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Approximate Coalitional Equilibria in the Bipolar World

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Part of the book series: Communications in Computer and Information Science ((CCIS,volume 974))

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Abstract

We study a discrete model of jurisdiction formation in the spirit of Alesina and Spolaore [1]. A finite number of agents live along a line. They can be divided into several groups. If a group is formed, then some facility is located at its median and every member x of a group S with a median m pays \(\frac{1}{|S|}+|x-m|\).

We consider the notion of coalitional stability: a partition is stable if no coalition wishes to form a new group decreasing the cost of all members. It was shown by Savvateev et al. [4] that no stable partition may exist even for 5 agents living at 2 points. We now study approximately stable partitions: no coalition wishes to form a new group decreasing all costs by at least \(\epsilon \).

In this work, we define a relative measure of partition instability and consider bipolar worlds where all agents live in just 2 points. We prove that the maximum possible value of this measure is approximately \(6.2\%\).

Supported with RFBR 16-01-00362 and RaCAF ANR-15-CE40-0016-01 grants.

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Notes

  1. 1.

    This is the type of stability developed by Aumann and Drèze [2].

  2. 2.

    This rule was also employed in the migrational models due to Bolton and Roland [6] and Jehiel and Scotchmer [10].

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Acknowledgments

We want to thank Alexei Savvateev for his support and advice during the work on this paper.

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Authors and Affiliations

  1. Moscow Institute of Physics and Technology, Dolgoprudny, Russia

    Andrei Golman & Daniil Musatov

  2. Russian Presidential Academy of National Economy and Public Administration, Moscow, Russia

    Daniil Musatov

  3. Caucasus Mathematical Center at Adyghe State University, Maykop, Russia

    Daniil Musatov

Authors
  1. Andrei Golman
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  2. Daniil Musatov
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Corresponding author

Correspondence to Daniil Musatov .

Editor information

Editors and Affiliations

  1. FRC CSC RAS, Dorodnitsyn Computing Centre, Moscow, Russia

    Yury Evtushenko

  2. University of Montenegro, Podgorica, Montenegro

    Milojica Jaćimović

  3. Krasovsky Institute of Mathematics and Mechanics, Yekaterinburg, Russia

    Michael Khachay

  4. Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia

    Yury Kochetov

  5. FRC CSC RAS, Dorodnitsyn Computing Centre, Moscow, Russia

    Vlasta Malkova

  6. FRC CSC RAS, Dorodnitsyn Computing Centre, Moscow, Russia

    Mikhail Posypkin

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Golman, A., Musatov, D. (2019). Approximate Coalitional Equilibria in the Bipolar World. In: Evtushenko, Y., Jaćimović, M., Khachay, M., Kochetov, Y., Malkova, V., Posypkin, M. (eds) Optimization and Applications. OPTIMA 2018. Communications in Computer and Information Science, vol 974. Springer, Cham. https://doi.org/10.1007/978-3-030-10934-9_36

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  • DOI: https://doi.org/10.1007/978-3-030-10934-9_36

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  • Online ISBN: 978-3-030-10934-9

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