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 ξ s ( x ) = x + Δ s ( x ) (where s = F − 1 ( π ) and Δ s 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, ξ s ( λ x ) = λ ι ( s ) ξ η ( 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 ξ s ( 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. [ABSTRACT FROM AUTHOR]