
Matching Problem ◽  
Matching Mechanism

We modify the stable matching problem by allowing agents' preferences to depend on the endogenous actions of agents on the other side of the market. Conventional matching theory results break down in the modified setup. In particular, every game that is induced by a stable matching mechanism (e.g., the Gale-Shapley mechanism) may have equilibria that result in matchings that are not stable with respect to the agents' endogenous preferences. However, when the Gale-Shapley mechanism is slightly modified, every equilibrium of its induced game results in a pairwise stable matching with respect to the endogenous preferences as long as they satisfy a natural reciprocity property. (JEL C78, D82)

Sign in / Sign up

Export Citation Format

Share Document