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# Attitude toward imprecise information

## Author

Listed:
• Thibault Gajdos

(CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

• Takashi Hayashi

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

• Jean-Marc Tallon

(CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, PSE - Paris School of Economics - ENPC - École des Ponts ParisTech - ENS Paris - École normale supérieure - Paris - PSL - Université Paris sciences et lettres - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique - EHESS - École des hautes études en sciences sociales - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

• Jean-Christophe Vergnaud

(CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

## 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.

## Suggested Citation

• Thibault Gajdos & Takashi Hayashi & Jean-Marc Tallon & Jean-Christophe Vergnaud, 2008. "Attitude toward imprecise information," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00451982, HAL.
• Handle: RePEc:hal:cesptp:halshs-00451982
DOI: 10.1016/j.jet.2007.09.002
Note: View the original document on HAL open archive server: https://halshs.archives-ouvertes.fr/halshs-00451982
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File URL: https://halshs.archives-ouvertes.fr/halshs-00451982/document

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## References listed on IDEAS

as
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Full references (including those not matched with items on IDEAS)

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### Keywords

Imprecise information; imprecision aversion; multiple priors; Steiner point;
All these keywords.

### JEL classification:

• D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty

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