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Attitude toward imprecise information

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Abstract

This paper presents an axiomatic model of decision making which incorporates objective but imprecise information. We axiomatize a decision criterion of the multiple priors (or maxmin expected utility) type. The model achieves two primary objectives. First, it explains how subjective belief varies with information. Second, it identifies an explicit attitude toward imprecision that underlies usual hedging axioms. Information is assumed to take the form of a probability-possibility set, that is, a set P of probability measures on the state space. The decision maker is told that the true probability law lies in P. She is assumed to rank pairs of the form (P,f) where P is a probability-possibility set and f is an act mapping states into outcomes. The representation result delivers multiple-priors utility at each probability-possibility set. There is a mapping that gives for each probability-possibility set the subjective set of priors. This allows both subjective expected utility when the subjective set of priors is reduced to a singleton and the other extreme where the decision maker takes the worst case scenario in the entire probability-possibility set. We show that the relation “more averse to imprecision” is characterized by inclusion of the sets of priors, irrespective of the utility functions that capture risk attitude. We characterize, under extra axioms, a more precise functional form, in which the subjective set of priors is obtained by (i) solving for the “mean value” of the probability-possibility set and (ii) shrinking the probability-possibility set toward the mean value to a degree determined by preference.

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File URL: ftp://mse.univ-paris1.fr/pub/mse/cahiers2006/V06081.pdf
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Bibliographic Info

Paper provided by Université Panthéon-Sorbonne (Paris 1) in its series Cahiers de la Maison des Sciences Economiques with number v06081.

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Length: 38 pages
Date of creation: Jul 2006
Date of revision:
Handle: RePEc:mse:wpsorb:v06081

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Keywords: Imprecise information; imprecision aversion; multiple priors; Steiner point.;

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References

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  1. Peter Klibanoff & Massimo Marinacci & Sujoy Mukerji, 2002. "A smooth model of decision making under ambiguity," ICER Working Papers - Applied Mathematics Series 11-2003, ICER - International Centre for Economic Research, revised Apr 2003.
  2. Thibault Gajdos & Jean-Marc Tallon & Jean-Christophe Vergnaud, 2002. "Decision Making with Imprecise Probabilistic Information," ICER Working Papers - Applied Mathematics Series 18-2003, ICER - International Centre for Economic Research, revised May 2003.
  3. Mukerji, Sujoy & Tallon, Jean-Marc, 2001. "Ambiguity Aversion and Incompleteness of Financial Markets," Review of Economic Studies, Wiley Blackwell, vol. 68(4), pages 883-904, October.
  4. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-87, May.
  5. Epstein, Larry G & Wang, Tan, 1994. "Intertemporal Asset Pricing Under Knightian Uncertainty," Econometrica, Econometric Society, vol. 62(2), pages 283-322, March.
  6. Wojciech Olszewski, 2007. "Preferences Over Sets of Lotteries -super-1," Review of Economic Studies, Oxford University Press, vol. 74(2), pages 567-595.
  7. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
  8. Tapking, Jens, 2004. "Axioms for preferences revealing subjective uncertainty and uncertainty aversion," Journal of Mathematical Economics, Elsevier, vol. 40(7), pages 771-797, November.
  9. F J Anscombe & R J Aumann, 2000. "A Definition of Subjective Probability," Levine's Working Paper Archive 7591, David K. Levine.
  10. Daniel Ellsberg, 2000. "Risk, Ambiguity and the Savage Axioms," Levine's Working Paper Archive 7605, David K. Levine.
  11. repec:hal:cesptp:halshs-00102346 is not listed on IDEAS
  12. Fabio Maccheroni & Massimo Marinacci & Aldo Rustichini, 2004. "Ambiguity Aversion, Robustness, and the Variational Representation of Preferences," Carlo Alberto Notebooks 12, Collegio Carlo Alberto, revised 2006.
  13. Epstein, Larry G, 1999. "A Definition of Uncertainty Aversion," Review of Economic Studies, Wiley Blackwell, vol. 66(3), pages 579-608, July.
  14. repec:hal:journl:halshs-00086021 is not listed on IDEAS
  15. Ghirardato, Paolo & Marinacci, M., 1997. "Ambiguity Made Precise: A Comparative Foundation and Some Implications," Working Papers 1026, California Institute of Technology, Division of the Humanities and Social Sciences.
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