Axiomatic evaluation of k-multiplicativity in ratio scaling: Investigating numerical distortion

It is a well established empirical observation that most human participants do not process the numerical instructions used in production or estimation tasks veridically. Luce and collaborators (e.g., Luce, 2002; Steingrimsson and Luce, 2007) have analyzed the kind of “numerical distortion” that appears to be operating. They stated the relationship between perceived and mathematical numbers to be described by a power function, if the empirically testable axiom of k-multiplicativity holds. This study examined the validity of k-multiplicativity by testing whether the stimulus intensities resulting from successive adjustments ×p×q multiplied by a constant factor k are equal to the stimulus intensity resulting from single adjustments ×r. Therefore, the data of three different ratio production experiments with a total of N=35 participants were (re-)analyzed. In Experiment I, integers were used as ratio production factors (p≥1), while in Experiment II, only fractions (p<1) were applied. In Experiment III, both p≥1 and p<1 were intermixed. In Experiments I and II, k-multiplicativity held for all n=20 participants. Experiment III revealed axiom violations for 13 of n=15 participants. The failure of 1-multiplicativity confirms the common observation that the participants' number representation is often not veridical. However, the validity of k-multiplicativity shows that the relationship between mathematical and perceived numbers follows a power function of the form W(p)=kpω with k≠1 and ω≠1. However, the numerical distortion differs for fractions compared to integers.