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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.

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  • 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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    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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    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. Chateauneuf, Alain & Eichberger, Jurgen & Grant, Simon, 2007. "Choice under uncertainty with the best and worst in mind: Neo-additive capacities," Journal of Economic Theory, Elsevier, vol. 137(1), pages 538-567, November.
    3. Alain Chateauneuf & Jean-Philippe Lefort, 2006. "Some Fubini theorems on product sigma-algebras for non-additive measures," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00130444, HAL.
    4. Philippe, Fabrice, 2000. "Cumulative prospect theory and imprecise risk," Mathematical Social Sciences, Elsevier, vol. 40(3), pages 237-263, November.
    5. Yaron Azrieli & Christopher P. Chambers & Paul J. Healy, 2020. "Incentives in experiments with objective lotteries," Experimental Economics, Springer;Economic Science Association, vol. 23(1), pages 1-29, March.
    6. Marinacci, Massimo, 1999. "Limit Laws for Non-additive Probabilities and Their Frequentist Interpretation," Journal of Economic Theory, Elsevier, vol. 84(2), pages 145-195, February.
    7. De Waegenaere, Anja & Wakker, Peter P., 2001. "Nonmonotonic Choquet integrals," Journal of Mathematical Economics, Elsevier, vol. 36(1), pages 45-60, September.
    8. repec:dau:papers:123456789/7324 is not listed on IDEAS
    9. Massimo Marinacci, 1995. "Decomposition and Representation of Coalitional Games," Discussion Papers 1152, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    10. Aouani, Zaier & Chateauneuf, Alain, 2008. "Exact capacities and star-shaped distorted probabilities," Mathematical Social Sciences, Elsevier, vol. 56(2), pages 185-194, September.
    11. Ghirardato, Paolo & Le Breton, Michel, 2000. "Choquet Rationality," Journal of Economic Theory, Elsevier, vol. 90(2), pages 277-285, February.
    12. Silvia Bortot & Ricardo Alberto Marques Pereira & Thuy H. Nguyen, 2015. "Welfare functions and inequality indices in the binomial decomposition of OWA functions," DEM Discussion Papers 2015/08, Department of Economics and Management.
    13. Simon Grant & Atsushi Kajii, 2005. "Probabilistically Sophisticated Multiple Priors," KIER Working Papers 608, Kyoto University, Institute of Economic Research.
    14. Einy, Ezra & Holzman, Ron & Monderer, Dov & Shitovitz, Benyamin, 1997. "Core Equivalence Theorems for Infinite Convex Games," Journal of Economic Theory, Elsevier, vol. 76(1), pages 1-12, September.
    15. 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.
    16. 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).
    17. Silvia Bortot & Ricardo Alberto Marques Pereira & Thuy Nguyen, 2015. "On the binomial decomposition of OWA functions, the 3-additive case in n dimensions," Working Papers 360, ECINEQ, Society for the Study of Economic Inequality.
    18. 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.
    19. Stauber, Ronald, 2019. "A strategic product for belief functions," Games and Economic Behavior, Elsevier, vol. 116(C), pages 38-64.
    20. Ronald Stauber, 2019. "A strategic product for belief functions," ANU Working Papers in Economics and Econometrics 2019-668, Australian National University, College of Business and Economics, School of Economics.
    21. Sebastian Maaß, 2006. "A Philosophical Foundation of Non-Additive Measure and Probability," Theory and Decision, Springer, vol. 60(2), pages 175-191, May.
    22. Silvia Bortot & Ricardo Alberto Marques Pereira, 2013. "The binomial Gini inequality indices and the binomial decomposition of welfare functions," Working Papers 305, ECINEQ, Society for the Study of Economic Inequality.
    23. Gajdos, Thibault, 2002. "Measuring Inequalities without Linearity in Envy: Choquet Integrals for Symmetric Capacities," Journal of Economic Theory, Elsevier, vol. 106(1), pages 190-200, September.
    24. Takao Asano & Yusuke Osaki, 2020. "Portfolio allocation problems between risky and ambiguous assets," Annals of Operations Research, Springer, vol. 284(1), pages 63-79, January.

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