{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T16:48:23Z","timestamp":1790700503881,"version":"4.1.0"},"reference-count":131,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2017,3,13]],"date-time":"2017-03-13T00:00:00Z","timestamp":1489363200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We demonstrate that behavioral probabilities of human decision makers share many common features with quantum probabilities. This does not imply that humans are some quantum objects, but just shows that the mathematics of quantum theory is applicable to the description of human decision making. The applicability of quantum rules for describing decision making is connected with the nontrivial process of making decisions in the case of composite prospects under uncertainty. Such a process involves deliberations of a decision maker when making a choice. In addition to the evaluation of the utilities of considered prospects, real decision makers also appreciate their respective attractiveness. Therefore, human choice is not based solely on the utility of prospects, but includes the necessity of resolving the utility-attraction duality. In order to justify that human consciousness really functions similarly to the rules of quantum theory, we develop an approach defining human behavioral probabilities as the probabilities determined by quantum rules. We show that quantum behavioral probabilities of humans do not merely explain qualitatively how human decisions are made, but they predict quantitative values of the behavioral probabilities. Analyzing a large set of empirical data, we find good quantitative agreement between theoretical predictions and observed experimental data.<\/jats:p>","DOI":"10.3390\/e19030112","type":"journal-article","created":{"date-parts":[[2017,3,13]],"date-time":"2017-03-13T10:26:18Z","timestamp":1489400778000},"page":"112","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":38,"title":["Quantum Probabilities as Behavioral Probabilities"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4833-0175","authenticated-orcid":false,"given":"Vyacheslav","family":"Yukalov","sequence":"first","affiliation":[{"name":"Department of Management, Technology and Economics, ETH Z\u00fcrich, Swiss Federal Institute of Technology, CH 8032 Z\u00fcrich, Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Didier","family":"Sornette","sequence":"additional","affiliation":[{"name":"Department of Management, Technology and Economics, ETH Z\u00fcrich, Swiss Federal Institute of Technology, CH 8032 Z\u00fcrich, Switzerland"},{"name":"Finance Institute, c\/o University of Geneva, 40 blvd. Du Pont d\u2019Arve, CH 1211 Geneva, Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2017,3,13]]},"reference":[{"key":"ref_1","unstructured":"Williams, C.P., and Clearwater, S.H. (1988). Explorations in Quantum Computing, Springer."},{"key":"ref_2","unstructured":"Nielsen, M.A., and Chuang, I.L. (2000). Quantum Computation and Quantum Information, Cambridge University Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1016\/S0370-1573(02)00266-1","article-title":"Fundamentals of quantum information theory","volume":"369","author":"Keyl","year":"2002","journal-title":"Phys. Rep."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2543","DOI":"10.1080\/09500340008232180","article-title":"Quantum games","volume":"47","author":"Eisert","year":"2000","journal-title":"J. Mod. Opt."},{"key":"ref_5","first-page":"394","article-title":"Quantum game theory","volume":"51","author":"Landsburg","year":"2004","journal-title":"Am. Math. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1016\/j.dss.2008.07.001","article-title":"A survey of quantum games","volume":"46","author":"Guo","year":"2008","journal-title":"Decis. Support. Syst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.jmp.2013.12.004","article-title":"A connection between quantum decision theory and quantum games: The Hamiltonian of strategic interaction","volume":"58","year":"2014","journal-title":"J. Math. Psychol."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Bohr, N. (1958). Atomic Physics and Human Knowledge, Wiley.","DOI":"10.1119\/1.1934707"},{"key":"ref_9","unstructured":"Bohr, N. (1985). Collected Works, Foundations of Quantum Physics, North-Holland."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Bohm, D., and Hiley, B. (1993). The Undivided Universe: An Ontological Interpretation of Quantum Theory, Routledge Chapman & Hall.","DOI":"10.1063\/1.2808635"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"190402","DOI":"10.1103\/PhysRevLett.101.190402","article-title":"Efficient hidden-variable simulation of measurements in quantum experiments","volume":"101","author":"Dakic","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3129","DOI":"10.1098\/rspa.1999.0443","article-title":"Quantum theory of probability and decision","volume":"455","author":"Deutsch","year":"1999","journal-title":"Proc. R. Soc. Lond. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.shpsb.2006.02.001","article-title":"Uncertainty and probability for branching selves","volume":"38","author":"Lewis","year":"2007","journal-title":"Stud. Hist. Philos. Mod. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1009","DOI":"10.1086\/605805","article-title":"Probability, self-location, and quantum branching","volume":"76","author":"Lewis","year":"2009","journal-title":"Philos. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"6867","DOI":"10.1016\/j.physleta.2008.09.053","article-title":"Quantum decision theory as quantum theory of measurement","volume":"372","author":"Yukalov","year":"2008","journal-title":"Phys. Lett. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1140\/epjb\/e2009-00245-9","article-title":"Physics of risk and uncertainty in quantum decision making","volume":"71","author":"Yukalov","year":"2009","journal-title":"Eur. Phys. J. B"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1142\/S0219525910002803","article-title":"Mathematical structure of quantum decision theory","volume":"13","author":"Yukalov","year":"2010","journal-title":"Adv. Complex Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1007\/s11238-010-9202-y","article-title":"Decision theory with prospect interference and entanglement","volume":"70","author":"Yukalov","year":"2011","journal-title":"Theor. Decis."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"105502","DOI":"10.1088\/1054-660X\/23\/10\/105502","article-title":"Quantum probabilities of composite events in quantum measurements with multimode states","volume":"23","author":"Yukalov","year":"2013","journal-title":"Laser Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1155","DOI":"10.1109\/TSMC.2014.2314283","article-title":"Manipulating decision making of typical agents","volume":"44","author":"Yukalov","year":"2014","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1111\/tops.12065","article-title":"Conditions for quantum interference in cognitive sciences","volume":"6","author":"Yukalov","year":"2014","journal-title":"Top. Cogn. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1450016","DOI":"10.1142\/S0219525914500167","article-title":"Self-organization in complex systems as decision making","volume":"17","author":"Yukalov","year":"2014","journal-title":"Adv. Compl. Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"01538","DOI":"10.3389\/fpsyg.2015.01538","article-title":"Preference reversal in quantum decision theory","volume":"6","author":"Yukalov","year":"2015","journal-title":"Front. Psychol."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"20150100","DOI":"10.1098\/rsta.2015.0100","article-title":"Quantum probability and quantum decision making","volume":"374","author":"Yukalov","year":"2016","journal-title":"Philos. Trans. R. Soc. A"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Cox, R.T. (1961). The Algebra of Probable Inference, Johns Hopkins Press.","DOI":"10.56021\/9780801869822"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1016\/j.aop.2013.11.005","article-title":"A discussion on the origin of quantum probabilities","volume":"340","author":"Holik","year":"2014","journal-title":"Ann. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"7349","DOI":"10.3390\/e17117349","article-title":"Quantum information as a non-Kolmogorovian generalization of Shannon\u2019s theory","volume":"17","author":"Holik","year":"2015","journal-title":"Entropy"},{"key":"ref_28","first-page":"115","article-title":"Natural information measures in Cox approach for contextual probabilistic theories","volume":"16","author":"Holik","year":"2016","journal-title":"Quant. Inf. Comput."},{"key":"ref_29","unstructured":"Von Neumann, J., and Morgenstern, O. (1953). Theory of Games and Economic Behavior, Princeton University."},{"key":"ref_30","unstructured":"Savage, L.J. (1954). The Foundations of Statistics, Wiley."},{"key":"ref_31","unstructured":"Arrow, K.J. (1971). Essays in the Theory of Risk Bearing, Markham."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"122","DOI":"10.2307\/1913738","article-title":"Risk aversion in the small and in the large","volume":"32","author":"Pratt","year":"1964","journal-title":"Econometrica"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/0022-0531(70)90038-4","article-title":"Increasing risk: A definition","volume":"2","author":"Rothschild","year":"1970","journal-title":"J. Econ. Theor."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1016\/0022-0531(71)90034-2","article-title":"Increasing risk: Its economic consequences","volume":"3","author":"Rothschild","year":"1971","journal-title":"J. Econ. Theor."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1086\/256692","article-title":"The utility analysis of choices involving risk","volume":"56","author":"Friedman","year":"1948","journal-title":"J. Political Econ."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1086\/257177","article-title":"The utility of wealth","volume":"60","author":"Markowitz","year":"1952","journal-title":"J. Political Econ."},{"key":"ref_37","unstructured":"Lindgren, B.W. (1971). Elements of Decision Theory, Macmillan."},{"key":"ref_38","unstructured":"White, D.I. (1976). Fundamentals of Decision Theory, Elsevier."},{"key":"ref_39","unstructured":"Rivett, P. (1980). Model Building for Decision Analysis, Wiley."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Berger, J.O. (1985). Statistical Decision Theory and Bayesian Analysis, Springer.","DOI":"10.1007\/978-1-4757-4286-2"},{"key":"ref_41","unstructured":"Marshall, K.T., and Oliver, R.M. (1995). Decision Making and Forecasting, McGraw-Hill."},{"key":"ref_42","unstructured":"Bather, J. (2000). Decision Theory, Wiley."},{"key":"ref_43","unstructured":"French, S., and Insua, D.R. (2000). Statistical Decision Theory, Arnold."},{"key":"ref_44","unstructured":"Raiffa, H., and Schlaifer, R. (2000). Applied Statistical Decision Theory, Wiley."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Weirich, P. (2001). Decision Space, Cambridge University Press.","DOI":"10.1017\/CBO9780511498602"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Gollier, C. (2001). Economics of Risk and Time, MIT Press.","DOI":"10.7551\/mitpress\/2622.001.0001"},{"key":"ref_47","unstructured":"Ariely, D. (2008). Predictably Irrational, Harper."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"503","DOI":"10.2307\/1907921","article-title":"Le comportement de l\u2019homme rationnel devant le risque: Critique des postulats et axiomes de l\u2019ecole Americaine","volume":"21","author":"Allais","year":"1953","journal-title":"Econometrica"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1037\/h0041692","article-title":"The prediction of decision among bets","volume":"50","author":"Edwards","year":"1955","journal-title":"J. Exp. Psychol."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1037\/h0038674","article-title":"Subjective probabilities inferred from decisions","volume":"69","author":"Edwards","year":"1962","journal-title":"Psychol. Rev."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"643","DOI":"10.2307\/1884324","article-title":"Risk, ambiguity, and the Savage axioms","volume":"75","author":"Ellsberg","year":"1961","journal-title":"Q. J. Econ."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Camerer, C.F., Loewenstein, G., and Rabin, R. (2003). Advances in Behavioral Economics, Princeton University.","DOI":"10.1515\/9781400829118"},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Durlauf, S.N., and Blume, L.E. (2008). New Palgrave Dictionary of Economics, Macmillan.","DOI":"10.1057\/b.9780631218234.2009.X"},{"key":"ref_54","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_55","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/0030-5073(78)90039-9","article-title":"Subjectively weighted utility: A descriptive extension of the expected utility model","volume":"21","author":"Karmarkar","year":"1978","journal-title":"Org. Behav. Hum. Perform."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/0030-5073(79)90016-3","article-title":"Subjectively weighted utility and the Allais paradox","volume":"24","author":"Karmarkar","year":"1979","journal-title":"Org. Behav. Hum. Perform."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"1065","DOI":"10.2307\/1912052","article-title":"A generalization of the quasilinear mean with applications to the measurement of income inequality and decision theory resolving the Allais paradox","volume":"51","author":"Chew","year":"1983","journal-title":"Econometrica"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1016\/0014-2921(94)00071-7","article-title":"Incorporating a stochastic element into decision theories","volume":"39","author":"Loomes","year":"1995","journal-title":"Eur. Econ. Rev."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1111\/j.1467-6435.1984.tb00748.x","article-title":"The economics of optimism and pessimism: A definition and some applications","volume":"37","author":"Hey","year":"1984","journal-title":"Kyklos"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"95","DOI":"10.2307\/1911158","article-title":"The dual theory of choice under risk","volume":"55","author":"Yaari","year":"1987","journal-title":"Econometrica"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1007\/BF00117641","article-title":"Ordinal independence in nonlinear utility theory","volume":"1","author":"Green","year":"1988","journal-title":"J. Risk Uncertain."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"139","DOI":"10.2307\/2938244","article-title":"Mixture symmetry and quadratic utility","volume":"59","author":"Chew","year":"1991","journal-title":"Econometrica"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11166-014-9185-0","article-title":"An experimental test of prospect theory for predicting choice under ambiguity","volume":"48","author":"Kothiyal","year":"2014","journal-title":"J. Risk Uncertain."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11166-014-9198-8","article-title":"The explanatory and predictive power of non-two-stage probability theories of decision making under ambiguity","volume":"49","author":"Hey","year":"2014","journal-title":"J. Risk Uncertain."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"1143","DOI":"10.3982\/ECTA6175","article-title":"Calibration results for non-expected utility theories","volume":"76","author":"Safra","year":"2008","journal-title":"Econometrica"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1017\/S026626710999023X","article-title":"The ambiguity aversion literature: A critical assessment","volume":"25","author":"Weinstein","year":"2009","journal-title":"Econ. Philos."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1017\/S0266267109990289","article-title":"The ambiguity aversion literature: A critical assessment","volume":"25","author":"Weinstein","year":"2009","journal-title":"Econ. Philos."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1038\/131421a0","article-title":"Light and life","volume":"131","author":"Bohr","year":"1933","journal-title":"Nature"},{"key":"ref_69","unstructured":"Von Neumann, J. (1955). Mathematical Foundations of Quantum Mechanics, Princeton University."},{"key":"ref_70","doi-asserted-by":"crossref","unstructured":"Baaquie, B.E. (2004). Quantum Finance, Cambridge University.","DOI":"10.1017\/CBO9780511617577"},{"key":"ref_71","doi-asserted-by":"crossref","unstructured":"Khrennikov, A. (2010). Ubiquitos Quantum Structure, Springer.","DOI":"10.1007\/978-3-642-05101-2"},{"key":"ref_72","doi-asserted-by":"crossref","unstructured":"Busemeyer, J.R., and Bruza, P. (2012). Quantum Models of Cognition and Decision, Cambridge University Press.","DOI":"10.1017\/CBO9780511997716"},{"key":"ref_73","doi-asserted-by":"crossref","unstructured":"Haven, E., and Khrennikov, A. (2013). Quantum Social Science, Cambridge University Press.","DOI":"10.1017\/CBO9781139003261"},{"key":"ref_74","doi-asserted-by":"crossref","unstructured":"Bagarello, F. (2013). Quantum Dynamics for Classical Systems, Wiley.","DOI":"10.1002\/9781118400586"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"1073","DOI":"10.3390\/e11041073","article-title":"Processing information in quantum decision theory","volume":"11","author":"Yukalov","year":"2009","journal-title":"Entropy"},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"062001","DOI":"10.1088\/0034-4885\/77\/6\/062001","article-title":"Physics and financial economics (1776-2014): Puzzles, Ising and agent-based models","volume":"77","author":"Sornette","year":"2014","journal-title":"Rep. Prog. Phys."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.mathsocsci.2015.02.004","article-title":"A survey of quantum-like approaches to decision making and cognition","volume":"75","author":"Ashtiani","year":"2015","journal-title":"Math. Soc. Sci."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"99","DOI":"10.2307\/1884852","article-title":"A behavioral model of rational choice","volume":"69","author":"Simon","year":"1955","journal-title":"Q. J. Econ."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1038\/scientificamerican0201-76","article-title":"The science of persuasion","volume":"284","author":"Cialdini","year":"2001","journal-title":"Sci. Am."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1146\/annurev.psych.59.103006.093710","article-title":"Neuroeconomics","volume":"59","author":"Loewenstein","year":"2008","journal-title":"Ann. Rev. Psychol."},{"key":"ref_81","doi-asserted-by":"crossref","unstructured":"Hens, T., and Bachmann, K. (2008). Behavioral Finance for Private Banking, Wiley.","DOI":"10.1002\/9781118467329"},{"key":"ref_82","unstructured":"Malvergne, Y., and Sornette, D. (2006). Extreme Financial Risks, Springer."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/0022-2496(90)90013-Y","article-title":"A historical and contemporary perspective on random scale representations of choice probabilities and reaction time","volume":"34","author":"Marley","year":"1990","journal-title":"J. Math. Psychol."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1007\/s11166-006-8289-6","article-title":"Cumulative prospect theory\u2019s functional menagerie","volume":"32","author":"Scott","year":"2006","journal-title":"J. Risk Uncertain."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"1291","DOI":"10.2307\/2951750","article-title":"Investigating generalizations of expected utility theory using experimental data","volume":"62","author":"Hey","year":"1994","journal-title":"Econometrica"},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"805","DOI":"10.2307\/2232669","article-title":"Regret theory: An alternative theory of rational choice under uncertainty","volume":"92","author":"Loomes","year":"1982","journal-title":"Econ. J."},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1023\/A:1014094209265","article-title":"A microeconomic test of alternative stochastic theories of risky choice","volume":"24","author":"Loomes","year":"2002","journal-title":"J. Risk Uncertain."},{"key":"ref_88","unstructured":"Heyer, D., and Niederee, R. (1989). Mathematical Psychology in Progress, Springer."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/0165-4896(92)90036-5","article-title":"Generalizing the concept of binary choice systems induced by rankings: One way of probabilizing deterministic measurement structures","volume":"23","author":"Heyer","year":"1982","journal-title":"Math. Soc. Sci."},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1006\/jmps.1996.0022","article-title":"Random utility representations of finite many relations","volume":"40","author":"Regenwetter","year":"1996","journal-title":"J. Math. Psychol."},{"key":"ref_91","unstructured":"Niederee, R., and Heyer, D. (1997). Choice, Decision and Measurement: Essays in Honor of R. Duncan Luce, Lawrence Erlbaum."},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"864","DOI":"10.1006\/jmps.2000.1357","article-title":"Random relations, random utilities, and random functions","volume":"45","author":"Regenwetter","year":"2001","journal-title":"J. Math. Psychol."},{"key":"ref_93","doi-asserted-by":"crossref","unstructured":"Regenwetter, M., Dana, J., and Davis-Stober, C.P. (2010). Testing transitivity of preferences on two-alternative forced choice data. Front. Psychol., 1.","DOI":"10.3389\/fpsyg.2010.00148"},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1016\/0014-2921(09)40007-4","article-title":"Experimental investigations of errors in decision making under risk","volume":"39","author":"Hey","year":"1995","journal-title":"Eur. Econ. Rev."},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1023\/A:1007829024107","article-title":"Which error story is best?","volume":"20","author":"Carbone","year":"2000","journal-title":"J. Risk Uncertain."},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1007\/s10683-005-5373-8","article-title":"Why we should not be silent about noise","volume":"8","author":"Hey","year":"2005","journal-title":"Exp. Econ."},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1007\/s10683-005-5372-9","article-title":"Modelling the stochastic component of behaviour in expriments: Some issues for the interpretation of the data","volume":"8","author":"Loomes","year":"2005","journal-title":"Exp. Econ."},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/0749-5978(87)90034-3","article-title":"The influence of positive affect on acceptable level of risk: The person with a large canoe has a large worry","volume":"39","author":"Isen","year":"1987","journal-title":"Org. Behav. Hum. Decis. Proc."},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1006\/obhd.1994.1003","article-title":"Risk-taking, framing effects, and affect","volume":"57","author":"Mano","year":"1994","journal-title":"Org. Behav. Hum. Decis. Proc."},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1006\/obhd.1998.2781","article-title":"The influence of framing on risky decisions: A meta-analysis","volume":"75","year":"1998","journal-title":"Org. Behav. Hum. Decis. Proc."},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1002\/bdm.441","article-title":"The role of competition and knowledge in the Ellsberg task","volume":"16","author":"Perner","year":"2003","journal-title":"J. Behav. Decis. Mak."},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/s11166-007-9020-y","article-title":"Individual and group decision making under risk: An experimental study of Bayesian updating and violations of first-order dominance","volume":"35","author":"Charness","year":"2007","journal-title":"J. Risk Uncertain."},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00199-007-0260-5","article-title":"Utility of gambling: Entropy modified linear weighted utility","volume":"36","author":"Luce","year":"2008","journal-title":"Econ. Theory"},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/s00199-007-0259-y","article-title":"Utility of gambling: Risk, paradoxes, and data","volume":"36","author":"Luce","year":"2008","journal-title":"Econ. Theory"},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1111\/1467-937X.00168","article-title":"Discrete choice with social interactions","volume":"68","author":"Brock","year":"2001","journal-title":"Rev. Econ. Stud."},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"193","DOI":"10.2307\/1907587","article-title":"A probabilistic theory of utility","volume":"26","author":"Luce","year":"1958","journal-title":"Econometrica"},{"key":"ref_107","unstructured":"Luce, R.D. (1959). Individual Choice Behavior: A Theoretical Analysis, Wiley."},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/0022-2496(77)90032-3","article-title":"The choice axiom after twenty years","volume":"15","author":"Luce","year":"1977","journal-title":"J. Math. Psychol."},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/0022-2496(77)90026-8","article-title":"The ralationship between Luce choice axiom, Thurstone theory of comparative judgement, and the double exponential distribution","volume":"15","author":"Yellott","year":"1977","journal-title":"J. Math. Psychol."},{"key":"ref_110","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1007\/BF00122574","article-title":"Advances in prospect theory: cumulative representation of uncertainty","volume":"5","author":"Tversky","year":"1992","journal-title":"J. Risk Uncertain."},{"key":"ref_111","unstructured":"Gescheider, G.A. (1997). Psychophysics: The Fundamentals, Lawrence Erlbaum Associates."},{"key":"ref_112","unstructured":"Henderson, D.R., and Hooper, C.L. (2006). Making Great Decisions in Business and Life, Chicago Park Press."},{"key":"ref_113","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1037\/h0070288","article-title":"A law of comparative judgment","volume":"4","author":"Thurstone","year":"1927","journal-title":"Psychol. Rev."},{"key":"ref_114","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1016\/0022-2496(67)90051-X","article-title":"Rational distance functions for multidimensional scaling","volume":"4","author":"Krantz","year":"1967","journal-title":"J. Math. Psychol."},{"key":"ref_115","doi-asserted-by":"crossref","first-page":"370","DOI":"10.1016\/0022-2496(71)90038-1","article-title":"Similarity between stumuli: An experimental test of the Luce and Restle choice models","volume":"8","author":"Rumhelhart","year":"1971","journal-title":"J. Math. Psychol."},{"key":"ref_116","first-page":"429","article-title":"Preference under risk in the presence of indistinguishable probabilities","volume":"13","author":"Lorentziadis","year":"2013","journal-title":"Oper. Res."},{"key":"ref_117","doi-asserted-by":"crossref","unstructured":"Devroye, L. (1986). Non-Uniform Random Variate Generation, Springer.","DOI":"10.1007\/978-1-4613-8643-8"},{"key":"ref_118","unstructured":"MacKay, D. (2003). Information Theory, Inference, and Learning, Cambridge University Press."},{"key":"ref_119","doi-asserted-by":"crossref","unstructured":"Cover, T.M., and Thomas, J.A. (2006). Elements of Information Theory, Wiley.","DOI":"10.1002\/047174882X"},{"key":"ref_120","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1214\/aoms\/1177729694","article-title":"On information and sufficiency","volume":"22","author":"Kullback","year":"1951","journal-title":"Ann. Math. Stat."},{"key":"ref_121","unstructured":"Kullback, S. (1959). Information Theory and Statistics, Wiley."},{"key":"ref_122","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1109\/TIT.1980.1056144","article-title":"Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy","volume":"26","author":"Shore","year":"1980","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_123","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1109\/TSSC.1968.300118","article-title":"The wigget problem revisited","volume":"4","author":"Tribus","year":"1968","journal-title":"IEEE Trans. Syst. Sci. Cybern."},{"key":"ref_124","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/0370-1573(91)90074-V","article-title":"Phase transitions and heterophase fluctuations","volume":"208","author":"Yukalov","year":"1991","journal-title":"Phys. Rep."},{"key":"ref_125","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1111\/j.1538-4632.2010.00800.x","article-title":"Space, scale and scaling in entropy maximizing","volume":"42","author":"Batty","year":"2010","journal-title":"Geogr. Anal."},{"key":"ref_126","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1126\/science.1257026","article-title":"Experimental observation of a generalized Gibbs ensemble","volume":"348","author":"Langen","year":"2015","journal-title":"Science"},{"key":"ref_127","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1142\/S0217979207035893","article-title":"Representative ensembles in statistical mechanics","volume":"21","author":"Yukalov","year":"2007","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_128","doi-asserted-by":"crossref","first-page":"062001","DOI":"10.1088\/1054-660X\/23\/6\/062001","article-title":"Theory of cold atoms: Basics of quantum statistics","volume":"23","author":"Yukalov","year":"2013","journal-title":"Laser Phys."},{"key":"ref_129","doi-asserted-by":"crossref","first-page":"1873","DOI":"10.3982\/ECTA10621","article-title":"Random choice as behavioral optimization","volume":"82","author":"Gul","year":"2014","journal-title":"Econometrica"},{"key":"ref_130","doi-asserted-by":"crossref","unstructured":"Murphy, R.O., and ten Brincke, R.H.W. (2017). Hierarchical maximum likelihood parameter estimation for cumulative prospect theory: Improving the reliability of individual risk parameter estimates. Manag. Sci., in press.","DOI":"10.1287\/mnsc.2016.2591"},{"key":"ref_131","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1063\/1.2995616","article-title":"Beauty and the quest for beauty in science","volume":"7","author":"Chandrasekhar","year":"1979","journal-title":"Phys. Today"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/3\/112\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:30:20Z","timestamp":1760207420000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/3\/112"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,3,13]]},"references-count":131,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,3]]}},"alternative-id":["e19030112"],"URL":"https:\/\/doi.org\/10.3390\/e19030112","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,3,13]]}}}