Strategyproof matching with regional minimum and maximum quotas

Designing Matching Mechanisms under General Distributional Constraints

“…We create 100 problem instances for each parameter setting. We follow these parameter settings according to Goto, Hashimoto, Iwasaki, Kawasaki, Ueda, Yasuda, and Yokoo (2014). Figure 5 shows the ratio of students with justified envy, and Figure 6 shows the ratio of student who claims an empty seat.…”

“…Thus, we can apply the generalized DA of Goto, Hashimoto, Iwasaki, Kawasaki, Ueda, Yasuda, and Yokoo (2014) and Kojima, Tamura, and Yokoo (2014). We show that even in such a setting, the ADA can be useful since it can improve students' welfare.…”

“…It must be noted that even when η(X ′ ) can be computed efficiently, there exists a case where computingη(X ′ ), or finding an admissible and maximal ζ, is difficult. For example, Goto, Hashimoto, Iwasaki, Kawasaki, Ueda, Yasuda, and Yokoo (2014) show that checking whether a feasible set of contracts exists or not is NP-complete when regional minimum/maximum quotas exist and a region can be any subset of schools. Here, calculating η(X ′ ), i.e., checking a given set of contracts X ′ is feasible, is easy, i.e., it can be done in a polynomial time.…”

“…Next, we consider a case where schools are hierarchically organized into regions and for each region, a regional minimum quota is imposed Goto, Hashimoto, Iwasaki, Kawasaki, Ueda, Yasuda, and Yokoo (2014). A hier-archical structure is ubiquitous in any organization (company, university, or Intuitively, the expected minimum count of ζ for r means the minimum number of students assigned to region r, when some more students are added to satisfy regional minimum quotas for all r ′ ⊆ r. Thus, if ζ already satisfies minimum quotas for all r ′ ⊆ r, i.e., ∑ c i ∈r ′ ζ i ≥ p r ′ holds, then emc(r ′ , ζ) = ∑ c i ∈r ′ ζ i holds.…”

“…Public schools are often required to satisfy balance on the composition of students, typically in terms of socioeconomic status Ehlers, Hafalir, Yenmez, and Yildirim (2014). Several mechanisms have been proposed Ehlers, Hafalir, Yenmez, and Yildirim (2014); Fragiadakis, Iwasaki, Troyan, Ueda, and Yokoo (2015); Goto, Hashimoto, Iwasaki, Kawasaki, Ueda, Yasuda, and Yokoo (2014); ; Kamada and Kojima (2015) for each of these various constraints, but previous studies have focused on tailoring mechanisms to specific settings, rather than providing a general framework.…”