Comparing discounting across different discount functions

library(tempodisco)

By default, tempodisco allows you to fit models using a variety of discount functions. However, this raises the question of how you can quantify and compare overall rates of discounting between two different discount functions. For example, it would be a mistake to directly compare $k$ values from the scaled exponential (Laibson, 1997) and nonlinear-time exponential (Ebert & Prelec, 2007) discount functions, since these mean different things. Instead, we need a model-agnostic measure of discounting. One of these is the ED50 (Yoon & Higgins, 2008), which is the delay at which the discount function returns a value of 0.5, i.e. the delay at which a delayed reward’s value is 50% of its face value. This can be computed using the ED50() function:

data("td_bc_single_ptpt")
mod1 <- td_bcnm(td_bc_single_ptpt, discount_function = 'scaled-exponential')
mod2 <- td_bcnm(td_bc_single_ptpt, discount_function = 'nonlinear-time-hyperbolic')

k1 <- coef(mod1)['k']
k2 <- coef(mod2)['k']
cat(sprintf('Percentage difference in k values: %.2f%%\n', 100*abs((k1 - k2)/((k1 + k2)/2))))
#> Percentage difference in k values: 142.68%

ed501 <- ED50(mod1)
ed502 <- ED50(mod2)
cat(sprintf('Percentage difference in ED50 values: %.2f%%\n', 100*abs(ed501 - ed502)/((ed501 + ed501)/2)))
#> Percentage difference in ED50 values: 14.50%

Another option is to use the model-based area under the curve (AUC) with the AUC function:

auc1 <- AUC(mod1)
auc2 <- AUC(mod2)
cat(sprintf('Percentage difference in AUC values: %.2f%%\n', 100*abs(auc1 - auc2)/((auc1 + auc2)/2)))
#> Percentage difference in AUC values: 18.36%

This latter method has the advantage of being well-defined for the “model-free” discount function, whereas the ED50 is not.