On the Functional Form of Temporal Discounting: An Optimized Adaptive Test

Abstract

The tendency to discount the value of future rewards has become one of the best-studied constructs in the behavioral sciences. Although hyperbolic discounting remains the dominant quantitative characterization of this phenomenon, a variety of models have been proposed and consensus around the one that most accurately describes behavior has been elusive. To help bring some clarity to this issue, we propose an Adaptive Design Optimization (ADO) method for fitting and comparing models of temporal discounting. We then conduct an ADO experiment aimed at discriminating among six popular models of temporal discounting. Rather than supporting a single underlying model, our results show that each model is inadequate in some way to describe the full range of behavior exhibited across subjects. The precision of results provided by ADO further identify specific properties of models, such as accommodating both increasing and decreasing impatience, that are mandatory to describe temporal discounting broadly.

Keywords: adaptive designs; design optimization; intertemporal choice; model selection; temporal discounting.

Fig. 1

Schematic illustration of the cyclical relationship between Design Optmization, data collection, and Bayesian updating in each mini-experiment. Reprinted from Myung et al. (ress, Figure 10).

Fig. 2

Depiction of the prior predictive distribution of discounting curves in each family. In each graph, the x-axis measures time and the y-axis measures the discount factor. Curves were generated by randomly selecting 100 parameter vectors from the priors for each model.

Fig. 3

MLE of the model with the lowest AIC for each participant

Fig. 4

Comparison of median estimates of each model in the entire sample versus the subgroups in which each model was best.

Fig. 5

The four overlaid distributions in each of panels r, s and ε are the marginal posterior distributions obtained in each model recovery simulation. The vertical line in each graph shows the true value of the parameter, so a tighter distribution around that line indicates a better estimate. In the bottom-right panel, the five overlaid discounting curves are the generating curve (red) and the best estimate of that curve based on each of the four simulations. Estimates are based on the CS model using the maximum a-posteriori probability (MAP) estimates of r and s. It may be difficult to differentiate all of the curves visually because the estimate obtained in the ADO simulation (solid black line) sits right on top of the generating curve (red line). That is to say, the discounting curve was recovered almost perfectly in the ADO simulation. The estimates obtained in the Geometric and Random simulations are also both quite close to the true curve, despite the fact that the parameter estimates are not very precise.