{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T01:46:07Z","timestamp":1775094367888,"version":"3.50.1"},"reference-count":21,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2014,6,30]],"date-time":"2014-06-30T00:00:00Z","timestamp":1404086400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Yang and Qiu proposed an expected utility-entropy (EU-E) measure of risk, which reflects an individual\u2019s intuitive attitude toward risk. Luce et al. have derived the numerical representations under behavioral axioms about preference orderings among gambles and their joint receipt, which further demonstrates the reasonability of the EU-E decision model as a normative one. In the paper, combining normalized expected utility and entropy together, we improve the EU-E measure of risk and decision model, and then propose the normalized EU-E measure of risk and decision model. The normalized EU-E measure of risk has some normative properties under certain conditions. Moreover, the normalized EU-E decision model can be a proper descriptive model to some extent. Using this model, two cases of common ratio effect and common consequence effect, which are the examples of certainty effects, can be explained in an intuitive way.<\/jats:p>","DOI":"10.3390\/e16073590","type":"journal-article","created":{"date-parts":[[2014,6,30]],"date-time":"2014-06-30T10:34:47Z","timestamp":1404124487000},"page":"3590-3604","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Normalized Expected Utility-Entropy Measure of Risk"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1711-0834","authenticated-orcid":false,"given":"Jiping","family":"Yang","sequence":"first","affiliation":[{"name":"School of Economics and Management, Beihang University, Beijing 100191, China"}]},{"given":"Wanhua","family":"Qiu","sequence":"additional","affiliation":[{"name":"School of Economics and Management, Beihang University, Beijing 100191, China"}]}],"member":"1968","published-online":{"date-parts":[[2014,6,30]]},"reference":[{"key":"ref_1","unstructured":"Von Neumann, J., and Morgenstern, O. (1944). The Theory of Games and Economic Behavior, Princeton University Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"263","DOI":"10.2307\/1914185","article-title":"Prospect theory: An analysis of decision under risk","volume":"47","author":"Kahneman","year":"1979","journal-title":"Econometrica"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1016\/0022-0531(77)90097-7","article-title":"The definition of risk: An extension","volume":"14","author":"Levy","year":"1977","journal-title":"J. Econ. Theory"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1016\/S0377-2217(96)00254-8","article-title":"Relative risk-value models","volume":"103","author":"Dyer","year":"1997","journal-title":"Eur. J. Oper. Res"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1287\/mnsc.45.4.519","article-title":"Measures of perceived risk","volume":"45","author":"Jia","year":"1999","journal-title":"Manag. Sci"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3862","DOI":"10.1016\/j.physa.2008.02.072","article-title":"Utility function estimation: The entropy approach","volume":"387","author":"Dionisio","year":"2008","journal-title":"Physica A"},{"key":"ref_7","first-page":"77","article-title":"Independence properties vis-\u00e0-vis several utility representations","volume":"58","author":"Marley","year":"2005","journal-title":"Risk. Decis"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1007\/s00010-008-2931-0","article-title":"On the utility of gambling: Extending the approach of Meginniss (1976)","volume":"76","author":"Ng","year":"2008","journal-title":"Aequationes Mathematicae"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Brams, S., Gehrlein, W., and Roberts, F. (2008). The Mathematics of Preference, Choice, and Order: Essays in Honor of Peter C. Fishburn, Springer.","DOI":"10.1007\/978-3-540-79128-7"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00199-007-0260-5","article-title":"Utility of gambling I: Entropy modified linear weighted utility","volume":"36","author":"Luce","year":"2008","journal-title":"Econ. Theory"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/s00199-007-0259-y","article-title":"Utility of gambling II: Risk, paradoxes, and data","volume":"36","author":"Luce","year":"2008","journal-title":"Econ. Theory"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"792","DOI":"10.1016\/j.ejor.2004.01.031","article-title":"A measure of risk and a decision-making model based on expected utility and entropy","volume":"164","author":"Yang","year":"2005","journal-title":"Eur. J. Oper. Res"},{"key":"ref_13","first-page":"381","article-title":"Several possible measures of risk","volume":"13","author":"Luce","year":"1981","journal-title":"Risk. Decis"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J"},{"key":"ref_15","unstructured":"Golan, A., Judge, G., and Miller, D. (1996). Maximum Entropy Econometrics: Robust Estimation with Limited Data, Wiley."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1080\/03081078608934927","article-title":"Normalized measures of entropy","volume":"12","author":"Kumar","year":"1986","journal-title":"Int. J. Gen. Syst"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"812","DOI":"10.1080\/01621459.1992.10475283","article-title":"A generalizable formulation of conditional logit with diagnostics","volume":"87","author":"Soofi","year":"1992","journal-title":"J. Am. Stat. Assoc"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"503","DOI":"10.2307\/1907921","article-title":"Le comportement de l\u2019homme rationnel devant le risqu\u00e9: Critique des postulats et axiomes de l\u2019ecole americaine","volume":"21","author":"Allais","year":"1953","journal-title":"Econometrica"},{"key":"ref_19","unstructured":"Savage, L.J. (1954). Foundations of Statistics, Wiley."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1257\/jep.1.1.121","article-title":"Choice under uncertainty: Problems solved and unsolved","volume":"1","author":"Machina","year":"1987","journal-title":"Econ. Perspect"},{"key":"ref_21","first-page":"55","article-title":"Empirical tests of weighted utility theory","volume":"50","author":"Chew","year":"1986","journal-title":"J. Math. Psychol"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/7\/3590\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:13:04Z","timestamp":1760217184000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/7\/3590"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,6,30]]},"references-count":21,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2014,7]]}},"alternative-id":["e16073590"],"URL":"https:\/\/doi.org\/10.3390\/e16073590","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,6,30]]}}}