A discrimination probability function cðx; yÞ obtained in the ''same-different'' paradigm assigns to every ordered pair of stimuli ðx; yÞ the probability with which they are judged to be different. This function is said to possess the regular minimality property if, for any stimulus pair ða; bÞ;
That is, b is the point of subjective equality for a if and only if a is the point of subjective equality for b: If the value of cða; bÞ across all such pairs ða; bÞ is not constant, the function is said to possess the nonconstant self-similarity property. A Thurstonian-type representation for cðx; yÞ (with independent images and deterministic decisions) is a model in which the two stimuli are mapped into two independent random variables PðxÞ and QðyÞ taking on their values in some ''perceptual'' space; and the decision whether the two stimuli are different is determined by the realizations of the two random variables in a given trial. Thurstonian-type representations can also be called ''random utility'' ones, provided one imposes no a priori restrictions on the structure of the perceptual space, the distributions of PðxÞ and QðyÞ; or the decision rules used. It is shown that (A) any cðx; yÞ has a Thurstoniantype representation; but (B) if cðx; yÞ possesses the regular minimality and nonconstant self-similarity properties, it cannot have a ''well-behaved'' Thurstonian-type representation, in which the probability with which PðxÞ; or QðyÞ; falls within a given subset of the perceptual space has appropriately defined bounded directional derivatives with respect to x (respectively, y). This regularity feature is likely to be found in most conceivable Thurstonian-type models constructed to fit empirical data.