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Corrigendum to Characterizations of Pareto-efficient, fair, and strategy-proof allocation rules in queueing problems (Characterizations of Pareto-efficient, fair, and strategy-proof allocation rules in queueing problems (2010) 68(1) (220–232), (S0899825609001547), (10.1016/j.geb.2009.07.003))
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Kay?, Ça?atay
Ramaekers, Eve
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2019
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Academic Press Inc.
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Abstract
We are grateful to Youngsub Chun, Manipushpak Mitra, and Suresh Mutuswami for pointing out that Pareto-efficiency, symmetry, and strategy-proofness are not sufficient to characterize the Largest Equally Distributed Pairwise Pivotal rule. The error occurs on page 227. Statement (3) in Theorem 3 should be as follows: Theorem 3 Let ? be a rule. (3) If ? satisfies Pareto-efficiency, equal treatment of equals in welfare, symmetry, and strategy-proofness, it is the Largest Equally Distributed Pairwise Pivotal rule. Proof Statement 3: Assume that ? satisfies the axioms of Theorem 3.3. Let [Formula presented] and [Formula presented]. By Pareto-efficiency, [Formula presented]. By Theorem 1, Pareto-efficiency and strategy-proofness imply that there is [Formula presented] such that for each [Formula presented], if [Formula presented], then [Formula presented] and if [Formula presented], then [Formula presented]. By equal treatment of equals in welfare, for each [Formula presented] with [Formula presented] and [Formula presented], we have [Formula presented], and [Formula presented] and [Formula presented]. By the logic of Statement 1, for each [Formula presented], we have [Formula presented]. Thus, [Formula presented]. By symmetry, [Formula presented]. ? Therefore, Theorem 3 could be rewritten as follows: Theorem 3 Let ? be a rule. (1) If ? satisfies Pareto-efficiency, equal treatment of equals in welfare, and strategy-proofness, it is a subcorrespondence of the Largest Equally Distributed Pairwise Pivotal rule.(2) If ? is a subcorrespondence of the Largest Equally Distributed Pairwise Pivotal rule, it satisfies Pareto-efficiency, no-envy, and strong strategy-proofness.(3) If ? satisfies Pareto-efficiency, equal treatment of equals in welfare, symmetry, and strategy-proofness, it is the Largest Equally Distributed Pairwise Pivotal rule.(4) If ? is the Largest Equally Distributed Pairwise Pivotal rule, it satisfies Pareto-efficiency, equal treatment of equals in welfare, anonymity, and strong strategy-proofness. In many models where the allocation rule is single-valued, equity in physical terms implies equity in welfare terms. In allocation problems of objects via lotteries, when preferences are strict, equal treatment of equals in physical terms and equal treatment of equals in welfare are equivalent (Bogomolnaia and Moulin, 2001). It is important to note that in our model, equity in physical terms is not related to equity in welfare terms, i.e., symmetry does not imply equal treatment of equals in welfare and vice versa. The rule that selects all efficient queues and sets each agent's transfer equal to zero satisfies symmetry, but not equal treatment of equals in welfare. Proper subcorrespondences of the rule that is the union of all single-valued balanced Groves rules associated with [Formula presented] satisfy equal treatment of equals in welfare, but not symmetry. Recently, Chun et al. (in press) prove that the Largest Equally Distributed Pairwise Pivotal rule is the only rule that satisfies Pareto-efficiency, equal treatment of equals in welfare, Pareto-indifference,1 and strong strategy-proofness.2 They make use of essential single-valuedness of the Largest Equally Distributed Pairwise Pivotal rule in their proof. For completeness, we would like to mention that Hashimoto and Saitoh (2012) show that under strategy-proofness, anonymity in welfare3 implies queue-efficiency and by using Statements (1) and (2) in Theorem 3, they prove that the Largest Equally Distributed Pairwise Pivotal rule is the only rule that satisfies budget balancedness, anonymity in welfare, and strategy-proofness. © 2016 Elsevier Inc.
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Corrigendum , to , Characterizations , Pareto-efficient , fair , strategy-proof , allocation , rules , queueing , problems
