Abstract
A (capacity-constrained) choice problem consists of a set of alternatives and a capacity. A (capacity-constrained) choice rule, at each choice problem, chooses from the alternatives without exceeding the capacity. A choice rule is lexicographic if there exists a list of priority orderings over potential alternatives such that at each choice problem, the set of chosen alternatives is obtainable by choosing the highest ranked alternative according to the first priority, then choosing the highest ranked alternative among the remaining alternatives according to the second priority, and proceeding similarly until the capacity is full or no alternative is left. Lexicographic choice rules have been useful in designing allocation mechanisms for school choice to accommodate diversity. We provide a characterization of lexicographic choice rules. We discuss some implications for the Boston school choice system. We also provide a characterization of deferred acceptance mechanisms that operate based on a lexicographic choice structure, instead of a priority structure.
Highlights
A choice problem consists of a choice set and a capacity
We show that a choice rule satisfies acceptance, gross substitutes, monotonicity, and capacity-wise weak axiom of revealed preference (CWARP) if and only if it is lexicographic: there is a list of priority orderings over alternatives such that at each problem, the set of chosen alternatives is obtainable by choosing the highest ranked alternative according to the first priority, choosing the highest ranked alternative among the remaining alternatives according to the second priority, and proceeding until the capacity is full or no alternative is left (Theorem 1)
Boston school district has been using a deferred acceptance mechanism based on a choice structure, where each school is endowed with a “capacity-wise lexicographic” choice rule, that is, at each capacity, the choice rule lexicographically operates based on a list containing as many priority orderings as the capacity, yet the lists for different capacity levels do not have to be related in any way
Summary
A (capacity-constrained) choice problem consists of a choice set (a set of alternatives) and a capacity. An example is the Boston school district which aims to give priority to neighborhood applicants for half of each school’s seats To achieve this goal, Boston school district has been using a deferred acceptance mechanism based on a choice structure, where each school is endowed with a “capacity-wise lexicographic” choice rule, that is, at each capacity, the choice rule lexicographically operates based on a list containing as many priority orderings as the capacity, yet the lists for different capacity levels do not have to be related in any way.3 Dur et al (2016) provides an analysis of how the order of the priority orderings in the choice rule of a school may affect the outcome in the Boston school choice context.
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