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A generalization of the Shapley–Ichiishi result

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  • Jeroen Kuipers

    ()

  • Dries Vermeulen

    ()

  • Mark Voorneveld

    ()

Abstract

The Shapley-Ichiishi result states that a game is convex if and only if the convex hull of marginal vectors equals the core. In this paper we generalize this result by distinguishing equivalence classes of balanced games that share the same core structure. We then associate a system of linear inequalities with each equivalence class, and we show that the system defines the class. Application of this general theorem to the class of convex games yields an alternative proof of the Shapley-Ichiishi result. Other applications range from computation of stable sets in non-cooperative game theory to determination of classes of TU games on which the core correspondence is additive (even linear). For the case of convex games we prove that the theorem provides the minimal defining system of linear inequalities. An example shows that this is not necessarily true for other equivalence classes of balanced games.

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Bibliographic Info

Article provided by Springer in its journal International Journal of Game Theory.

Volume (Year): 39 (2010)
Issue (Month): 4 (October)
Pages: 585-602

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Handle: RePEc:spr:jogath:v:39:y:2010:i:4:p:585-602

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Related research

Keywords: TU games; Core; Linearity regions; Computation of Q-sets;

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  1. Curiel, I. & Pederzoli, G. & Tijs, S.H., 1989. "Sequencing games," Open Access publications from Tilburg University urn:nbn:nl:ui:12-154243, Tilburg University.
  2. Curiel, Imma & Pederzoli, Giorgio & Tijs, Stef, 1989. "Sequencing games," European Journal of Operational Research, Elsevier, vol. 40(3), pages 344-351, June.
  3. Hillas, John, 1990. "On the Definition of the Strategic Stability of Equilibria," Econometrica, Econometric Society, vol. 58(6), pages 1365-90, November.
  4. Vermeulen, A J & Potters, J A M & Jansen, M J M, 1996. "On Quasi-Stable Sets," International Journal of Game Theory, Springer, vol. 25(1), pages 43-49.
  5. Granot, D, et al, 1996. "The Kernel/Nucleolus of a Standard Tree Game," International Journal of Game Theory, Springer, vol. 25(2), pages 219-44.
  6. S. C. Littlechild & G. Owen, 1973. "A Simple Expression for the Shapley Value in a Special Case," Management Science, INFORMS, vol. 20(3), pages 370-372, November.
  7. Aumann, Robert J. & Maschler, Michael, 1985. "Game theoretic analysis of a bankruptcy problem from the Talmud," Journal of Economic Theory, Elsevier, vol. 36(2), pages 195-213, August.
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