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Canonical Representation of Set Functions

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  • Itzhak Gilboa
  • David Schmeidler

Abstract

The representation of a cooperative transferable utility game as a linear combination of unanimity games may be viewed as an isomorphism between not-necessarily additive set functions on the players space and additive ones on the coalitions space. We extend the unanimity-basis representation to general (infinite) spaces of players, study spaces of games of games which satisfy certain properties and provide some conditions for sigma-additivity of the resulting additive set function (on the space of coalitions). These results also allow us to extend some representations of the Choquet integral from finite to infinite spaces.

Suggested Citation

  • Itzhak Gilboa & David Schmeidler, 1992. "Canonical Representation of Set Functions," Discussion Papers 986, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  • Handle: RePEc:nwu:cmsems:986
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    File URL: http://www.kellogg.northwestern.edu/research/math/papers/986.pdf
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    References listed on IDEAS

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    1. Gilboa Itzhak & Schmeidler David, 1993. "Updating Ambiguous Beliefs," Journal of Economic Theory, Elsevier, vol. 59(1), pages 33-49, February.
    2. Gilboa, Itzhak & Lehrer, Ehud, 1991. "Global Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(2), pages 129-147.
    3. Itzhak Gilboa & David Schmeidler, 1992. "Additive Representation of Non-Additive Measures and the Choquet Integral," Discussion Papers 985, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
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    Cited by:

    1. Gilboa, Itzhak & Samuelson, Larry & Schmeidler, David, 2013. "Dynamics of inductive inference in a unified framework," Journal of Economic Theory, Elsevier, vol. 148(4), pages 1399-1432.
    2. Ghirardato, Paolo, 1997. "On Independence for Non-Additive Measures, with a Fubini Theorem," Journal of Economic Theory, Elsevier, vol. 73(2), pages 261-291, April.
    3. Alain Chateauneuf & Jean-Philippe Lefort, 2006. "Some Fubini theorems on sigma-algebras for non additive measures," Cahiers de la Maison des Sciences Economiques b06086, Université Panthéon-Sorbonne (Paris 1).
    4. Ozan Candogan & Ishai Menache & Asuman Ozdaglar & Pablo A. Parrilo, 2011. "Flows and Decompositions of Games: Harmonic and Potential Games," Mathematics of Operations Research, INFORMS, vol. 36(3), pages 474-503, August.
    5. Massimo Marinacci, 1996. "Decomposition and Representation of Coalitional Games," Mathematics of Operations Research, INFORMS, vol. 21(4), pages 1000-1015, November.
    6. Aouani, Zaier & Chateauneuf, Alain, 2008. "Exact capacities and star-shaped distorted probabilities," Mathematical Social Sciences, Elsevier, vol. 56(2), pages 185-194, September.
    7. Ghirardato, Paolo & Le Breton, Michel, 2000. "Choquet Rationality," Journal of Economic Theory, Elsevier, vol. 90(2), pages 277-285, February.
    8. Simon Grant & Atsushi Kajii, 2005. "Probabilistically Sophisticated Multiple Priors," KIER Working Papers 608, Kyoto University, Institute of Economic Research.

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