# Attitude toward imprecise information

## Author Info

• Thibault Gajdos

() (CES - Centre d'économie de la Sorbonne - CNRS : UMR8174 - Université Paris I - Panthéon-Sorbonne)

• Takashi Hayashi

() (Department of Economics, University of Texas at Austin - University of Texas at Austin)

• Jean-Marc Tallon

() (CES - Centre d'économie de la Sorbonne - CNRS : UMR8174 - Université Paris I - Panthéon-Sorbonne, EEP-PSE - Ecole d'Économie de Paris - Paris School of Economics - Ecole d'Économie de Paris)

• Jean-Christophe Vergnaud

(CES - Centre d'économie de la Sorbonne - CNRS : UMR8174 - Université Paris I - Panthéon-Sorbonne)

## Abstract

This paper presents an axiomatic model of decision making under uncertainty which incorporates objective but imprecise information. 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 $P$ and is assumed to rank pairs of the form $(P,f)$ where $f$ is an act mapping states into outcomes. The key representation result delivers maxmin expected utility where the min operator ranges over a set of probability priors --just as in the maxmin expected utility (MEU) representation result of \cite{GILB/SCHM/89}. However, unlike the MEU representation, the representation here also delivers a mapping, $\varphi$, which links the probability-possibility set, describing the available information, to the set of revealed priors. The mapping $\varphi$ is shown to represent the decision maker's attitude to imprecise information: under our axioms, the set of representation priors is constituted as a selection from the probability-possibility set. This allows both expected utility when the selected set is a singleton and extreme pessimism when the selected set is the same as the probability-possibility set, i.e. , $\varphi$ is the identity mapping. We define a notion of comparative imprecision aversion and show it is characterized by inclusion of the sets of revealed probability distributions, irrespective of the utility functions that capture risk attitude. We also identify an explicit attitude toward imprecision that underlies usual hedging axioms. Finally, we characterize, under extra axioms, a more specific functional form, in which the set of selected probability distributions 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 preferences.

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## Bibliographic Info

Paper provided by HAL in its series Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) with number halshs-00451982.

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Date of revision:
Publication status: Published, Journal of Economic Theory, 2008, 140, 1, 23-56
Handle: RePEc:hal:cesptp:halshs-00451982

Note: View the original document on HAL open archive server: http://halshs.archives-ouvertes.fr/halshs-00451982
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Web page: http://hal.archives-ouvertes.fr/

## Related research

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

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Find related papers by JEL classification:
• D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty

## References

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1. Sujoy Mukerji & Peter Klibanoff, 2002. "A Smooth Model of Decision,Making Under Ambiguity," Economics Series Working Papers 113, University of Oxford, Department of Economics.
2. Fabio Maccheroni & Massimo Marinacci & Aldo Rustichini, 2006. "Ambiguity Aversion, Robustness, and the Variational Representation of Preferences," Econometrica, Econometric Society, vol. 74(6), pages 1447-1498, November.
3. Raphaël Giraud, 2006. "Objective Imprecise Probabilistic Information, Second Order Beliefs and Ambiguity Aversion: an Axiomatization," UniversitÃ© Paris1 PanthÃ©on-Sorbonne (Post-Print and Working Papers) halshs-00102346, HAL.
4. Thibault Gajdos & Jean-Marc Tallon & Jean-Christophe Vergnaud, 2004. "Decision Making with Imprecise Probabilistic Information," UniversitÃ© Paris1 PanthÃ©on-Sorbonne (Post-Print and Working Papers) halshs-00086021, HAL.
5. repec:oxf:wpaper:046 is not listed on IDEAS
6. 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.
7. Epstein, Larry G & Wang, Tan, 1994. "Intertemporal Asset Pricing Under Knightian Uncertainty," Econometrica, Econometric Society, vol. 62(2), pages 283-322, March.
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. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-87, May.
10. Daniel Ellsberg, 2000. "Risk, Ambiguity and the Savage Axioms," Levine's Working Paper Archive 7605, David K. Levine.
11. Mukerji, S. & Tallon, J.-M., 1999. "Ambiguity Aversion and Incompleteness of Financial Markets," Papiers d'Economie MathÃÂ©matique et Applications 1999-28, UniversitÃ© PanthÃ©on-Sorbonne (Paris 1).
12. Wojciech Olszewski, 2007. "Preferences Over Sets of Lotteries," Review of Economic Studies, Wiley Blackwell, vol. 74(2), pages 567-595, 04.
13. F J Anscombe & R J Aumann, 2000. "A Definition of Subjective Probability," Levine's Working Paper Archive 7591, David K. Levine.
14. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
15. repec:hal:journl:halshs-00086021 is not listed on IDEAS
16. Epstein, Larry G, 1999. "A Definition of Uncertainty Aversion," Review of Economic Studies, Wiley Blackwell, vol. 66(3), pages 579-608, July.
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