Aggregation theorems and multidimensional stochastic choice models

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.

Evaluating and Combining Subjective Probability Estimates

“…We limit our brief review only to those aspects of the literature that bear on our problem. Clemen (1989) and Genest and Zidek (1986) have provided excellent reviews with annotated bibliographies that cover virtually all but the most recent contributions (see also Marley, 1991). Clemen considered probability distributions as one of many types of variables that might be combined, while Genest and Zidek focused on them almost exclusively.…”

Some issues related to the topological aggregation of preferences

“…See, for instance, Tanguiane (1979) or Marley (1991). To deal with a "continuous" case, a different approach, using probability measures on X, seems to be more adequate.…”

“…(See Tanguiane 1979or Marley 1991 2 It is well known (see Chichilnisky 1980) the following social choice paradox: In a preference space represented by X= S 1 (the unit circle with the usual topology), "two persons could never agree", in the sense that there is no aggregation rule F: X× X--*X which is continuous, anonymous, and unanimous. The above definitions and problems constitute our framework on the topological approach to the aggregation of preferences.…”

Violations of Transitivity: Implications for a Theory of Contextual Choice

“…If attribute sensitivity is context dependent, however, a model capable of predicting violations of ordinal preference is possible. A behavioral alternative to probabilistic, multidimensional choice theories may therefore be feasible, because it was empirical violations of ordinal preference (and SST) that convinced many psychologists to abandon deterministic, psychophysical models (e.g., Luce, 1959) in favor of probabilistic, multidimensional models (Tversky, 1972; see also Luce, 1977;Marley, 1991).…”