It is around 1980 that new era for decision making under risk/uncertainty began to uncover numerous alternative representations which generalize the traditional (subjective) expected utility maximization. The initial major contributors include Chew and MacCrimmon (1979), Chew (1983), Fishburn (1982), Kahneman and Tversky (1979), Machina (1982), Quiggin (1981), and Schmeidler (1988) (first appeared in 1981 as a discussion paper). One of Fishburn's works in this area is the discovery of an axiomatic structure of SSB (skew-symmetric bilinear) preferences in decision making under risk, and its numerical representation, dubbed SSB utility (see Fishburn, 1982). Since then, he published a series of papers which study SSB preferences and their numerical representations in various contexts in decision making under risk/uncertainty (see a survey, Fishburn, 1988b).
This paper further explores representational aspects of SSB preferences particularly in decision making under uncertainty and discusses their necessary and sufficient axiomatizations. Three representational forms will be examined. One of them is known as an SSA (skew-symmetric additive) representation first explored by Fishburn, 1984a. The other two are new in the literature, one of which seems to be a more natural application of SSB utility to decision making under uncertainty than SSA representation. A characteristic feature of the first two representations is nonseparability of utilities for decision outcomes. The last one is a generalization of subjective expected utility (SEU) which replaces subjective probabilities with non-separable representation of comparative beliefs first discovered by Fishburn (1983a and b).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Chew, S.H. & McCrimmon, K.R. (1979). Alpha-nu choice theory: a generalization of expected utility theory. Unpublished mimeograph.
Chew, S.H. (1983). A generalization of the quasilinear mean with applications to the measurement of income inequality and decision theory resolving the Allais paradox.Econometrica51, 1065–1092.
Fishburn, P.C. (1981). Subjective expected utility: a review of normative theories.Theory and Decision,13, 139–199.
Fishburn, P.C. (1982). Nontransitive measurable utility.Journal of Mathematical Psychology,26, 31–67.
Fishburn, P.C. (1983a). Ellsberg revisited: a new look at comparative probability.Annals of Statistics,11, 1047–1059.
Fishburn, P.C. (1983b). A generalization of comparative probability on finite sets.Journal of Mathematical Psychology,27, 298–310.
Fishburn, P.C. (1984a). SSB utility theory and decision-making under uncertainty.Mathematical Social Sciences,8, 63–94.
Fishburn, P.C. (1984b). Multiattribute nonlinear utility theory.Management Science,30, 1301–1310.
Fishburn, P.C. (1988a). Nontransitive measurable utility for decision under uncertainty.Journal of Mathematical Economics,18, 187–207.
Fishburn, P.C. (1988b).Nonlinear preference and utility theory. Baltimore: Johns Hopkins University Press.
Fishburn, P.C. & LaValle, I.H. (1987a). A nonlinear, nontransitive and additive probability model for decisions under uncertainty.Annals of Statistics,15, 830–844.
Fishburn, P.C. & LaValle, I.H. (1987b). Transitivity is equivalent to independence for state-additive SSB utilities.Journal of Economic Theory,44, 202–208.
Fishburn, P.C. & Nakamura, Y. (1991). Nontransitive measurable utility with constant threshold.Journal of mathematical Psychology,35, 471–500.
Kahneman, D. & Tversky, A. (1979). Prospect theory; an analysis of decision under risk.Econometrica,47, 263–291.
Machina, M.J. (1982). Expected utility analysis without the independence axiom.Econometrica,50, 277–323.
Nakamura, Y. (1990). Bilinear utility and a threshold structure for nontransitive preferences.Mathematics of Social Sciences,19, 1–21.
Nakamura, Y. (1997).A generalization of subjective expected utility without additivity and transitivity. IPPS discussion paper No. 719, University of Tsukuba.
Nakamura, Y. (1998). Skew-symmetric additive representations of preferences.Journal of Mathematical Economics,30, 367–387.
Nakamura, Y, (2001). Totally convex preferences for gambles.Mathematical Social Sciences,42, 295–305.
Quiggin, J. (1981). A theory of anticipated utility.Journal of Economic Behavior and Organization,3, 323–343.
Schmeidler, D. (1988). Subjective probability and expected utility without additivity.Econometrica,57, 571–587.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Nakamura, Y. (2009). SSB Preferences: Nonseparable Utilities or Nonseparable Beliefs. In: Brams, S.J., Gehrlein, W.V., Roberts, F.S. (eds) The Mathematics of Preference, Choice and Order. Studies in Choice and Welfare. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79128-7_3
Download citation
DOI: https://doi.org/10.1007/978-3-540-79128-7_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79127-0
Online ISBN: 978-3-540-79128-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)