Measurement Foundations for Multiattribute Psychophysical Theories Based on First Order Polynomials

The class of first order polynomial measurement representations is defined, and a method for proving the existence of such representations is described. The method is used to prove the existence of first order polynomial generalizations of expected utility theory, difference measurement, and additive conjoint measurement. It is then argued that first order polynomial representations provide a deep and far reaching characterization of psychological invariance for subjective magnitudes of multiattributed stimuli. To substantiate this point, two applications of first order polynomial representation theory to the foundations of psychophysics are described. First, Relation theory, a theory of subjective magnitude proposed by Shepard (Journal of Mathematical Psychology, 1981, 24, 21-57) and Krantz (Journal of Mathematical Psychology, 1972, 9, 168-199), is generalized to a theory of magnitude for multiattributed stimuli. The generalization is based on a postulate of context invariance for the constituent uniattribute magnitudes of a multiattribute magnitude. Second, the power law for subjective magnitude is generalized to a multiattribute version of the power law. Finally, it is argued that a common logical pattern underlies multiattribute generalizations of psychological theories to first order polynomial representations. This abstract pattern suggests a strategy for theory construction in multiattribute psychophysics.