Unite and Conquer: A Multiplicative Inequality for Choice Probabilities - Sattath, Shmuel; Tversky, Amos (Econometrica, 1976)
Two probabilistic theories of choice behavior (the model of independent random utility and the model of elimination by aspects) imply a testable property, the multiplicative inequality, according to which the probability of selecting an alternative x from an offered set A u B is at least as large as the product of the probabilities of selecting x from A and from B. WHEN FACED WITH a choice among complex alternatives (e.g., commodity bundles, investment plans, job offers) people often exhibit inconsistency. That is, they do not always select the same alternative under seemingly identical conditions. In order to accommodate this fact and obtain an adequate conception of choice behavior, psychologists and economists (e.g., Thurstone [18 and 19], GeorgescuRoegen [8], Luce [12], and Marschak [14]) developed models of choice in which the traditional concept of preference is replaced by the notion of choice probability. These models were investigated by many authors (e.g., Davidson and Marschak [5], Debreu [6], Chipman [4], Luce and Suppes [13], and Tversky [20]) from both mathematical and experimental standpoints, and they have also been applied to various aspects of economic theory such as equilibrium analysis (Hildenbrand [10] and Bhattacharya and Majumdar [2]) and consumer behavior (GeorgescuRoegen [9], Quandt [17], Mossin [16], and McFadden and Richter [15]). Two general forms of probabilistic choice models, called random utility and constant utility, were investigated (see Luce and Suppes [13]). In the random utility form, the subjective values undergo random fluctuations, and the alternative with the highest momentary value is selected. In the constant utility form, choice probability is expressed as a function of some (constant) scale values. Thus, the two forms differ regarding the locus of the random element in the choice process. The random utility form attributes uncertainty to the determination of subjective value, whereas the constant utility form attributes uncertainty to the application of the decision rule. (Some choice models, however, can be expressed in either form.) This paper investigates one prominent example of each form: the independent random utility model and the model of elimination by aspects. It shows that both models satisfy two testable properties which provide (fairly tight) upper and lower bounds for all choice probabilities. To formulate the results, we introduce the following definitions: