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# Entropy and the value of information for investors

## Author

Listed:
• Antonio Cabrales

• Olivier Gossner

(PSE - Paris-Jourdan Sciences Economiques - ENS Paris - École normale supérieure - Paris - INRA - Institut National de la Recherche Agronomique - EHESS - École des hautes études en sciences sociales - ENPC - École des Ponts ParisTech - CNRS - Centre National de la Recherche Scientifique, PSE - Paris School of Economics, LSE - London School of Economics and Political Science)

• Roberto Serrano

(Department of Economics - Brown University, Institute IMDEA Software [Madrid])

## Abstract

Consider any investor who fears ruin when facing any set of investments that satisfy no-arbitrage. Before investing, he can purchase information about the state of nature in the form of an information structure. Given his prior, information structure $\alpha$ is more informative than information structure $\beta$ if, whenever he is willing to buy $\beta$ at some price, he is also willing to buy $\alpha$ at that price. We show that this informativeness ordering is complete and is represented by the decrease in entropy of his beliefs, regardless of his preferences, initial wealth, or investment problem. We also show that no prior-independent informativeness ordering based on similar premises exists.

## Suggested Citation

• Antonio Cabrales & Olivier Gossner & Roberto Serrano, 2011. "Entropy and the value of information for investors," PSE Working Papers halshs-00648884, HAL.
• Handle: RePEc:hal:psewpa:halshs-00648884
Note: View the original document on HAL open archive server: https://halshs.archives-ouvertes.fr/halshs-00648884
as

File URL: https://halshs.archives-ouvertes.fr/halshs-00648884/document

## References listed on IDEAS

as
1. Thierry Post & Martijn J. van den Assem & Guido Baltussen & Richard H. Thaler, 2008. "Deal or No Deal? Decision Making under Risk in a Large-Payoff Game Show," American Economic Review, American Economic Association, vol. 98(1), pages 38-71, March.
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6. Azrieli, Yaron & Lehrer, Ehud, 2008. "The value of a stochastic information structure," Games and Economic Behavior, Elsevier, vol. 63(2), pages 679-693, July.
7. Sergiu Hart, 2011. "Comparing Risks by Acceptance and Rejection," Journal of Political Economy, University of Chicago Press, vol. 119(4), pages 617-638.
8. Susan Athey & Jonathan Levin, 1998. "The Value of Information In Monotone Decision Problems," Working papers 98-24, Massachusetts Institute of Technology (MIT), Department of Economics.
9. Olivier Gossner, 2011. "Simple Bounds on the Value of a Reputation," Econometrica, Econometric Society, vol. 79(5), pages 1627-1641, September.
10. Olivier Gossner & Tristan Tomala, 2006. "Empirical Distributions of Beliefs Under Imperfect Observation," Mathematics of Operations Research, INFORMS, vol. 31(1), pages 13-30, February.
11. Blume, Lawrence & Easley, David, 1992. "Evolution and market behavior," Journal of Economic Theory, Elsevier, vol. 58(1), pages 9-40, October.
Full references (including those not matched with items on IDEAS)

## Citations

Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
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Cited by:

1. Alexis Toda, 2015. "Bayesian general equilibrium," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 58(2), pages 375-411, February.
2. Antonio Jiménez-Martínez, 2015. "A model of belief influence in large social networks," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 59(1), pages 21-59, May.
3. Antony Millner & Daniel Heyen, 2017. "Valuing predictability," GRI Working Papers 260, Grantham Research Institute on Climate Change and the Environment.
4. Luc Bridet & Margaret Leighton, 2015. "The Major Decision: Labor Market Implications of the Timing of Specialization in College," Discussion Paper Series, Department of Economics 201510, Department of Economics, University of St. Andrews.
5. Kyungmin Kim & Benjamin Lester & Braz Camargo, 2012. "Subsidizing Price Discovery," 2012 Meeting Papers 338, Society for Economic Dynamics.
6. Antonio Cabrales & Olivier Gossner & Roberto Serrano, 2017. "A normalized value for information purchases," Working Papers 2017-51, Center for Research in Economics and Statistics, revised 05 Jan 2017.
7. repec:spr:annopr:v:262:y:2018:i:2:d:10.1007_s10479-016-2256-7 is not listed on IDEAS
8. Antonio Cabrales & Olivier Gossner & Roberto Serrano, 2012. "The Appeal of Information Transactions," Working Papers 2012-13, Brown University, Department of Economics.
9. Stark, Oded & Jakubek, Marcin & Falniowski, Fryderyk, 2014. "Reconciling the Rawlsian and the utilitarian approaches to the maximization of social welfare," Economics Letters, Elsevier, vol. 122(3), pages 439-444.
10. Bellalah, Mondher, 2016. "Shadow costs of incomplete information and short sales in the valuation of the firm and its assets," The North American Journal of Economics and Finance, Elsevier, vol. 37(C), pages 406-419.
11. Agostino Manduchi, 2013. "Non-neutral information costs with match-value uncertainty," Journal of Economics, Springer, vol. 109(1), pages 1-25, May.
12. Volha Audzei, 2015. "Information Acquisition and Excessive Risk: Impact of Policy Rate and Market Volatility," ACTA VSFS, University of Finance and Administration, vol. 9(2), pages 115-135.
13. Braz Camargo, 2011. "Career Concerns: A Human Capital Perspective," 2011 Meeting Papers 1274, Society for Economic Dynamics.
14. Mariana Cunha & António Osório & Ricardo Ribeiro, 2016. "Endogenous product design and quality with rationally inattentive consumers," Working Papers de Economia (Economics Working Papers) 03, Católica Porto Business School, Universidade Católica Portuguesa.
15. Li, Jian & Zhou, Junjie, 2016. "Blackwell's informativeness ranking with uncertainty-averse preferences," Games and Economic Behavior, Elsevier, vol. 96(C), pages 18-29.
16. Lindbeck, Assar & Weibull, Jörgen, 2017. "Investment, Rational Inattention, and Delegation," Working Paper Series 1171, Research Institute of Industrial Economics.

### Keywords

Informativeness; Information structures; Entropy; Decision under uncertainty; Investment; Blackwell ordering;

### JEL classification:

• C00 - Mathematical and Quantitative Methods - - General - - - General
• C43 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Index Numbers and Aggregation
• D00 - Microeconomics - - General - - - General
• D80 - Microeconomics - - Information, Knowledge, and Uncertainty - - - General
• D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
• G00 - Financial Economics - - General - - - General
• G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

### NEP fields

This paper has been announced in the following NEP Reports:

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