This paper introduces a novel framework for modeling diversity using the theory of majorization and applies it to school choice problems. We consider a school with limited capacity that must select students from an applicant pool, where each student is classified into types (e.g., by race, gender, or socioeconomic status). Within this setting, we ask: How should we compare the diversity of student groups? And what admissions procedure balance the school's goal of admitting students with the highest priority (e.g., academic merit, proximity to the school) and achieving a diverse student body?

We provide optimal solutions to an institution that has distributional objectives when choosing from a set of applications based on merit (or priority). For example, in college admissions, administrators may want to admit a diverse class in addition to choosing students with the highest qualifications. We provide a family of choice rules that maximize merit subject to attaining a level of the distributional objective. We study the desirable properties of choice rules in this family and use them to find all subsets of applications on the Pareto frontier of the distributional objective and merit. In addition, we provide two novel characterizations of matroids. A full version is available at https://arxiv.org/abs/2301.00237.

Path independence is arguably one of the most important choice rule properties in economic theory. We show that a choice rule is path independent if and only if it is rationalizable by a utility function satisfying ordinal concavity, a concept closely related to concavity notions in discrete mathematics. We also provide a rationalization result for choice rules that satisfy path independence and the law of aggregate demand. A full version is available at https://arxiv.org/abs/2303.00892.

In a matching market with distributional objectives, we study the existence of mechanisms that weakly improve the distributional objective (compared to a status-quo matching), and satisfy constrained efficiency, individual rationality, and strategy-proofness.

A priority system has traditionally been the protocol of choice for the allocation of scarce life-saving resources during public health emergencies. COVID-19 revealed the limitations of this allocation rule. Many argue that priority systems abandon ethical values, such as equity, by discriminating against disadvantaged communities. We show that a restrictive feature of the traditional priority system largely drives these limitations. Following minimalist market design, an institution design paradigm that integrates research and policy efforts, we formulate pandemic allocation of scarce life-saving resources as a new application of market design. Interfering only with the restrictive feature of the priority system to address its shortcomings, we formulate a reserve system as an alternative allocation rule. Our theoretical analysis develops a general theory of reserve design. We relate our analysis to debates during COVID-19 and describe the impact of our paper on policy and practice. This paper was accepted by Axel Ockenfels, behavioral economics and decision analysis.