Scoring discount rates

We assume the commonly used hyperbolic discount function (Mazur, 1987), which models present subjective value V as a function of a reward magnitude R at a delay D, V = R ⋅ 1/(1 + k ⋅ D). Here, k is the discount rate, which has units of days^− 1. Because k is known to be very positively skewed, we expressed our prior beliefs as normally distributed over log(k).

We estimated the full posterior distribution over log discount rates given the data, P(log(k)|data), where the data consisted of the raw trial data from the delay discounting experiment. This was done for each participant × condition combination separately and independently. Data columns were: A and B the reward values for the immediate and delayed choices, respectively, a delay for the immediate choice D^A = 0 and a delay D^B for the delayed choice and R for the participant’s response. The following probabilistic model was used:

where t is a trial, corresponding to a row in the raw data table, and Φ(⋅) is the cumulative normal distribution. In practice, P(log(k)|data) was computed using Markov chain Monte Carlo methods described by Vincent (2016).

Because our Bayesian parameter estimation procedure can produce model predictions for the probability of choosing the delayed reward on each trial (P_t), we are able to assess the ability of the discount function to account for behavioral data using signal detection theory (Wickens, 2002). The area under the receiver operating characteristic curve gives the model’s ability to correctly predict the inter-temporal choice data—a value of 1 means perfect prediction, 0.5 means chance level, and between 0.5 and 0 means below chance level.