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Sufficient Conditions for the Unique Stable Sets in Three Agent Pillage Games

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  • Manfred Kerber
  • Colin Rowat

Abstract

Pillage games [Jordan, 2006, "Pillage and Property", JET] have two features that make them richer than cooperative games in either characteristic or partition function form: they allow power externalities between coalitions; they allow resources to contribute to coalitions' power as well as to their utility. Extending von Neumann and Morgenstern's analysis of three agent games in characteristic function form to anonymous pillage games, we find: when the core is non-empty, it must take one of five forms; all such games with an empty core represent the same dominance relation. When a stable set exists, and the game also satisfies a continuity and a responsiveness axiom, it is unique and contains no more than 15 elements, a tight bound. By contrast, stable sets in three agent games in characteristic or partition function form may not be unique, and may contain continua. Finally, we provide an algorithm for computing the stable set, and can easily decide non-existence. Thus, in addition to offering attractive modelling possibilities, pillage games seem well behaved and analytically tractable, overcoming a difficulty that has long impeded use of cooperative game theory's flexibility.

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Bibliographic Info

Paper provided by Department of Economics, University of Birmingham in its series Discussion Papers with number 12-11.

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Length: 32 pages
Date of creation: Nov 2012
Date of revision:
Handle: RePEc:bir:birmec:12-11

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Postal: Edgbaston, Birmingham, B15 2TT
Web page: http://www.economics.bham.ac.uk
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Keywords: co-operative game theory; stable sets; algorithm; core;

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References

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  1. Lucas, William F., 1992. "Von Neumann-Morgenstern stable sets," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 1, chapter 17, pages 543-590 Elsevier.
  2. Alan F. Breardon & Colin Rowat, 2010. "Stable Sets in multi-good pillage games are small," Discussion Papers 10-05, Department of Economics, University of Birmingham.
  3. Thomas F. Hellmann & Noam Wasserman, 2011. "The First Deal: The Division of Founder Equity in New Ventures," NBER Working Papers 16922, National Bureau of Economic Research, Inc.
  4. Vincent Anesi, 2006. "Committees with Farsighted Voters: A New Interpretation of Stable Sets," Social Choice and Welfare, Springer, vol. 27(3), pages 595-610, December.
  5. Stergios Skaperdas, 1996. "Contest success functions (*)," Economic Theory, Springer, vol. 7(2), pages 283-290.
  6. Manfred Kerber & Colin Rowat, 2011. "A Ramsey bound on stable sets in Jordan pillage games," International Journal of Game Theory, Springer, vol. 40(3), pages 461-466, August.
  7. Geoffroy de Clippel & Roberto Serrano, 2008. "Bargaining, coalitions and externalities: A comment on Maskin," Working Papers 2008-11, Instituto MadrileƱo de Estudios Avanzados (IMDEA) Ciencias Sociales.
  8. Michele Piccione & Ariel Rubinstein, 2007. "Equilibrium in the Jungle," Economic Journal, Royal Economic Society, vol. 117(522), pages 883-896, 07.
  9. Ray, D. & Vohra, R., 1993. "Equilibrium Binding Agreements," Papers 21, Boston University - Department of Economics.
  10. Manfred Kerber & Colin Rowat, 2009. "Stable Sets in Three Agent Pillage Games," Discussion Papers 09-07, Department of Economics, University of Birmingham.
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