Acting and Reflecting pp 143-170 | Cite as

# Decisions Without Ordering

## Abstract

We review the axiomatic foundations of subjective utility theory with a view toward understanding the implications of each axiom. We consider three different approaches, namely, the construction of utilities in the presence of canonical probabilities, the construction of probabilities in the presence of utilities, and the simultaneous construction of both probabilities and utilities. We focus attention on the axioms of independence and weak ordering. The independence axiom is seen to be necessary in order to prevent a form of Dutch Book in sequential problems.

Our main focus is to examine the implications of not requiring the weak order axiom. We assume that gambles are partially ordered. We consider both the construction of probabilities when utilities are given and the construction of utilities in the presence of canonical probabilities. In the first case we find that a partially ordered set of gambles leads to a set of probabilities with respect to which the expected utility of a preferred gamble is higher than that of a dispreferred gamble. We illustrate some comparisons with theories of upper and lower probabilities. In the second case, we find that a partially ordered set of gambles leads to a set of lexicographic utilities each of which ranks preferred gambles higher than dispreferred gambles.

## Keywords

Partial Order Utility Theory Indifference Curve Expect Utility Theory Weak Order## Preview

Unable to display preview. Download preview PDF.

## References

- Allais, M. (1979) “The So-Called Allais Paradox and Rational Decisions Under Uncertainty,” in Allais and Hagen (eds.)
*Expected Utility Hypotheses and the Allais Paradox*. D. Reidel: Dordrecht.Google Scholar - Anscombe, F.J. and Aumann, R.J. (1963) “A definition of subjective probability,”
*Annals of Math. Stat*, 34, 199–205.CrossRefGoogle Scholar - Aumann, R.J. (1962) “Utility theory without the completeness axiom,”
*Econometrica*, 30, 445–462.CrossRefGoogle Scholar - Aumann, R.J. (1964) “Utility theory without the completeness axiom: a correction,”
*Econometrica*, 32, 210–212.CrossRefGoogle Scholar - Bell, D. and Raiffa, H. (1979) “Decision Regret: A Component of Risk Aversion,” MS., Harvard University.Google Scholar
- Chew Soo Hong and MacCrimmon, K.R. (1979) “Alpha-Nu choice theory: a generalization of expected utility theory,” working paper, University of British Columbia.Google Scholar
- Cox, D.R. (1958) “Some Problems Connected with Statistical Inference,”
*Annals of Math, Stat*., 29, 357–363.CrossRefGoogle Scholar - deFinetti, B. (1937) “La prevision: ses lois logiques, ses sources subjectives,”
*Annals de l’Institut Henri Poincare*, 7, 1–68.Google Scholar - Fishburn, P.C. (1970)
*Utility Theory for Decision Making*. Kriefer Publishing Co.: N.Y.Google Scholar - Fishburn, P.C. (1981) “An Axiomatic Characterization of Skew-Symmetric Bilinear Functionals, with applications to utility theory,”
*Economic Letters*, 8, 311–313.CrossRefGoogle Scholar - Fishburn, P.C. (1983) “Nontransitive Measurable Utility,”
*J. Math. Psych*., 26, 31–67.CrossRefGoogle Scholar - Good, I.J. (1952) “Rational Decisions,”
*J. Royal Stat. Soc. B*, 14, 107–114.Google Scholar - Hausner, M. (1954) “Multidimensional utilities,” in R.M. Thrall, C.H. Coombs, and R.L. Davis (eds.),
*Decision processes*. Wiley: N.Y.Google Scholar - Herstein, I.N. and Milnor, J. (1953) “An axiomatic approach to measurable utility,”
*Econometrica*, 21, 291–297.CrossRefGoogle Scholar - Jeffreys, H. (1971)
*Theory of Probability*, 3rd ed. Oxford University Press: Oxford.Google Scholar - Kadane, J. and Sedransk, N. (1980) “Toward a More Ethical Clinical Trial,”
*Bayesian Statistics*. Bernardo et al (eds.) University Press: Valencia.Google Scholar - Kadane, J., et al (1990)
*A New Design for Clinical Trials*. Wiley: forthcoming.Google Scholar - Kahneman, D. and Tversky, A. (1979) “Prospect Theory: An Analysis of Decision Under Risk,”
*Econometrica*, 47, 263–291.CrossRefGoogle Scholar - Kannai, Y. (1963) “Existence of a utility in infinite dimensional partially ordered spaces,”
*Israel J. of Math*., 1, 229–234.CrossRefGoogle Scholar - Klee, V.L. (1955) “Separation Properties of Convex Cones,”
*Proc. Amer. Math. Soc*., 6, 313–318.CrossRefGoogle Scholar - Levi, I. (1974) “On Indeterminate Probabilities,”
*J. Phil*., 71, 391–418.CrossRefGoogle Scholar - Levi, I. (1980)
*The Enterprise of Knowledge*, MIT Press: Cambridge.Google Scholar - Levi, I. (1986) “The Paradoxes of Allais and Ellsberg,”
*Economics and Philosophy*, 2, 23–53.CrossRefGoogle Scholar - Lindley, D.V. (1972)
*Bayesian Statistics: A Review*. SIAM: Philadelphia.Google Scholar - Loomes, G. and Sudgen, R. (1982) “Regret Theory: An Alternative Theory of Rational Choice Under Uncertainty,”
*Economic J*., 92, 805–824.CrossRefGoogle Scholar - McClennen, E.F. (1983) “Sure Thing Doubts,” in B. Stigum and F. Wenstop (eds.),
*Foundations of Utility and Risk Theory with Applications*. D. Reidel: Dordrecht.Google Scholar - Machina, M. (1982) “‘Expected Utility’ Analysis Without the Independence Axiom,”
*Econometrica*, 50, 277–323.CrossRefGoogle Scholar - Machina, M. (1983) “The Economic Theory of Individual Behavior Toward Risk: Theory, Evidence and New Directions,” Dept. of Economics, U.C.S.D.: San Diego, CA 92093. Tech. Report #433.Google Scholar
- Ramsey, F.P. (1931) “Truth and Probability,” in
*The Foundations of Mathematics and other essays*. Keg an, Paul, Trench, Trubner, and Co. Ltd.: London.Google Scholar - Samuelson, P. (1950) “Probability and the Attempts to Measure Utility,”
*Economic Review*, 1, 167–173.Google Scholar - Savage, L.J. (1954)
*The Foundations of Statistics*. Wiley: N.Y.Google Scholar - Schick, F. (1984)
*Having Reasons*. Princeton Univ. Press: Princeton.Google Scholar - Seidenfeld, T. (1988) “Decision Theory without Independence or Without Ordering, What is the difference?” with discussion,
*Economics and Philosophy*, 4, 267–315.CrossRefGoogle Scholar - Sen, A.K. (1977) “Social Choice Theory: A Re-examination,”
*Econometrica*, 45, 53–89.CrossRefGoogle Scholar - Shimony, A. (1955) “Coherence and the axioms of probability,”
*J. Symbolic Logic*, 20, 1–28.CrossRefGoogle Scholar - Smith, C.A.B. (1961) “Consistency in Statistical Inference and Decision,”
*J. Royal Stat. Soc.B*, 23, 1–25.Google Scholar - Szpilrajn, E. (1930) “Sur l’extension de l’ordre partiel,”
*Fundamenta Mathematicae*, 16, 386–389.Google Scholar - Suppes, P. (1974) “The Measurement of Belief,”
*J. Royal Stat. Soc. B*, 36, 160–175.Google Scholar - von Neumann, J. and Morgenstern, O. (1947)
*Theory of Games and Economic Behavior*, 2nd ed. Princeton Univ. Press: Princeton.Google Scholar - Walley, P. and Fine, T. (1979) “Varieties of modal (classificatory) and comparative probability,”
*Synthese*, 41, 321–374.CrossRefGoogle Scholar - Wolfenson, M. and Fine, T. (1982) “Bayes-like decision making with upper and lower probabilities,”
*J. Amer. Stat. Assoc*., 77, 80–88.CrossRefGoogle Scholar