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Additive representations of non-additive measures and the choquet integral

  • Modeling Beliefs and Preferences
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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 larger 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 integrand;

  • the Choquet integral with respect to any capacity is the difference 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 larger space.

We also discuss the interpretation of 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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Authors and Affiliations

  1. Kellogg Graduate School of Management, Northwestern University, 60208, Evanston, IL, USA

    Itzhak Gilboa

  2. Department of Economics, Ohio State University, 43210, Columbus, OH, USA

    David Schmeidler

  3. Department of Statistics and Operations Research, Tel Aviv University, 69978, Tel Aviv, Israel

    David Schmeidler

Authors
  1. Itzhak Gilboa
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  2. David Schmeidler
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Additional information

We wish to thank Elchanan Ben-Porath, Dieter Denneberg, Didier Dubois, Gerald Hanweck, Jean-Yves Jaffray, Ehud Kalai, Morton Kamien, Ehud Lehrer, and two anonymous referees for comments and discussions. We are especially grateful to Bart Lipman who pointed out to us a few mistakes in an earlier version. NSF Grants Nos. SES-9113108 (Gilboa) and SES-9111873 (Schmeidler) are gratefully acknowledged.

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Gilboa, I., Schmeidler, D. Additive representations of non-additive measures and the choquet integral. Ann Oper Res 52, 43–65 (1994). https://doi.org/10.1007/BF02032160

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