Multidimensional Fechnerian scaling: Pairwise comparisons, regular minimality, and nonconstant self-similarity. (Q1867365)

In this paper the Multidimensional Fechnerian Scaling (MDFS) Theory co-developed by the author (see the background references quoted in the paper) is extended to take into account the fact that when multidimensional stimuli are presented pairwise, they often belong to different areas (e.g.\ time intervals or locations). The basic axioms of MDFS are modified to suit this case. In particular, a symmetry axiom is introduced roughly stating that if \(y_0\) minimizes the probability of being distinguished from \(x_0\) among all \(y\), then \(x_0\) minimizes the analogous probability among all \(x\) compared to \(y_0\). In addition, an axiom is introduced recognizing the empirically substantiated observation that the probability of distinguishing \(x\) from itself is not the same for all \(x\). The main result of the paper is that under these new axioms the psychometric order of the stimulus space cannot exceed one. That means that around its minima, the discrimination probability function (the function \(\phi(x,y)\) giving the probability that stimulus \(x\) is distinguished from \(y\)) may have pencil-sharp or needle-sharp shape, but it cannot be flat. More intuitively, under the stated axioms nearby stimuli are well separated in the observer's percetption.