Hostname: page-component-745bb68f8f-cphqk Total loading time: 0 Render date: 2025-01-08T10:20:51.450Z Has data issue: false hasContentIssue false
  • English
  • Français

Testable Conditions for Triads of Paired Comparison Choices

Published online by Cambridge University Press:  01 January 2025

H. William Morrison
H. William Morrison*
Affiliation:
IBM Thomas J. Watson Research Center
Get access Rights & Permissions [Opens in a new window]

Abstract

Forty-two choice models, each representing stimuli by one-dimensional probability distributions, are obtained by relaxing the assumptions of Thurstone's Case V Law of Comparative Judgment. The models which imply or fail to imply each of nine testable probabilistic conditions are determined. Stochastic transitivity is vulnerable in most of these models. The results suggest discarding weak stochastic transitivity, and in its place using the conjunction of weak stochastic transitivity and the triangular condition. However, unless it is possible to predict which stimuli will produce violations of the conditions, none of the conditions can be rejected on the basis of too frequent intransitive triads of choices.

Type
Original Paper
Information
Psychometrika , Volume 28 , Issue 4 , December 1963 , pp. 369 - 390
Copyright
Copyright © 1963 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

I am grateful to the referees, and to J. H. Griesmer and C. H. Coombs for a number of helpful criticisms and comments. J. H. Griesmer suggested a proof of Theorem 9, and a referee suggested a more direct proof of Theorem 5. A proof of Theorem 5 and the result concerning a maximally intransitive judge were given in [18].

References

Arrow, K. J. Social choice and individual values, New York: Wiley, 1951.Google Scholar
Block, H. D. and Marschak, J. Random orderings and stochastic theories of responses. In Olkin, I., Ghurye, S. G., Hoeffding, W., Madow, W. G. and Mann, H. B. (Eds.), Contributions to probability and statistics. Stanford: Stanford Univ. Press, 1960, 97–132.Google Scholar
Chipman, J. S. Stochastic choice and subjective probability. Econometrica, 1958, 26, 613–613 (Abstract).Google Scholar
Chipman, J. S. Stochastic choice and subjective probability. In Willner, D. (Eds.), Decisions, values and groups (Vol. 1). Oxford: Pergamon, 1960, 70–95.CrossRefGoogle Scholar
Coombs, C. H. On the use of inconsistency of preferences in psychological measurement. J. exp. Psychol., 1958, 55, 1–7.CrossRefGoogle ScholarPubMed
Coombs, C. H. Inconsistency of preferences as a measure of psychological distance. In Churchman, C. W. and Ratoosh, P. (Eds.), Measurement: Definitions and theories. New York: Wiley, 1959, 221–232.Google Scholar
Coombs, C. H., Greenberg, M. and Zinnes, J. A double law of comparative judgment for the analysis of preferential choice and similarities data. Psychometrika, 1961, 26, 165–171.CrossRefGoogle Scholar
Davidson, D. and Marschak, J. Experimental tests of stochastic decision theories. In Churchman, C. W. and Ratoosh, P. (Eds.), Measurement: Definitions and theories. New York: Wiley, 1959, 233–269.Google Scholar
Edwards, W. The theory of decision making. Psychol. Bull., 1954, 51, 380–416.CrossRefGoogle ScholarPubMed
Georgescu-Roegen, N. Threshold in choice and the theory of demand. Econometrica, 1958, 26, 157–168.CrossRefGoogle Scholar
Guilbaud, G. Sur une difficulté de la théorie du risque. Colloques International du Centre National de la Recherche Scientifique (Econométrie), 1953, 40, 19–25.Google Scholar
Luce, R. D. Individual choice behavior, New York: Wiley, 1959.Google Scholar
Luce, R. D. A choice theory analysis of similarity judgments. Psychometrika, 1961, 26, 151–163.CrossRefGoogle Scholar
Kendall, M. G. Rank correlation methods, London: Griffin, 1948.Google Scholar
McGarvey, D. C. A theorem on the construction of voting paradoxes. Econometrica, 1953, 21, 608–610.CrossRefGoogle Scholar
Marschak, J. Binary-choice constraints and random utility indicators. In Arrow, K. J., Karlin, S. and Suppes, P. (Eds.), Mathematical models in the social sciences. Stanford: Stanford Univ. Press, 1959, 312–329.Google Scholar
Menger, K. Probabilistic theories of relations. Proc. nat. Acad. Sci., 1951, 37, 178–180.CrossRefGoogle ScholarPubMed
Morrison, H. W. Intransitivity of paired comparison choices. Unpublished doctoral dissertation, University of Michigan, 1962.Google Scholar
Quandt, R. E. A probabilistic theory of consumer behavior. Quart. J. Econ., 1956, 70, 507–536.CrossRefGoogle Scholar
Thurstone, L. L. A law of comparative judgment. Psychol. Rev., 1927, 34, 273–286.CrossRefGoogle Scholar
Vail, S. V. A stochastic model of utilities. Unpublished manuscript, No. 24, University of Michigan Seminar on the Applications of Mathematics to the Social Sciences, April 23, 1953.Google Scholar