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Refinement Derivatives and Values of Games

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  • Luigi Montrucchio
  • Patrizia Semeraro

Abstract

A definition of set-wise differentiability for set functions is given through refining the partitions of sets. Such a construction is closely related to the one proposed by Rosenmuller (1977) as well as that studied by Epstein (1999) and Epstein and Marinacci (2001). We present several classes of TU games which are differentiable and study differentiation rules. The last part of the paper applies refinement derivatives to the calculation of value of games. Following Hart and Mas-Colell (1989), we define a value operator through the derivative of the potential of the game. We show that this operator is a truly value when restricted to some appropriate spaces of games. We present two alternative spaces where this occurs: the spaces pM( ) and POT2. The latter space is closely related to Myerson's balanced contribution axiom.

Suggested Citation

  • Luigi Montrucchio & Patrizia Semeraro, 2006. "Refinement Derivatives and Values of Games," Carlo Alberto Notebooks 9, Collegio Carlo Alberto.
    • Handle: RePEc:cca:wpaper:9
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    References listed on IDEAS

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    1. Neyman, Abraham, 2002. "Values of games with infinitely many players," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 3, chapter 56, pages 2121-2167, Elsevier.
    2. Massimo Marinacci & Luigi Montrucchio, 2005. "Stable cores of large games," International Journal of Game Theory, Springer;Game Theory Society, vol. 33(2), pages 189-213, June.
    3. MERTENS, Jean-François, 1980. "Values and derivatives," LIDAM Reprints CORE 435, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Dov Monderer, 1990. "A Milnor Condition for Nonatomic Lipschitz Games and Its Applications," Mathematics of Operations Research, INFORMS, vol. 15(4), pages 714-723, November.
    5. Calvo, Emilio & Santos, Juan Carlos, 1997. "Potentials in cooperative TU-games," Mathematical Social Sciences, Elsevier, vol. 34(2), pages 175-190, October.
    6. Dov Monderer & Ezra Einy & Diego Moreno, 1998. "The least core, kernel and bargaining sets of large games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 11(3), pages 585-601.
    7. Epstein, Larry G. & Marinacci, Massimo, 2001. "The Core of Large Differentiable TU Games," Journal of Economic Theory, Elsevier, vol. 100(2), pages 235-273, October.
    8. Marinacci, Massimo & Montrucchio, Luigi, 2003. "Subcalculus for set functions and cores of TU games," Journal of Mathematical Economics, Elsevier, vol. 39(1-2), pages 1-25, February.
    9. Hart, Sergiu & Mas-Colell, Andreu, 1989. "Potential, Value, and Consistency," Econometrica, Econometric Society, vol. 57(3), pages 589-614, May.
    10. Larry G. Epstein, 1999. "A Definition of Uncertainty Aversion," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 66(3), pages 579-608.
    11. Sergiu Hart & Dov Monderer, 1997. "Potentials and Weighted Values of Nonatomic Games," Mathematics of Operations Research, INFORMS, vol. 22(3), pages 619-630, August.
    12. Massimo Marinacci & Luigi Montrucchio, 2003. "Ultramodular functions," ICER Working Papers - Applied Mathematics Series 13-2003, ICER - International Centre for Economic Research.
    13. Jean-François Mertens, 1980. "Values and Derivatives," Mathematics of Operations Research, INFORMS, vol. 5(4), pages 523-552, November.
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    Cited by:

    1. Massimiliano Amarante & Luigi Montrucchio, 2010. "The bargaining set of a large game," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 43(3), pages 313-349, June.
    2. Francesca Centrone, 2016. "Representation of Epstein-Marinacci derivatives of absolutely continuous TU games," Economics Bulletin, AccessEcon, vol. 36(2), pages 1149-1159.

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    More about this item

    Keywords

    TU games; large games; non-additive set functions; value; derivatives;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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