Abstract
We search for impartiality in the allocation of objects when monetary transfers are not possible. Our main focus is anonymity. The standard definition requires that if agents’ names are permuted, their assignments be permuted in the same way. Since no rule satisfies this definition, we introduce weaker variants, “anonymity for distinct preferences,” “pairwise-anonymity for distinct preferences,” “pairwise-anonymity for fully differentiated profiles,” and “independence of others’ permutations.” We show that for more than two agents and two objects, no rule is pairwise-anonymous for distinct preferences and Pareto-efficient (Theorem 1), no rule is pairwise-anonymous for distinct preferences and independent of others’ permutations (Theorem 2), and no rule is pairwise-anonymous for fully differentiated profiles and strategy-proof (Theorem 3).
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Notes
Sometimes several objects can be equally ranked in preferences. However, allowing such indifferences makes it difficult to construct desirable rules. See Ehlers (2002).
For this result, standard anonymity can even be strengthened to allow both agents’ and objects’ names to be permuted.
To be more concrete, the core and the no-trade rules are the only rules satisfying standard anonymity, strategy-proofness, individual rationality and nonbossiness. For the definitions of individual rationality and nonbossiness, see Miyagawa (2002).
The process of the Random Priority rule is as follows: Take an agent priority, and based on it, each agent receives his favorite object. Next, take another agent priority, and based on it, each agent receives his favorite object. Then, take the average probability of the allocations derived through the above processes.
Even if each agent has single-peaked preferences, this incompatibility still holds (Kasajima 2013).
We discuss the case of \(|N|\ne |K|\) in Sect. 5.
Preference relation \(R_{i}\) is complete if for each \(a,b\in K\), \(a\,R_{i}\,b\) or \(b\,R_{i}\,a\), transitive if for each \(a,b,c\in K\), \( a\,R_{i}\,b\) and \(b\,R_{i}\,c\) imply \(a\,R_{i}\,c\), antisymmetric if for each \(a,b\in K\), \(a\,R_{i}\,b\) and \(b\,R_{i}\,a\) imply \(a=b\).
Equal treatment of equals: For each \(R\in {\mathcal {R}}^N\) and each pair \(\{i,j\}\subset N\), if \(R_i=R_j\), \(f_i(R)=f_j(R)\).
To be exact, pairwise-anonymity for distinct preferences implies independence of others’ permutations for \(n=3\), so in this case no rule is pairwise-anonymous for distinct preferences.
References
Abdulkadiroğlu A, Sönmez T (1999) House allocation with existing tenants. J Econ Theory 88(2):233–260
Anno H (2015) A short proof for the characterization of the core in housing markets. Econ Lett 126:66–67
Bogomolnaia A, Moulin H (2001) A new solution to the random assignment problem. J Econ Theory 100(2):295–328
Ehlers L (2002) Coalitional strategy-proof house allocation. J Econ Theory 105(2):298–317
Hatfield JW (2009) Strategy-proof, efficient, and nonbossy quota allocations. Soc Choice Welf 33(3):505–515
Kasajima Y (2013) Probabilistic assignment of indivisible goods with single-peaked preferences. Soc Choice Welf 41(1):203–215
Kesten O, Yazici A (2012) The Pareto dominant strategy-proof and equitable rule for problems with indivisible goods. Econ Theory 50(2):463–488
Miyagawa E (2002) Strategy-proofness and the core in house allocation problems. Games Econ Behav 38(2):347–361
Pápai S (2000a) Strategyproof assignment by hierarchical exchange. Econometrica 68(6):1403–1433
Pápai S (2000b) Strategyproof multiple assignment using quotas. Rev Econ Des 5(1):91–105
Shapley L, Scarf H (1974) On cores and indivisibility. J Math Econ 1(1):23–37
Svensson L (1999) Strategy-proof allocation of indivisible goods. Soc Choice Welf 16(4):557–567
Takamiya K (2001) Coalition strategy-proofness and monotonicity in Shapley-Scarf housing markets. Math Soc Sci 41(2):201–213
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I am very grateful to Shigehiro Serizawa for his significant contribution to this work, and to William Thomson for reading this paper repeatedly and giving me many comments to improve the paper. I also thank the associate editor, two anonymous referees, Masaki Aoyagi, Agustin Bonifacio, Patrick Harless, Kazuhiko Hashimoto, Morimitsu Kurino, Noriaki Matsushima, James Schummer, Tayfun Sonmez, Wataru Tamura, Utku Unver, Jun Wako, participants at Frontiers of Market Design, Osaka University Applied Microeconomic Theory Workshop, the 11th Meeting of Society for Social Choice and Welfare, the Osaka-Rochester Theory Workshop, PET 14 Seattle, 2014 Autumn Meeting of Japanese Economic Association, 2014 SSK Workshop on Distributive Justice in Honor of Professor William Thomson, Aomori Public University Workshop on mechanism design, environment and social welfare, ISI-ISER Young Economists Workshop, and especially Shuhei Morimoto and Koji Takamiya for their comments. This work is supported by Grant-in-Aid for JSPS Fellows and the Joint Usage/Research Center at the Institute of Social and Economic Research at Osaka University. All remaining errors are mine.
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Kondo, H. Notions of anonymity for object assignment: impossibility theorems. Rev Econ Design 23, 113–126 (2019). https://doi.org/10.1007/s10058-019-00223-1
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DOI: https://doi.org/10.1007/s10058-019-00223-1