Entropy-Related Measures of the Utility of Gambling

  • R. Duncan Luce
  • Anthony J. Marley
  • Che Tat Ng
Part of the Studies in Choice and Welfare book series (WELFARE)

The first author has known Peter for a very long time, dating back some 45 years to when we met at a colloquium he gave at the University of Pennsylvania. After that our paths crossed fairly often. For example, in the early 1970s, he spent a year at the Institute for Advanced Study where Luce spent three years until the attempt to establish a program in scientific social science was abandoned for a more literary approach favored by the humanists and, surprisingly, the mathematicians then at the Institute. The second author has learnt a tremendous amount about both substantive and technical issues from Peter's work, beginning with Peter's book “Utility Theory for Decision Making” (Fishburn, 1970), which he reviewed for Contemporary Psychology (see Marley, 1972).

Peter's volume on interval orders (Fishburn, 1985) was a marvelous development of various ideas related to the algebra of imperfect discrimination that elaborated the first author's initial work on semiorders (Luce, 1956).


Expect Utility Utility Theory Pure Consequence Cumulative Prospect Theory Expect Value 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Aczél, J., & Daróczy, Z. (1975). On Measures of Information and Their Characterizations. New York, NY: Academic Press.Google Scholar
  2. Aczél, J., & Daróczy, Z. (1978). A mixed theory of information. I. Symmetric, recursive and measurable entropies of randomized systems of events. R.A.I.R.O. Informatique théorique (Theoretical Computer Science), 12, 149–155.Google Scholar
  3. Aczél, J., & Kannappan, P. (1978). A mixed theory of information. III. Inset entropies of degree β. Information and Control, 39, 315–322.CrossRefGoogle Scholar
  4. Bleichrodt, H., & Schmidt, U. (2002). A context-dependent model of the gambling effect. Management Science, 48, 802–812.CrossRefGoogle Scholar
  5. Cho, Y.-H., Luce, R.D., & Truong, L. (2002). Duplex decomposition and general segregation of lotteries of a gain and a loss: An empirical evaluation. Organization Behavior and Human Decision Processes, 89, 1176–1193.CrossRefGoogle Scholar
  6. Conlisk, J. (1993). The utility of gambling. Journal of Risk and Uncertainty, 6, 255–275.CrossRefGoogle Scholar
  7. Davidson, K.R., & Ng, C.T. (1981). Cocycles on cancellative semigroups. Utilitas Mathematica, 20, 27–34.Google Scholar
  8. Diecidue, E., Schmidt, U., & Wakker, P. P. (2004). The utility of gambling reconsidered. Journal of Risk and Uncertainty, 29, 241–259.CrossRefGoogle Scholar
  9. Ebanks, B.R. (1982). The general symmetric solution of a functional equation arising in the mixed theory of information. Comptes rendus mathématiques de l'Académie des sciences (Mathematical Reports of the Academy of Science), 4, 195–200.Google Scholar
  10. Ebanks, B.R., Kannappan, P.L., & Ng, C.T. (1988). Recursive inset entropies of multiplicative type on open domains. Aequationes Mathematicae, 36, 268–293.CrossRefGoogle Scholar
  11. Ellsberg, D. (1961). Risk, ambiguity and the Savage axioms. Quarterly Journal of Economics, 75, 643–669.CrossRefGoogle Scholar
  12. Fishburn, P.C. (1970). Utility Theory for Decision Making. New York, NY: Wiley.Google Scholar
  13. Fishburn, P.C. (1980). A simple model of the utility of gambling. Psychometrika, 45, 435–338.CrossRefGoogle Scholar
  14. Fishburn, P.C. (1985). Interval Orders and Interval Graphs. New York, NY: Wiley.Google Scholar
  15. Fishburn, P.C. (1988). Nonlinear Preference and Utility Theory. Baltimore, MD: Johns Hopkins.Google Scholar
  16. Fishburn, P.C., & Luce, R.D. (1995). Joint receipt and Thaler's hedonic editing rule. Mathematical Social Sciences, 29, 33–76.CrossRefGoogle Scholar
  17. Havrda, J., & Charvát, F. (1967). Quantification method of classification processes. Concept of structural α-entropy. Kybernetika, 3, 30–35.Google Scholar
  18. Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econo-metrica, 47, 263–291.Google Scholar
  19. Krantz, D.H., Luce, R.D., Suppes, P., & Tversky, A. (1971). Foundations of Measurement,I. New York, San Diego: Academic Press.Google Scholar
  20. Le Menestrel, M.(2001). A process approach to the utility for gambling. Theory and Decision, 50, 249–262.CrossRefGoogle Scholar
  21. Luce, R.D. (1956). Semiorders and a theory of utility discrimination. Econometrica, 24, 178–191.CrossRefGoogle Scholar
  22. Luce, R.D. (1997). Associative joint receipts. Mathematical Social Sciences, 34, 51–74.CrossRefGoogle Scholar
  23. Luce, R.D. (2000). Utility of Gains and Losses: Measurement-theoretical and Experimental Approaches. Mahwah, NJ: Lawrence Erlbaum Associates, Errata: Luce web page at http//www. Scholar
  24. Luce, R.D., & Fishburn, P.C. (1991). Rank- and sign-dependent linear utility models for finite first-order gambles. Journal of Risk and Uncertainty, 4, 29–59.CrossRefGoogle Scholar
  25. Luce, R.D., & Fishburn, P.C. (1995). A note on deriving rank-dependent utility using additive joint receipts. Journal of Risk and Uncertainty, 11, 5–16.CrossRefGoogle Scholar
  26. Luce, R.D., & Marley, A.A.J. (2000). On elements of chance. Theory and Decision, 49, 97–126.CrossRefGoogle Scholar
  27. Luce, R.D., Ng, C.T., Marley, A.A.J., & Aczél, J. (2008a) Utility of gambling I: Entropy-modified linear weighted utility. Economic Theory, 36, 1–33.CrossRefGoogle Scholar
  28. Luce, R.D., Ng, C.T., Marley, A.A.J., & Aczél, J. (2008b) Utility of gambling II: Risk, Paradoxes, and Data. Economic Theory, 36, 165–187.CrossRefGoogle Scholar
  29. Erratum to Luce, Ng, Marley, & Aczél (2008a, b): See luce/luce.html.Google Scholar
  30. Marley, A.A.J. (1972). The logic of decisions: A review of P. C. Fishburn's Utility Theory for Decision Making. Contemporary Psychology, 17, 379–380.Google Scholar
  31. Marley, A.A.J., & Luce, R.D.D (2005). Independence properties vis-à-vis several utility representations. Theory and Decision, 58, 77–143.CrossRefGoogle Scholar
  32. Meginniss, J.R. (1976). A new class of symmetric utility rules for gambles, subjective marginal probability functions, and a generalized Bayes' rule. Proceedings of the American Statistical Association, Business and Economic Statistics Section, 471–476.Google Scholar
  33. Ng, C.T., Luce, R.D., & Marley, A.A.J. (2008a). Utility of gambling: extending the approach of Meginniss (1976). Aequationes Mathematicae.Google Scholar
  34. Ng, C.T., Luce, R.D., & Marley, A.A.J. (2008b). Utility of gambling when events are valued: An application of inset entropy. Theory and Decision.Google Scholar
  35. Ng, C.T., Luce, R.D., & Marley, A.A.J. (2008c, submitted). Utility of gambling under p-additive joint receipt.Google Scholar
  36. Pope, R. (1995). Toward a more precise decision framework. Theory and Decision, 39, 241–265.CrossRefGoogle Scholar
  37. Ramsey, F.P. (1931). The Foundations of Mathematics and Other Logical Essays. New York, NY: Harcourt, Brace. In H. E. Kyburg, & H. E. Smokler (Eds.) Studies in Subjective Probability, Chap. VII. New York, NY: Wiley, 61–92 (1964).Google Scholar
  38. Savage, L.J. (1954). The foundations of probability. New York, NY: Wiley.Google Scholar
  39. Shannon, C.E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27, 379–423, 623–656.Google Scholar
  40. Slovic, P. (1967). The relative influence of probabilities and payoffs upon perceived risk of a gamble. Psychonomic Science, 9, 223–224.Google Scholar
  41. Slovic, P., & Lichtenstein, S. (1968). The importance of variance preferences in gambling decisions. Journal of Experimental Psychology, 78, 646–654.CrossRefGoogle Scholar
  42. von Neumann, J., & Morgenstern, O. (1947). Theory of Games and Economic Behavior, 2nd Ed. Princeton, NJ: Princeton University.Google Scholar
  43. Wakker, P.P. (1990). Characterizing optimism and pessimism directly through comonotonicity. Journal of Economic Theory, 52, 453–463.CrossRefGoogle Scholar
  44. Yang, J., & Qiu, W. (2005). A measure of risk and a decision-making model based on expected utility and entropy. European Journal of Operations Research, 164, 792–799.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • R. Duncan Luce
    • 1
  • Anthony J. Marley
    • 2
  • Che Tat Ng
    • 3
  1. 1.Institute for Mathematical Behavioral SciencesUniversity of CaliforniaIrvineUSA
  2. 2.Department of PsychologyUniversity of VictoriaVictoriaCanada
  3. 3.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada

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