Conditions under which affine representations are also Fechnerian under the power law of similarity on the Weber sensitivities

In a recent paper, Hsu, Iverson, and Doble (2010) examined some properties of a (weakly balanced) affine representation for choices, Ψ(x,y)=F(u(x)−u(y)σ(y)), and showed that using the Fechner method of integrating jnds, one can reconstruct the scales u and σ from the behavior of (Weber) sensitivities ξs(x)=x+Δs(x) (where s=F−1(π) and Δs is the jnd) in a neighborhood of s=0. Following Iverson (2006b), in this article we impose a power law of similarity on the sensitivities, ξs(λx)=λι(s)ξη(λ,s)(x), and study its impact on u and σ in the affine representation. Especially, we specify the conditions for the first- and second-order derivatives of ξs(x) with respect to s (and evaluated at s→0) under which the affine representation degenerates to a Fechnerian one. We also link the results to the solutions in Iverson (2006b), which was worked out within the Fechnerian framework.