Comparing Incommensurable Quantities: Intertemporal vs. Risky Choices with Single Outcomes

The equate-to-differentiate (ETD) model posits that individuals tend to equate a less significant difference between options on one dimension and thus leave the greater one-dimensional difference to be differentiated as the determinant for the preferred option. However, when confronted with an ostensibly "simple" choice between two risky options with single-nonzero outcomes or between two intertemporal options with single-dated outcomes, we face an insurmountable barrier against the ETD model's explanation and prediction of these choices. The reason is that determining which intra-dimensional difference (∆Payoff_A,B or ∆Probability_A,B/∆Delay_A,B) between Option A and Option B is greater is meaningless and is considered to be a challenge in the physical world. To address this challenge and evaluate whether such decisions are indeed governed by the ETD process, the present study developed a visual analogue scale designed to capture individuals' subjective comparisons across dimensions of different units. Across two studies, we demonstrate that the analogically measured intra-dimensional comparison reliably and consistently predicts choice patterns attributed to separate anomalies: the common difference effect and unit effect in intertemporal decisions, and subproportionality and the peanuts effect in risky decisions. These findings suggest that both types of decisions may share a common cognitive mechanism based on dimensional evaluation, despite involving distinct informational metrics (time vs. probability). By enabling direct measurement of dimension-wise comparisons, our analogue scale-though unconventional-offers a novel methodological tool for exploring the underlying structure of seemingly "simple" decisions. The implications of this work extend to the development of unified models capable of integrating intertemporal and risky decision-making under a shared explanatory framework.