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Updated Apr 12, 2012 by vincent....@gmail.com

Application to inter-temporal choices

We consider a binary inter-temporal choice task in which a subject has to choose between

  • an impulsive choice X€ now
  • a delayed choice of 50€ whose delay is fixed in advance.

We suppose the subject performs choices according to the hyperbolic discount model of section 4. We want to infer the inverse temperature and the delay discounting parameter and start with relatively flat priors on those parameters.

To fully define the experiment, value of the impulsive choice is left to define. I will here apply the optimization method described in the previous section in order to optimally choose the values of the impulsive choice and compare it to a random design.

Simulating behavior

We use a model of a subject to perform the task. The model we choose is the actual model that we assume the subject to have (hyperbolic discount + softmax). We set its parameters to :

  • λ=0.01
  • β=3

Design comparison

We consider 40 consecutive trials. Delays for the delayed option are chosen in [0,52]. The delayed choice value is fixed to 50€

The aim of this section is to compare the following designs :

  • Impulsive values are sampled independently from a uniform distribution on [1€; 50€].
  • Impulsive values are chosen using the optimal design every two trials. (the others being sampled independently from u[1€; 50€] )

I run 30 simulations for each of the two designs and plot

  • the value of the criteria (mean and standard deviation) as a function of the number of trials.

  • The final expected parameters for each simulation

From the first plot, it can be seen that after around 10 trials, the optimization methods leads to much more confidence in the estimates. The second plot suggests that the estimation is unbiased in both designs, with the exception of a few outliers (4 for the optimized design, 2 for the random design)

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