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On the existence and uniqueness of pairwise stable networks

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  • Tim Hellmann

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Abstract

In this paper we show how externalities between links affect the existence and uniqueness of pairwise stable (PS) networks. For this we introduce the properties ordinal convexity (concavity) and ordinal strategic complements (substitutes) of utility functions on networks. It is shown that there exists at least one PS network if the profile of utility functions is ordinal convex and satisfies the ordinal strategic complements property. On the other hand, ordinal concavity and ordinal strategic substitutes are sufficient for some uniqueness properties of PS networks. Additionally, we elaborate on the relation of the link externality properties to definitions in the literature. Copyright Springer-Verlag 2013

Suggested Citation

  • Tim Hellmann, 2013. "On the existence and uniqueness of pairwise stable networks," International Journal of Game Theory, Springer;Game Theory Society, vol. 42(1), pages 211-237, February.
  • Handle: RePEc:spr:jogath:v:42:y:2013:i:1:p:211-237 DOI: 10.1007/s00182-012-0335-9
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    References listed on IDEAS

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    1. Sudipta Sarangi & Robert P. Gilles, 2005. "Stable Networks and Convex Payoffs," Departmental Working Papers 2005-13, Department of Economics, Louisiana State University.
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      • Dutta, Bhaskar & Mutuswami, Suresh, 1996. "Stable Networks," Working Papers 971, California Institute of Technology, Division of the Humanities and Social Sciences.
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    5. Francis Bloch & Matthew Jackson, 2006. "Definitions of equilibrium in network formation games," International Journal of Game Theory, Springer;Game Theory Society, vol. 34(3), pages 305-318, October.
    6. Berno Buechel & Tim Hellmann, 2012. "Under-connected and over-connected networks: the role of externalities in strategic network formation," Review of Economic Design, Springer;Society for Economic Design, pages 71-87.
    7. Bloch, Francis & Jackson, Matthew O., 2007. "The formation of networks with transfers among players," Journal of Economic Theory, Elsevier, pages 83-110.
    8. Pascal Billand & Christophe Bravard & Sudipta Sarangi, 2011. "Local Spillovers, Convexity and the Strategic Substitutes Property in Networks," Post-Print halshs-00574265, HAL.
    9. Subhadip Chakrabarti & Robert Gilles, 2007. "Network potentials," Review of Economic Design, Springer;Society for Economic Design, pages 13-52.
    10. Subhadip Chakrabarti & Robert Gilles, 2007. "Network potentials," Review of Economic Design, Springer;Society for Economic Design, vol. 11(1), pages 13-52, June.
    11. Venkatesh Bala & Sanjeev Goyal, 2000. "A Noncooperative Model of Network Formation," Econometrica, Econometric Society, vol. 68(5), pages 1181-1230, September.
    12. Monderer, Dov & Shapley, Lloyd S., 1996. "Potential Games," Games and Economic Behavior, Elsevier, vol. 14(1), pages 124-143, May.
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    Citations

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    Cited by:

    1. Dawid, Herbert & Hellmann, Tim, 2014. "The evolution of R&D networks," Journal of Economic Behavior & Organization, Elsevier, vol. 105(C), pages 158-172.
    2. Hellmann, Tim & Staudigl, Mathias, 2014. "Evolution of social networks," European Journal of Operational Research, Elsevier, vol. 234(3), pages 583-596.
    3. Philipp Möhlmeier & Agnieszka Rusinowska & Emily Tanimura, 2013. "A degree-distance-based connections model with negative and positive externalities," Documents de travail du Centre d'Economie de la Sorbonne 13040, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    4. Vincent Boucher, 2017. "The Estimation of Network Formation Games with Positive Spillovers," Cahiers de recherche 1710, Centre de recherche sur les risques, les enjeux économiques, et les politiques publiques.
    5. Bryan S. Graham, 2015. "Methods of Identification in Social Networks," Annual Review of Economics, Annual Reviews, vol. 7(1), pages 465-485, August.
    6. Miyauchi, Yuhei, 2016. "Structural estimation of pairwise stable networks with nonnegative externality," Journal of Econometrics, Elsevier, pages 224-235.
    7. Gauer, Florian & Hellmann, Tim, 2017. "Strategic formation of homogeneous bargaining networks," Center for Mathematical Economics Working Papers 529, Center for Mathematical Economics, Bielefeld University.
    8. Hellmann, Tim & Landwehr, Jakob, 2014. "Stable Networks in Homogeneous Societies," Center for Mathematical Economics Working Papers 517, Center for Mathematical Economics, Bielefeld University.
    9. repec:hal:journl:halshs-01316936 is not listed on IDEAS
    10. Philipp Möhlmeier & Agnieszka Rusinowska & Emily Tanimura, 2016. "Competition for the access to and use of information in networks," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01316936, HAL.
    11. Gauer, Florian, 2015. "Strategic Formation of Homogeneous Bargaining Networks," Annual Conference 2015 (Muenster): Economic Development - Theory and Policy 112943, Verein für Socialpolitik / German Economic Association.
    12. repec:eee:gamebe:v:106:y:2017:i:c:p:51-74 is not listed on IDEAS
    13. Philipp Möhlmeier & Agnieszka Rusinowska & Emily Tanimura, 2016. "Competition for the access to and use of information in networks," Documents de travail du Centre d'Economie de la Sorbonne 16033, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    14. Dawid, Herbert & Hellmann, Tim, 2014. "The evolution of R&D networks," Center for Mathematical Economics Working Papers 467, Center for Mathematical Economics, Bielefeld University.
    15. Philippe Bich & Lisa Morhaim, 2017. "On the existence of Pairwise stable weighted networks," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01564591, HAL.

    More about this item

    Keywords

    Network formation; Pairwise stability; Existence; Uniqueness; Convexity; Strategic complements; A14; C72; D85;

    JEL classification:

    • A14 - General Economics and Teaching - - General Economics - - - Sociology of Economics
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D85 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Network Formation

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