Consistency of Monomial and Difference Representations of Functions Arising from Empirical PhenomenaU

Choice probabilities in the behavioral sciences are often analyzed from the standpoint of a difference representation such as P x, x, y.sFwu x, x.yg y.x. Here, x and y are real, positive vector variables, x is a positive real variable, P x, x, y.is the probability of choosing alternative x, x.over alternative y, and u, g and F are real valued, continuous functions, strictly increasing in all arguments. In some situations e.g. in psychophysics., the researchers are more interested in the functions u and g than in the function F. In such cases, they investigate the choice phenomenon by estimating empirically the value x such that P x, x, y.sr, for some values of r, and for many values of the variables involved in x and y. In other words, they study the function j satisfying j x, y; r.sxmP x, x, y.sr. A reasonable model to consider for the function j is offered by the monomial representation j x, y; r s ny1 xyh i r . m . is1 i js1 yjz j r .C r., in which the hi’s, the zj’s and C are functions of r. In this paper we investigate the consistency of these difference and monomial representations. The main result is that, under some background conditions, if both the difference and the monomial representations are assumed, then: i. all functions hi 1FiFny1. must be *We thank Bruce Bennett, Jean-Paul Doignon, and Geoff Iverson for their reactions, and Yung-Fong Hsu for pointing out a gap in a previous draft of our proof of Theorem 3.2. We are also grateful to the Institute for Mathematical Behavioral Sciences for its hospitality to the first author. This research has been supported by the Natural Sciences and Engineering Research Council of Canada Grant No. OGP 0164211, and by NSF Grant SBR 930-7420. 632 0022-247Xr99 $30.00 Copyright Q 1999 by Academic Press All rights of reproduction in any form reserved. MONOMIAL AND DIFFERENCE REPRESENTATIONS 633 constant; ii. if one of the functions zj is nonconstant, then all of them must be of the form zj r.sujexpwd Fy1 r.x, for some constants uj)0 1FjFm.and d /0, where Fy1 is the inverse of the function F of the difference representation. Surprisingly, F can be chosen rather arbitrarily.