Matching by Adjustment: If X Matches Y, Does Y Match X?

When dealing with pairwise comparisons of stimuli in two fixed observation areas (e.g., one stimulus on the left, one on the right), we say that the stimulus space is regular well-matched if (1) every stimulus is matched by some stimulus in another observation area, and this matching stimulus is determined uniquely up to matching equivalence (two stimuli being equivalent if they always match or do not match any stimulus together); and (2) if a stimulus is matched by another stimulus then it matches it. The regular well-matchedness property has non-trivial consequences for several issues, ranging from the ancient “sorites” paradox to “probability-distance hypothesis” to modeling of discrimination probabilities by means of Thurstonian-type models. We have tested the regular well-matchedness hypothesis for locations of two dots within two side-by-side circles, and for two side-by-side “flower-like” shapes obtained by superposition of two cosine waves with fixed frequencies in polar coordinates. In the location experiment the two coordinates of the dot in one circle were adjusted to match the location of the dot in another circle. In the shape experiment the two cosine amplitudes of one shape were adjusted to match the other shape. The adjustments on the left and on the right alternated in long series according to the “ping-pong” matching scheme developed in Dzhafarov (2006b, J. Math. Psychol., 50, 74–93). The results have been found to be in a good agreement with the regular well-matchedness hypothesis.