Supplement to “ Improving matching under hard distributional constraints ”

In this supplementary appendix, we relax one of the key features of the DQDA mechanism: that the reduction sequence be exogenous to the submitted preferences. We define a new mechanism, the endogenous-reduction DQDA (EDQDA) mechanism that allows the reduction sequence to change depending on what preferences are submitted. Intuitively, this should allow the mechanism to respond more to changes in demand, and thus allocate seats even more flexibly than DQDA. There are two costs associated with this approach. First, EDQDA loses the important strategyproofness property satisfied by ACDA and DQDA (and standard DA) with no floor constraints; and second, EDQDA will no longer Pareto dominate ACDA. However, we are able to show that EDQDA will be approximately strategyproof in large markets (in a formal sense defined below). In addition, we use simulations to study the magnitude of the welfare gains from our dynamic quotas mechanisms. While some students may be worse off under EDQDA, EDQDA tends to make students better off “on average”, in the sense that in the simulations, the rank distribution of EDQDA will first-order stochastically dominate that of ACDA. To define EDQDA, we use a slightly different definition of a reduction sequence. A reduction sequence is now written as ρ = {(s1, θ1), . . . , (sK , θK)}, where each (sk, θk) ∈ S × Θ. ρ is a baseline order for reducing the ceilings, but, unlike for DQDA, an entry will be skipped if all floors for the corresponding type have already been met. In addition, we will only reduce the type-specific ceilings, and not the capacities. The same entry may appear multiple times in ρ.1