The Choice Axiom after Twenty Years

This survey is divided into three major sections. The first concerns mathematical results about the choice axiom and the choice models that devoIve from it. For example, its relationship to Thurstonian theory is satisfyingly understood; much is known about how choice and ranking probabilities may relate, although little of this knowledge seems empirically useful; and there are certain interesting statistical facts. The second section describes attempts that have been made to test and apply these models. The testing has been done mostly, though not exclusively, by psychologists; the applications have been mostly in economics and sociology. Although it is clear from many experiments that the conditions under which the choice axiom holds are surely delicate, the need for simple, rational underpinnings in complex theories, as in economics and sociology, leads one to accept assumptions that are at best approximate. And the third section concerns alternative, more general theories which, in spirit, are much like the choice axiom. Perhaps I had best admit at the outset that, as a commentator on this scene, I am qualified no better than many others and rather less well than some who have been working in this area recently, which I have not been. My pursuits have led me along other, somewhat related routes. On the one hand, I have contributed to some of the recent, purely algebraic aspects of fundamental measurement (for a survey of some of this material, see Krantz, Lute, Suppes, & Tversky, 1971). And on the other hand, I have worked in the highly probabilistic area of psychophysical theory; but the empirical materials have led me away from axiomatic structures, such as the choice axiom, to more structural, neural models which are not readily axiomatized at the present time. After some attempts to apply choice models to psychophysical phenomena (discussed below in its proper place), I was led to conclude that it is not a very promising approach to, these data, and so I have not been actively studying any aspect of the choice axiom in over 12 years. With that understood, let us begin.