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Additive Representation of Non-Additive Measures and the Choquet Integral

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  • Itzhak Gilboa
  • David Schmeidler

Abstract

This paper studies some new properties of set functions (and, in particular, "non-additive probabilities" or "capacities") and the Choquet integral with respect to such functions, in the case of a finite domain. We use an isomorphism between non-additive measures on the original space (of states of the world) and additive ones on a large space (of events), and embed the space of real-valued functions on the former in the corresponding space on the latter. This embedding gives rise to the following results: the Choquet integral with respect to any totally monotone capacity is an average over minima of the inegrand; the Choquet integral with respect to any capacity is the differences between minima of regular integrals over sets of additive measures; under fairly general conditions one may define a "Radon-Nikodym derivative" of one capacity with respect to another; the "optimistic" pseudo-Bayesian update of a non-additive measure follows from the Bayesian update of the corresponding additive measure on the large space. We also discuss the interpretation o these results and the new light they shed on the theory of expected utility maximization with respect to non-additive measures.

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Bibliographic Info

Paper provided by Northwestern University, Center for Mathematical Studies in Economics and Management Science in its series Discussion Papers with number 985.

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Date of creation: Apr 1992
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Handle: RePEc:nwu:cmsems:985

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Cited by:
  1. Marcello Basili & Carlo Zappia, 2005. "Ambiguity and uncertainty in Ellsberg and Shackle," Department of Economics University of Siena 460, Department of Economics, University of Siena.
  2. Mingli Zheng & Sajid Anwar, 2005. "Rational Legal Decision-Making, Value Judgment, and Efficient Precaution in Tort Law," Journal of Institutional and Theoretical Economics (JITE), Mohr Siebeck, Tübingen, vol. 161(3), pages 411-, September.
  3. Itzhak Gilboa & David Schmeidler, 1992. "Canonical Representation of Set Functions," Discussion Papers 986, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  4. Scott E. Page, 1998. "Uncertainty, Difficulty, and Complexity," Research in Economics 98-08-076e, Santa Fe Institute.
  5. Borglin, Anders & Flåm, Sjur, 2007. "Rationalizing Constrained Contingent Claims," Working Papers 2007:12, Lund University, Department of Economics.
  6. Marcello Basili & Carlo Zappia, 2007. "The weight of argument and non-additive measures: a note," Department of Economic Policy, Finance and Development (DEPFID) University of Siena 003, Department of Economic Policy, Finance and Development (DEPFID), University of Siena.
  7. Stefano Ficco & Vladimir A. Karamychev, 2004. "Information Overload in Multi-Stage Selection Procedures," Tinbergen Institute Discussion Papers 04-077/1, Tinbergen Institute.
  8. Carlo Zappia, 2008. "Non-Bayesian decision theory ante-litteram: the case of G. L. S. Shackle," Department of Economic Policy, Finance and Development (DEPFID) University of Siena 0408, Department of Economic Policy, Finance and Development (DEPFID), University of Siena.

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