Abstract
In many choice situations, the options are multidimensional. Numerous probabilistic models have been developed for such choices between multidimensional options and for the parallel choices determined by one or more components of such options. In this paper, it is assumed that a functional relation exists between the choice probabilities over the multidimensional options and the choice probabilities over the associated component unidimensional options. It is shown that if that function satisfies a marginalization property then it is essentially an arithmetic mean, and if the function satisfies a likelihood independence property then it is a weighted geometric mean. The results are related to those on the combination of expert opinion, and various probabilistic models in the choice literature are shown to have the geometric mean form.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aczel, J.: 1966, Lectures on Functional Equations and Their Applications. New York: Academic Press.
Aczel, J.: 1984, ‘On weighted synthesis of judgements’, Aequationes Mathematicae, 27, 288–307.
Aczel, J., Kannappan, Pl., Ng, C. T., and Wagner, C.: 1983, ‘Functional equations and inequalities in “rational group decision making”,’ In E.F. Beckenbach and W. Walter (Eds.), General Inequalities 3, Basel: Birkäuser Verlag.
Aczel, J., Ng, C. T., and Wagner, C.: 1984, ‘Aggregation theorems for allocation problems’, SIAM J. Algebraic Discrete Methods, 5, 1–8.
Aczel, J., and Dhombres, J.: 1989, Functional Equations in Several Variables. New York: Cambridge University Press.
Alsina, C.: 1989, ‘Synthesizing judgements given by probability distribution functions’, Manuscript, Department Mathemàtiques, Universidad Politècnica de Catalunya.
Bordley, R. F.: 1982, ‘A multiplicative formula for aggregating probability assessments’, Management Science, 28, 1137–1148.
Corbin, R. and Marley, A. A. J.: 1974, ‘Random utility models with equality: An apparent, but not actual, generalization of random utility models’, J. Mathematical Psychology, 11, 274–293.
Debreau, G.: 1960, ‘Review of R.D. Luce “Individual choice behaviour: A theoretical analysis”,’ American Economic Review, 50, 186–188.
Dempster, A. P.: 1967, ‘Upper and lower probabilities induced by a multivalued mapping’, Ann. Mathematical Statistics, 38, 325–339.
Falmagne, J. C.: 1981, ‘On a recurrent misuse of a classical functional equation result’, J. Mathematical Psychology, 23, 190–193.
Genest, C.: 1984, ‘Pooling operators with the marginalization property’, Can. J. Statistics, 12, 153–163.
Genest, C., Weerahandi, S., and Zidek, J. V.: 1984, ‘Aggregating opinion pools through logarithmic pooling’, Theory and Decision, 17, 61–70.
Genest, C. and Zidek, J. V.: 1986, ‘Combining probability distributions: A critique and an annotated bibliography’, Statistical Science, 1, 114–148.
Hoijtink, H.: 1989, ‘A latent trait model for dichotomous choice data’, Manuscript, University of Groningen, Netherlands.
Luce, R. D.: 1959, Individual Choice Behaviour, New York: Wiley.
Luce, R. D.: 1977, ‘The choice axiom after twenty years’, J. Mathematical Psychology, 15, 215–233.
Luce, R. D. and Suppes, P.: 1965, ‘Preference, utility, and subjective probability’, In R. D. Luce, R. R. Bush, and E. Galanter (Eds.), Handbook of Mathematical Psychology, III. New York: Wiley, pp. 230–270.
Marley, A. A. J.: 1989, ‘A random utility family that includes many of the “classical” models and has closed form choice probabilities and choice reaction times’, Brit. J. of Mathematical and Statistical Psychology, 42, 13–36.
Marley, A. A. J.: 1991a, ‘Context dependent probabilistic choice models based on measures of binary advantage’, Mathematical Social Sciences (to appear).
Marley, A. A. J.: 1991b, ‘Developing and characterizing multidimensional Thurstone and Luce models for identification and preference’, In F. G. Ashby (Ed.), Multidimensional Models of Perception and Cognition, Hillsdale, NJ: Erlbaum.
Marshall, A. W. and Olkin, I.: 1988, ‘Families of multivariate distributions’, J. American Statistical Association, 83, 834–841.
McConway, K. J.: 1981, ‘Marginalization and linear opinion pools’, J. American Statistical Association, 76, 410–414.
McFadden, D.: 1978, ‘Modelling the choice of residential location’, In A. Karlquist et al. (Eds), Spatial Interaction Theory and Planning Models, Amsterdam: North Holland.
Rotondo, J.: 1986, ‘A generalization of Luce's choice axiom and a new class of choice models’, Psychometric Society, Abstract, 1986.
Schmidt, F. F.: 1984, ‘Consensus, respect, and weighted averaging’, Synthese, 62, 25–46.
Shafer, G.: 1976, A Mathematical Theory of Evidence, N.J.: Princeton University Press.
Small, K. A.: 1986, ‘A discrete choice model for ordered alternatives’, Econometrika, 55, 409–425.
Strauss, D.: 1981, ‘Choice by features: An extension of Luce's choice model to account for similarities’, Brit. J. Mathematical and Statistical Psychology, 34, 50–61.
Tversky, A.: 1972a, ‘Elimination by aspects: A theory of choice’, Psychological Review, 79, 281–299.
Tversky, A.: 1972b, ‘Choice by elimination’, J. Mathematical Psychology, 9, 341–367.
Wagner, C. G.: 1989, ‘Consensus for belief functions and related uncertainty measures’, Theory and Decision, 26, 295–304.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Marley, A.A.J. Aggregation theorems and multidimensional stochastic choice models. Theor Decis 30, 245–272 (1991). https://doi.org/10.1007/BF00132446
Issue Date:
DOI: https://doi.org/10.1007/BF00132446