Stable partitions in many division problems: the proportional and the sequential dictator solutions
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We study how to partition a set of agents in a stable way when each coalition in the partition has to share a unit of a perfectly divisible good, and each agent has symmetric single-peaked preferences on the unit interval of his potential shares. A rule on the set of preference profiles consists of a partition function and a solution. Given a preference profile, a partition is selected and as many units of the good as the number of coalitions in the partition are allocated, where each unit is shared among all agents belonging to the same coalition according to the solution. A rule is stable at a preference profile if no agent strictly prefers to leave his coalition to join another coalition and all members of the receiving coalition want to admit him. We show that the proportional solution and all sequential dictator solutions admit stable partition functions. We also show that stability is a strong requirement that becomes easily incompatible with other desirable properties like efficiency, strategy-proofness, anonymity, and non-envyness.
KeywordsDivision problem Symmetric single-peaked preferences Stable partition
We thank two referees of this journal for helpful comments. The work of G. Bergantiños is partially supported by research Grant ECO2011-23460 from the Spanish Ministry of Science and Innovation and FEDER. J. Massó acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2011-0075) and through Grant ECO2008-0475-FEDER (Grupo Consolidado-C), and from the Generalitat de Catalunya, through the prize “ ICREA Academia” for excellence in research and Grants SGR 2009-419 and 2014-515. The work of A. Neme is partially supported by the Universidad Nacional de San Luis, through Grant 319502, and by the Consejo Nacional de Investigaciones Científicas y T écnicas (CONICET), through Grant PIP 112-200801-00655. Part of this research was done while A. Neme visited the UAB thanks to the Generalitat de Catalunya Grant SGR 2009-419.
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