A probabilistic, dynamic, and attribute-wise model of intertemporal choice

Abstract

Most theoretical and empirical research on intertemporal choice assumes a deterministic and static perspective, leading to the widely adopted delay discounting models. As a form of preferential choice, however, intertemporal choice may be generated by a stochastic process that requires some deliberation time to reach a decision. We conducted 3 experiments to investigate how choice and decision time varied as a function of manipulations designed to examine the delay duration effect, the common difference effect, and the magnitude effect in intertemporal choice. The results, especially those associated with the delay duration effect, challenged the traditional deterministic and static view and called for alternative approaches. Consequently, various static or dynamic stochastic choice models were explored and fit to the choice data, including alternative-wise models derived from the traditional exponential or hyperbolic discount function and attribute-wise models built upon comparisons of direct or relative differences in money and delay. Furthermore, for the first time, dynamic diffusion models, such as those based on decision field theory, were also fit to the choice and response time data simultaneously. The results revealed that the attribute-wise diffusion model with direct differences, power transformations of objective value and time, and varied diffusion parameter performed the best and could account for all 3 intertemporal effects. In addition, the empirical relationship between choice proportions and response times was consistent with the prediction of diffusion models and thus favored a stochastic choice process for intertemporal choice that requires some deliberation time to make a decision.

Figure 1

Screenshots of the experimental software. The top panel shows the display in Experiments 1 and 2 and the bottom panel shows the display in Experiment 3. The SS options were always shown on the left and the LL options were always on the right in the first two experiments, while in Experiment 3 the positions of options were randomized across trials.

Figure 2

Observed choice proportions and predicted average choice probabilities of the LL options for various intertemporal effects in the first session of Experiment 1. The left column shows the results of two typical participants (top two panels) and the average results across participants (bottom panel) in Session 1; the right column shows the corresponding predictions of the best model in Experiment 1.

Figure 3

Average mean response times for question groups with different actual choice proportions of the LL options in Experiment 1. Questions within each group had the same values on the target attribute of a specific intertemporal effect. Error bars show 95% confidence intervals.

Figure 4

Predictions of the best model in Experiment 1. The left panel shows a scatterplot of the actual choice proportions and the predicted average choice probabilities of LL options within question groups; the right panel shows a scatterplot of the actual mean response times and predicted mean response times within question groups. Each point in the scatterplots is associated with a group of questions with the same values on the target attribute for a specific intertemporal effect answered by an individual participant in a single session.

Figure 5

Observed choice proportions and predicted average choice probabilities of the LL options for various intertemporal effects in Experiment 2. The left column shows the results of two typical participants (top two panels) and the average results across participants (bottom panel); the right column shows the corresponding predictions of the best model in Experiment 2.

Figure 6

Average mean response times for question groups with different actual choice proportions of the LL options in Experiment 2. Questions within each group had the same values on the target attribute of a specific intertemporal effect. Error bars show 95% confidence intervals.

Figure 7

Observed choice proportions and predicted average choice probabilities of the LL options for various intertemporal effects in Experiment 3. The left column shows the results of two typical participants (top two panels) and the average results across participants (bottom panel); the right column shows the corresponding predictions of the best model in Experiment 3.

Figure 8

Average mean response times for question groups with different actual choice proportions of the LL options in Experiment 3. Each question group contained identical questions for a specific intertemporal effect. Error bars show 95% confidence intervals.