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Social Choice and Welfare

Erschienen in:

01.10.2014 | Original Paper

Characterizations of the sequential priority rules in the assignment of object types

verfasst von: Nanyang Bu

Erschienen in: Social Choice and Welfare | Ausgabe 3/2014

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Abstract

We study the problem of assigning object types without monetary transfer. Each type has a number of copies. Each individual is assigned at most one copy of a type. We examine some desirable axioms and study the sequential priority rules. Our main result is that the sequential priority rules are the only rules that satisfy weak non-wastefulness, weak neutrality, strategy-proofness, resource monotonicity, and bilateral consistency.

Vorheriger Artikel Asymmetrically fair rules for an indivisible good problem with a budget constraint

Nächster Artikel Bayesian implementation with partially honest individuals

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This version of consistency is proposed by Kojima and Unver (2011).

Rules of this kind are often referred to as “serial dictatorships”.

There are two versions of the DA-rule, namely, the student-proposing DA-rule and the school-proposing DA-rule. Since the school-proposing DA-rule is not well-behaved, we following the convention of this type of literature and focus only on the student-proposing DA-rule. For the same reason, we are only interested in the student-proposing IA-rule. To simplify terminology, we drop the prefix of “student-proposing” for both of the rules.

I follow Thomson (2011) to refer to the “Boston mechanism” (Abdulkadiroğlu and Sönmez 2003) as the immediate acceptance rule.

Priority-indexed DA-rules have been characterized on the basis of several sets of properties (Ehlers and Klaus 2006, 2012; Kojima and Manea 2010). Priority-indexed IA-rules have also been studied and characterized (Kojima and Unver 2011; Bu 2013).

This property is often referred to as “individual rationality”.

Although the population is fixed, we can imagine an individual “leaves with his assignment” by letting the individual have a preference where the null object is on the top and reducing the supply of his initially assigned object by one.

Thomson (2010) is a survey on consistency. For the school choice problem, necessary and sufficient conditions for the DA-rule to satisfy versions of consistency have been identified (Ergin 2002; Klaus and Klijn 2013).

A requirement, called “non-bossiness”, states that a change in one’s announcement should affect some other individual’s assignments only if it also affects his own assignment. Here, consistency implies non-bossiness but bilateral consistency does not.

In the context of school choice, the DA-rule is defined by means of the following algorithm: at step 1, each student applies to his top acceptable school; each school temporarily accepts those among its applicants who have the highest priority up to its capacity, and rejects all other applicants. For each \(l > 1\), at step \(l\), each student rejected at step \(l-1\) applies to his next best acceptable school. Each school temporarily accepts students with highest priorities up to its capacity among its new applicants and those temporarily accepted at step \(l-1\), and rejects the others. The algorithm ends when no student is rejected. All acceptances are finalized then.

In the context of school choice, the IA-rule is defined by means of the following algorithm: at step 1, each student applies to his top acceptable school; each school accepts those among its applicants who have the highest priority up to its capacity, and rejects all other applicants. For each \(l > 1\), at step \(l\), each student rejected at step \(l-1\) applies to his \(l\)th acceptable school. Each school accepts those who have the highest priority up to its number of seats left after step \(l-1\) among its new applicants, and rejects the others. The algorithm ends when no student is rejected.

Lemma 1 is also called the “Elevator Lemma” (Thomson 2010).

We say \(\prec _a\) is asymmetric if for each pair \(\{i,j\} \subseteq N\), \(i \prec _a j\) implies \(j \nprec _a i\). We say \(\prec _a\) is total if for each pair \(\{i,j\} \subseteq N\), \(i \ne j\) implies \(i \prec _a j\) or \(j \prec _a i\). We say \(\prec _a\) is transitive if for each triple \(\{i,j,k\} \subseteq N\), \(i \prec _a j\) and \(j \prec _a k\) imply \(i \prec _a k\).

Abdulkadiroğlu A, Sönmez T (2003) School choice: a mechanism design approach. Am Econ Rev 93:729–747CrossRef

Abizada A, Chen S (2012) Allocating tasks when some may get canceled or additional ones may arrive, Working paper

Bu N (2013) A new fairness notion in the assignment of indivisible resources, Working paper

Chun Y, Thomson W (1988) Monotonicity properties of bargaining solutions when applied to economics. Math Soc Sci 15:11–27CrossRef

Ehlers L, Klaus B (2006) Efficient priority rules. Games Econ Behav 55:372–384CrossRef

Ehlers L, Klaus B (2012) Strategy-proofness makes the difference: deferred-acceptance with responsive priorities. Cahier de recherches économiques du DEEP No. 12:06

Ergin H (2000) Consistency in House Allocation Problems. J Math Econ 34:77–97CrossRef

Ergin H (2002) Efficient resource allocation on the basis of priorities. Econometrica 70:2489–2498CrossRef

Gale D, Shapley LS (1962) College admissions and the stability of marriage. Am Math Mon 69:9–15CrossRef

Klaus B, Klijn F (2013) Local and global consistency properties for student placement. J Math Econ 49:222–229CrossRef

Kojima F, Manea M (2010) Axioms for deferred acceptance. Econometrica 78:633–653CrossRef

Kojima F, Unver MU (2011) The “Boston” School-Choice Mechanism, Boston College Working Paper in Economics, No: 729

Roemer JE (1986) Equality of resources implies equality of welfare. Quart J Econ 101:751–784CrossRef

Shapley L, Scarf H (1974) On cores and indivisibility. J Math Econ 1:23–37CrossRef

Svensson L-G (1999) Strategy-proof allocation of indivisible goods. Soc Choice Welf 16:557–567CrossRef

Thomson W (2010) Consistent allocation rules, unpublished manuscript

Thomson W (2011) Strategy-proof allocation rules, book manuscript

Titel: Characterizations of the sequential priority rules in the assignment of object types
verfasst von: Nanyang Bu
Publikationsdatum: 01.10.2014
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 3/2014
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-014-0791-1

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