Abstract
Holman and Marley have shown that Thurstone's Case V model becomes equivalent to the Choice Axiom if its discriminal processes are assumed to be independent double exponential random variables instead of normal ones. It is shown here that for pair comparisons, this representation is not unique; other discriminal process distributions (specifiable only in terms of their characteristic functions) also yield a model equivalent to the Choice Axiom. However, none of these models is equivalent to the Choice Axiom for triple comparisons: There the double exponential representation is unique. It is also shown that within the framework of Thurstone's theory, the double exponential distribution, and hence the Choice Axiom, is implied by a weaker assumption, called “invariance under uniform expansions of the choice set.”
Highlights
The Relationship between Lute’s Choice Axiom, Thurstone’s Theory of Comparative Judgment, and the Double Exponential Distri butionl
As premises for a model of choice behavior, Lute’s (1959) Choice Axiom and Thurstone’s (1927) Theory of Comparative Judgment seem at first glance to be different, but quite unrelated
In his 1959monograph, Lute included a table showing that for pair-comparison experiments, the predictions of the Choice Axiom are virtually identical to those of Case V of Thurstone’s theory. (Recall that in Case V the discriminal processescorresponding to a set of objects o1, oz,... take the form ur + X, ua + X,..., where u1, ua,... are constants and x, x, )... are independent identically distributed normal random variables.) He i I thank J
Summary
As premises for a model of choice behavior, Lute’s (1959) Choice Axiom and Thurstone’s (1927) Theory of Comparative Judgment seem at first glance to be different, but quite unrelated. (where a and b are arbitrary constants, a > 0; see Fig. lB), Xi - Xj will be logistic, and the resulting model is equivalent to the Choice Axiom for any choice experiment, not for pair comparisons.(This result is proved here in Section 2.6.) Holman and Marley did not show that the double exponential is the ~ZJJ distribution with this property. In presenting their result, Lute and Suppes (1965). In order to solve the original uniquenessproblem, it was natural to generalize it, and a good deal of the paper is devoted to the uniquenessproperties of arbitrary discriminal processdistributions
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