On Iverson's law of similarity

TL;DR

This work analyzes Iverson's law of similarity, $\xi_{s}(\lambda x)=\gamma(\lambda, s)\xi_{\eta(\lambda, s)}(x)$, in psychophysics by treating it as a functional-equation problem and exploring when the sensitivity function $\xi$ can be expressed in reduced, one-variable forms. It shows that if there exists $s^{*}$ with $\lambda \mapsto \eta(\lambda, s^{*})$ invertible, then $g(s)\xi_{s}(x)=\Phi(f(s)x)$, and if $\eta$ is constant on a subset of $S$, then $\xi_{s}(x)=\kappa(s)x^{\rho(s)}$, linking Iverson's law to Falmagne's power law. The paper then investigates gain-control and parallel/anchored representations, and provides a complete description of anchored, balanced, and parallel psychometric families. In the multiplicatively translational setting, it derives $\eta(\lambda, s)=H(\lambda H^{-1}(s))$ and shows that power-law exponents $\phi(s)$ yield $\xi_{s}(x)=x^{\phi(s)}F(xH^{-1}(s))$, with monotonic $\phi$ forcing local constancy and leading to subtractive representations; shift-invariance scenarios are also examined. Overall, the results delineate the structural forms permissible under Iverson's law and offer groundwork for empirical discrimination among competing representations.

Abstract

Iverson (2006) proposed the law of similarity \[ ξ_{s}(λx)= γ(λ, s)ξ_{η(λ, s)}(x) \] for the sensitivity functions $ξ_{s}\, (s\in S)$. Compared to the former models, the generality of this one lies in that here $γ$ and $η$ can also depend on the variables $λ$ and $s$. In the literature, this model (or its special cases) is usually considered together with a given psychophysical representation (e.g. Fechnerian, subtractive, or affine). Our goal, however, is to study at first Iverson's law of similarity on its own. At first we show that if certain mild assumptions are fulfilled, then $ξ$ can be written in a rather simple form containing only one-variable functions. The obtained form proves to be very useful when we assume some kind of representation. Motivated by Hsu and Iverson (2016), in the second part of the third section we study the above model assuming that the mapping $η$ is multiplicatively translational. First, we show how these mappings can be characterized. Later on we turn to the examination of the so-called power law. According to our results, the corresponding function $ξ$ then does not have a Fechnerian representation, but it do have a subtractive representation. As an application of the results of the subsection, we close the paper with the study of the shift invariance property.

Theorems & Definitions (42)

• Lemma 1
• Lemma 2
• Remark 1
• Remark 2
• Remark 3
• Theorem 1
• proof
• Remark 4
• Remark 5
• Proposition 1