×

Combining additive representations on subsets into an overall representation. (English) Zbl 1153.91389

Summary: Many traditional conjoint representations of binary preferences are additively decomposable, or additive for short. An important generalization arises under rank-dependence, when additivity is restricted to cones with a fixed ranking of components from best to worst (comonotonicity), leading to configural weighting, rank-dependent utility, and rank- and sign-dependent utility (prospect theory). This paper provides a general result showing how additive representations on an arbitrary collection of comonotonic cones can be combined into one overall representation that applies to the union of all cones considered. The result is applied to a new paradigm for decision under uncertainty developed by Duncan Luce and others, which allows for violations of basic rationality properties such as the coalescing of events and other framing conditions. Through our result, a complete preference foundation of a number of new models by Luce and others can be obtained. We also show how additive representations on different full product sets can be combined into a representation on the union of these different product sets.

MSC:

91B08 Individual preferences
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Birnbaum, M. H., The nonadditivity of personality impressions, Journal of Experimental Psychology, 102, 543-561 (1974)
[2] Birnbaum, M. H., Tests of branch splitting and branch-splitting independence in Allais paradoxes with positive and mixed consequences, Organizational Behavior and Human Decision Processes, 102, 154-173 (2007)
[3] Blackorby, C.; Bossert, W.; Donaldson, D., Population ethics and the existence of value functions, Journal of Public Economics, 82, 301-308 (2001)
[4] Chateauneuf, A.; Wakker, P. P., From local to global additive representation, Journal of Mathematical Economics, 22, 523-545 (1993) · Zbl 0804.90004
[5] Chew, S. H.; Wakker, P. P., The comonotonic sure-thing principle, Journal of Risk and Uncertainty, 12, 5-27 (1996) · Zbl 0848.90007
[6] Ebert, U., Social welfare, inequality, and poverty when needs differ, Social Choice and Welfare, 23, 415-448 (2004) · Zbl 1109.91372
[7] Gilboa, I., Expected utility with purely subjective non-additive probabilities, Journal of Mathematical Economics, 16, 65-88 (1987) · Zbl 0632.90008
[8] Green, J. R.; Jullien, B., Ordinal independence in non-linear utility theory, Journal of Risk and Uncertainty. Journal of Risk and Uncertainty, Journal of Risk and Uncertainty, 2, 119-387 (1989), (erratum)
[9] Krantz, D. H.; Luce, R. D.; Suppes, P.; Tversky, A., (Additive and polynomial representations. Additive and polynomial representations, Foundations of measurement, Vol. I (1971), Academic Press: Academic Press New York), (2nd ed.) 2007, Dover Publications, New York · Zbl 0232.02040
[10] Köbberling, V.; Wakker, P. P., Preference foundations for nonexpected utility: A generalized and simplified technique, Mathematics of Operations Research, 28, 395-423 (2003) · Zbl 1082.91042
[11] Köbberling, V.; Wakker, P. P., A simple tool for qualitatively testing, quantitatively measuring, and normatively justifying Savage’s subjective expected utility, Journal of Risk and Uncertainty, 28, 135-145 (2004) · Zbl 1097.91507
[12] Luce, R. D., Rank-dependent, subjective expected-utility representations, Journal of Risk and Uncertainty, 1, 305-332 (1988)
[13] Luce, R. D., Rational versus plausible accounting equivalences in preference judgments, Psychological Science, 1, 225-234 (1990)
[14] Luce, R. D., Utility of gains and losses: Measurement-theoretical and experimental approaches (2000), Lawrence Erlbaum Publishers: Lawrence Erlbaum Publishers London · Zbl 0997.91500
[15] Luce, R. D.; Fishburn, P. C., Rank- and sign-dependent linear utility models for finite first-order gambles, Journal of Risk and Uncertainty, 4, 29-59 (1991) · Zbl 0743.90009
[16] Luce, R. D.; Marley, A. A.J., Ranked additive utility representations of gambles: Old and new axiomatizations, Journal of Risk and Uncertainty, 30, 21-62 (2005) · Zbl 1109.91325
[17] Marley, A. A.J.; Luce, R. D., Independence Properties vis-à-vis several utility representations, Theory and Decision, 58, 77-143 (2005) · Zbl 1137.91382
[18] Marley, A. A.J.; Luce, R. D.; Kocsis, I., A solution to a problem raised in Luce and Marley (2005), Journal of Mathematical Psychology, 52, 64-68 (2008) · Zbl 1134.91366
[19] Quiggin, J., A theory of anticipated utility, Journal of Economic Behaviour and Organization, 3, 323-343 (1982)
[20] Savage, L. J., The foundations of statistics (1954), Wiley: Wiley New York, (2nd ed.) 1972, Dover Publications, New York · Zbl 0121.13603
[21] Schmeidler, D., Subjective probability and expected utility without additivity, Econometrica, 57, 571-587 (1989) · Zbl 0672.90011
[22] Segal, U., Anticipated utility: A measure representation approach, Annals of Operations Research, 19, 359-373 (1989) · Zbl 0707.90023
[23] Starmer, C.; Sugden, R., Probability and Juxtaposition effects: An experimental investigation of the common ratio effect, Journal of Risk and Uncertainty, 2, 159-178 (1989)
[24] Tversky, A.; Kahneman, D., Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty, 5, 297-323 (1992) · Zbl 0775.90106
[25] von Winterfeldt, D.; Edwards, W., Decision analysis and behavioral research (1986), Cambridge University Press: Cambridge University Press Cambridge, UK
[26] Wakker, P. P., The algebraic versus the topological approach to additive representations, Journal of Mathematical Psychology, 32, 421-435 (1988) · Zbl 0663.92024
[27] Wakker, P. P., Additive representations on rank-ordered sets II. The topological approach, Journal of Mathematical Economics, 22, 1-26 (1993) · Zbl 0894.92041
[28] Weymark, J. A., Generalized Gini inequality indices, Mathematical Social Sciences, 1, 409-430 (1981) · Zbl 0477.90019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.