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.

This paper studies an object allocation problem, which involves assigning objects to agents while taking into account agents’ endowments, object capacities and agents’ preferences. We focus on the Pareto efficient and individually rational allocations. The goal is to study the Pareto efficient and individually rational allocations that maximize the number of individuals improving upon their initial endowments. We present the Reallocating Cycles Algorithm for addressing this problem. Next, we discuss several extensions where our algorithm can be used with a minor modification. Extensions include an affirmative action setting where priority is given to the most disadvantaged individuals.

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.

We develop a revealed preference framework to test whether an aggregate allocation of indivisible objects satisfies Pareto efficiency and individual rationality (PI) without observing individual preferences. Exploiting the type-based preferences of Echenique et. al. (2013), we derive necessary and sufficient conditions for PI-rationalizability. We show that an allocation is PI-rationalizable if and only if its allocation graph is acyclic, and equivalently if its associated bipartite graph contains no alternating cycles. The bipartite representation admits a matroid structure, enabling a simple greedy algorithm to measure the severity of PI violations and identify the minimal set of individual–object assignments whose removal restores rationalizability. Our results yield the first complete revealed preference test for PI in matching markets and provide an implementable tool for empirical applications.