Revisiting the connection between the Luce’s choice rule and signal detection theory

In many decision tasks, we have a set of alternative choices and are faced with the problem of how to take our latent preferences or beliefs about each alternative and make a single choice. For example, we must decide which item is ‘old’ in a forced-choice memory study; or which cereal we prefer in a supermarket; or which color a word is in a Stroop task. Modeling how people go from latent strengths for each alternative to a single choice is thus a critical component of nearly all cognitive and decision models. Most models follow one of two traditions to establish this link. Modern psychophysics and memory researchers make use of signal detection theory, in the tradition of Fechner (1860) and Thurstone (1929), assuming that latent strengths are perturbed by noise, and the highest resulting signal is selected (e.g., Wixted, 2020). By contrast, many modern cognitive modeling and machine learning approaches use the softmax rule to give some weight to non-maximal-strength alternatives (Luce choice axiom; Luce, 1959). Despite the prominence of these two theories of choice, current approaches rarely address the connection between them, and the choice of one or the other appears more motivated by the tradition in the relevant literature than by theoretical or empirical reasons to prefer one theory to the other. The goal of the current work is to revisit this topic by elucidating which of these two models provides a better characterization of latent processes in K-alternative decision tasks, with a particular focus on memory tasks. In line with previous work (e.g., Luce and Suppes, 1966; Yellot, 1977), we find via both simulation and mathematical proofs that the softmax and signal detection link functions can mimic each other with high fidelity in all circumstances. However, we show that while the softmax parameter varies across task structures using the same stimuli (i.e., changes when K is varied), the parameter d’ of the signal-detection model is stable. The results of these studies are consistent with the results of Treisman and Faulkner (1985) in a novel suite of memory tasks. Together, our findings indicate that replacing softmax with signal-detection link models would yield more generalizable predictions across changes in task. More ambitiously, the invariance of signal detection model parameters across different tasks suggests that the mechanisms of these models (i.e., the corruption of signals by stochastic noise) may be more than just a mathematical convenience but reflect something real about human decision-making.