Consistency of monomial and difference representations of functions arising from empirical phenomena

Choice probabilities in the behavioral sciences are often analyzed from the standpoint of a difference representation such as P(x, x, y) = F[u(x, x) - g(y)]. 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 ii, g and F are real valued, continuous functions, strictly increasing in all arguments. In some situations (e.g. in psychoph ysics), the researchers are more interested in the functions u and g than i n the function F. In such cases, they investigate the choice phenomenon by estimating empirically the value x such that P(x, x,y)= rho, for some value s of rho, and for many values of the variables involved in x and y. In othe r words, they study the function xi satisfying xi(x,y; rho) - x double left right arrow P(x, x,y) = rho. A reasonable model to consider for the functi on xi is offered by the monomial representation xi(x, y; rho) = Pi(i = 1)(n - 1) x(i)(-eta i(rho)) Pi(j = 1)(m) y(j)(zeta j (rho))C(rho), in which the eta(i)'s, the zeta(j)'s and C are functions of rho. In this pa per we investigate the consistency of these difference and monomial represe ntations. The main result is that, under some background conditions, if bot h the difference and the monomial representations are assumed, then: (i) al l functions eta(i) (1 less than or equal to i less than or equal to n - 1) must be constant; (ii) if one of the functions zeta(j) is nonconstant, then all of them must be of the form zeta(j)(rho) = theta(j)exp[delta F-1(rho)] , for some constants theta(j) > 0 (1 less than or equal to j less than or e qual to m) and delta not equal 0, where F-1 is the inverse of the function F of the difference representation. Surprisingly, F can be chosen rather ar bitrarily. (C) 1999 Academic Press.