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Stable partitions in many division problems: the proportional and the sequential dictator solutions

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Abstract

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.

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Notes

  1. For axiomatic characterizations of solutions see for example Barberà et al. (1997), Ching (1992, 1994), Dagan (1996), Ehlers (2002a, b), Herrero and Villar (2000), Schummer and Thomson (1997), Sönmez (1994), and Thomson (1994, 1995, 1997, 2003).

  2. Kar and Kibris (2008) consider the efficiency of such rules in a setting where the number of units to share is fixed rather than endogenous. They show that for the domain of single-peaked preferences and for well behaved solutions (efficient, non dictatorial, strategy proof, resource monotonic and consistent), it is not possible to find a partition such that the final allocation is efficient.

  3. In particular, in Gensemer et al. (1996) they show that a migration equilibrium might fail to exists if the solution applied to the local economies satisfies two of the following three properties: Pareto efficiency, strategy proofness and no-envy. In Gensemer et al. (1998) they show that a migration equilibrium might fail to exists whenever the solution applied to the local economies is either the Proportional, the Sequential Dictator, the Uniform or the Egalitarian rules.

  4. Conley and Konichi (2000) propose another stability concept, “migration-proof Tiebout equilibrium”, which is weaker than the migration equilibrium. They show that for sufficiently large populations of homogeneous agents, a migration-proof Tiebout equilibrium exists, is unique and asymptotically efficient. See also Sertel (1992) for an alternative notion of stability and efficiency of partitions, in which a membership property rights code is used in an abstract setting.

  5. Kar and Kibris (2008) show that in the domain of symmetric singled-peaked preferences, and when the number of units of the good to be shared is fixed, whenever a (local) solution is efficient, there exists a partition such that the final allocation is efficient. At the end of the paper we further discuss on the importance of assuming that agents have symmetric single-peaked preferences.

  6. Note that for both the proportional solution and the sequential dictator solution a migration equilibrium (Gensemer et al. (1996, 1998)) might fail to exists even when we restrict the preferences to be symmetric single-peaked.

  7. Note that the definition of solution allows for each subset \(S\subseteq N\) to have its own distinct solution \(f^{S}\). As we have already said in the Introduction, we restrict our analysis to the case in which a unique solution is applied to every coalition in the partition. This requirement implies that the same principles are used across coalitions and can be interpreted as a consistency requirement.

  8. If there exist \(i\) and \(k\) such that (1) and (2) hold we say \(i\) wants to leave \(S_{\pi }(i)\) to join \(S_{k}\) and all agents in \(S_{k}\) want to admit \( i\).

  9. Observe that our algorithm is more general since it can be applied to any solution. If it converges, the outcome is a stable partition for the solution.

  10. Kar and Kibris (2008) have a similar result in a setting where the number of goods to be shared is fixed. In our case however, the number of goods, and hence the number of coalitions in the partition, is endogenous and depend on the preferences of the agents.

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Acknowledgments

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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Authors and Affiliations

  1. Research Group in Economic Analysis, Facultad de Económicas, Universidad de Vigo, 36310, Vigo, Pontevedra, Spain

    Gustavo Bergantiños

  2. Departament d’Economia i d’Història Econòmica, Universitat Autònoma de Barcelona and Barcelona GSE, 08193, Bellaterra, Barcelona, Spain

    Jordi Massó

  3. Department of Economics and Nuffield College, University of Oxford, New Road, Oxford, OX1 1NF, UK

    Inés Moreno de Barreda

  4. Instituto de Matemática Aplicada de San Luis and COCINET, Universidad Nacional de San Luis, San Luis, Argentina

    Alejandro Neme

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  1. Gustavo Bergantiños
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  2. Jordi Massó
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  3. Inés Moreno de Barreda
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  4. Alejandro Neme
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Correspondence to Inés Moreno de Barreda.

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Bergantiños, G., Massó, J., Moreno de Barreda, I. et al. Stable partitions in many division problems: the proportional and the sequential dictator solutions. Theory Decis 79, 227–250 (2015). https://doi.org/10.1007/s11238-014-9467-7

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