Lexicographic choice under variable capacity constraints

Battal Doğan,Kemal Yıldız,Serhat Doğan

doi:10.1111/jpet.12482

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https://doi.org/10.1111/jpet.12482

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Abstract

AbstractIn several matching markets, to achieve diversity, agents' priorities are allowed to vary across an institution's available seats, and the institution is let to choose agents in a lexicographic fashion based on a predetermined ordering of the seats, called a (capacity‐constrained) lexicographic choice rule. We provide a characterization of lexicographic choice rules and a characterization of deferred acceptance mechanisms that operate based on a lexicographic choice structure under variable capacity constraints. We discuss some implications for the Boston school choice system and show that our analysis can be helpful in applications to select among plausible choice rules.

Highlights

Many real‐life resource allocation problems involve the allocation of an object that is available in a limited number of identical copies, called the capacity of the object
In Theorem 1, we show that a choice rule satisfies capacity‐filling, gross substitutes, monotonicity, and the irrelevance of accepted alternatives if and only if it is lexicographic: 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, first, the highest ranked alternative according to the first priority ordering, choosing the highest ranked alternative among the remaining alternatives according to the second priority ordering, and proceeding until the capacity is full or no alternative is left
Note that in the proof Theorem 1, we showed that a lexicographic choice rule satisfies Capacity‐wise weak axiom of revealed preference (CWARP) as well

Summary

| INTRODUCTION

Many real‐life resource allocation problems involve the allocation of an object that is available in a limited number of identical copies, called the capacity of the object. Ehlers and Klaus (2016), in their Theorem 3, characterize deferred acceptance mechanisms based on a choice structure satisfying capacity‐filling, gross substitutes, and monotonicity, with the following properties of mechanisms: unavailable‐type‐invariance (if the positions of the unavailable types are shuffled at a profile, the allocation should not change); weak nonwastefulness (no agent receives the null object while he prefers an object, i.e., not exhausted to the null object), resource‐monotonicity (increasing the capacities of some objects does not hurt any agent), truncation‐invariance (if an agent truncates his preference relation in such a way that his allotment remains acceptable under the truncated preference relation, the allocation should not change), and strategy‐proofness (no agent can benefit by misreporting his preferences). A capacity‐wise lexicographic choice rule satisfies capacity‐filling, gross substitutes, monotonicity, the capacity‐wise weak axiom of revealed preference, and the Boston requirement for (≻w, ≻o) if and only if it is (capacity‐constrained) lexicographic for a priority profile (≻1, ..., ≻n) such that i. We will compare the four choice rules with respect to our set of choice rule properties

Compromise Choice Rule

| CONCLUSION