Echenique, Lee, Shum, and Yenmez (2013) established the testable revealed preference restrictions for stable aggregate matching with transferable (TU) and non-transferable utility (NTU) and for extremal stable matchings. In this paper, we rephrase their restrictions in terms of properties on a corresponding bipartite graph. From this, we obtain a simple condition that verifies whether a given aggregate matching is rationalisable. For matchings that are not rationalisable, we provide a simple greedy algorithm that computes the minimum number of matches that needs to be removed to obtain a rationalisable matching. We also show that the related problem of finding the minimum number of types that we need to remove in order to obtain a rationalisable matching is NP-complete.

We consider three popular affirmative action policies in school choice: quota-based, priority-based, and reserve-based affirmative actions. The Boston mechanism (BM) is responsive to the latter two policies in that a stronger priority-based or reserve-based affirmative action makes some minority student better off. However, a stronger quota-based affirmative action may yield a Pareto inferior outcome for the minority under the BM. These positive results disappear once we look for a stronger welfare consequence on the minority or focus on BM equilibrium outcomes.

We ask what the reallocation of indivisible objects reveals about the market that produced it. A central authority assigns the objects, after which recipients exchange them among themselves. Preferences are never observed, and individuals of the same type have identical preferences. A reallocation is rationalizable as Pareto efficient and individually rational exactly when a directed graph built from the data is acyclic, and as weak core-stable exactly when a mixed graph admits an orientation acyclic within types and free of alternating cycles. Deciding the latter is NP-complete, and on the single-crossing domain, although the test becomes more restrictive, it turns out to be more useful in practice. Both properties are less demanding than the strict core, so their rejection eliminates a broader set of possibilities. We apply the tests to administrative data on social housing lettings in England.

This paper studies the problem of reallocating objects to agents while taking into account agents' endowments, object capacities and agents’ preferences. The goal is to find a Pareto efficient and individually rational allocation that minimizes the number of individuals who need to change from their initial allocation to the final one. We call this problem as MINDIST. We establish NP-completeness result for MINDIST. We also show that MINDIST remains NP-complete when we restrict individual preferences to be binary, meaning that each individual can rank at most two objects in the preferences. Finally, we present an integer programming formulation to solve small to moderately sized instances of the NP-hard problems.

We introduce a novel notion of fairness, inspired by the equality of opportunity literature, into the school choice setting, endowed with a measure of the match qualities between students and schools. In this framework, fairness considerations are made by a social evaluator based on the match quality distribution. We impose the standard notion of stability as a minimal desideratum and study matchings that satisfy our notion of fairness and an eciency requirement based on aggregate match quality. To overcome some of the identied incompatibilities, we relax the fairness and eciency denitions, and embed them in a family of linear social welfare functions. We then propose an algorithm that maximize social welfare over the set of stable matchings. Finally, we illustrate our approach with an application to the allocation of Italian high school students in the 2021/2022 academic year.