IOURNAL OF MATHEMATICAL PSYCHOLOGY 31, 83-92 (1987) Stimulus- Response with Reciprocal Kernels: The Rise and Fall of Sensation THOMAS L. SAATY AND Lurs G. VARGAS The paper presents a theory for constructing response scales based on the reciprocal property of paired comparisons of stimuli from the same sensory continuum. Reciprocal paired comparisons detine the pair estimator function K(s, I), the kernel of a Fredholm integral equation of the second kind. The solution of this equation yields a response scale of the form Ye- I”, where s is the stimulus scale. The response scale provides a basis for a theory to describe how images and sensation patterns are formed from stimuli. ( 1987 Academx Press. Inc 1. INTRODUCTION A basic concept at the core of understanding is the reciprocal property. Essen- tially this property asserts that when comparing two stones according to weight with the aid of the hands and one stone is judged to be live times heavier than the other, then the other is automatically one fifth the weight of the first. Both stones participate in the judgment, the smaller one serving as a unit of reference. All our senses are capable of making such comparisons and so is our mind in comparing abstract ideas with respect to common properties. From such comparisons among objects in pairs, one derives a relative scale of measurement among the objects. Reciprocal comparison is an inherent ability of all human beings which enables us to scale things encountered in practice. The logical question then is: Is our judgment sufficiently accurate to ensure that if stone A is live times heavier than stone B and stone B is three times heavier than stone C that we would judge A to be fifteen times heavier than C? Most likely not. To improve the accuracy of the scale derived from the paired comparisons we should also compare A with C. We would then need to say something about the inconsistency in performing comparisons. In general if av indicates the relative dominance of object i over objectj when comparing n objects in pairs, the com- parisons are said to be consistent if the following relation holds: adaik = aik for all i,j, k = 1, 2 ,..., n. (1) The reciprocal property mentioned earlier is given by aii = l/a, and follows from consistency but does not imply it and is thus a weaker condition that we can safely 83 0022-2496187 $3.00 Copyright ((,’ 1987 by Academic Press, Inc. All rights of reproduction in any form reserved.