An analysis of discounting model selection methods: Assessing the generalization of discounting models

How the subjective value of an outcome changes as a function of time, probability, or effort has been an active area of psychological and economic research for decades. The exact functional form of how a commodity is discounted has been debated, and there have been numerous forms proposed. One of the challenges when trying to determine the functional form of discounting data is how models are compared, what modeling methods are used, how many data points are used, and what comparison metrics were used. Thus, we sought to replicate and extend previous research comparing discounting model selection methods by simulating discounting data from five functional forms: the Mazur hyperbolic model (Mazur, 1987), Rachlin hyperboloid (Rachlin, 2006), Myerson-Green hyperboloid (Myerson & Green, 1995), Samuelson exponential model (Samuelson, 1937), and beta-delta model (Laibson, 1997). With each of these models we manipulated the number (i.e., density) of data points, used two forms of modeling, and assessed the degree to which each model generalizes to data it has not used in the fitting process. Model comparisons were conducted using the Akaike information criterion (AIC), Bayesian information criterion (BIC), and leave-one-out cross validation (LOOCV). In general, AIC, BIC, and LOOCV selected the correct model, whereas the Rachlin model had the lowest error across folds of LOOCV when relying on multilevel modeling.