Available discount functions

library(tempodisco)

tempodisco implements many lesser-known discount functions beyond the smaller set of popular functions used by Franck et al., 2015. The full list is as follows:

Name	Functional form	Notes
exponential (Samuelson, 1937)	$f(t; k) = e^{-k t}$
hyperbolic (Mazur, 1987)	$f(t; k) = \frac{1}{1 + kt}$
scaled-exponential (Laibson, 1997)	$f(t; k, w) = w e^{-k t}$	Also known as quasi-hyperbolic or beta-delta and written as $f(t; \beta, \delta) = \beta e^{-\delta t}$
nonlinear-time-exponential (Ebert & Prelec, 2007)	$f(t; k, s) = e^{-k t^s}$	Also known as constant sensitivity
inverse-q-exponential (Green & Myerson, 2004)	$f(t; k, s) = \frac{1}{(1 + k t)^s}$	Also known as generalized hyperbolic (Loewenstin & Prelec), hyperboloid (Green & Myerson, 2004), or q-exponential (Han & Takahashi, 2012)
nonlinear-time-hyperbolic (Rachlin, 2006)	$f(t; k, s) = \frac{1}{1 + k t^s}$	Also known as power-function (Rachlin, 2006)
dual-systems-exponential (Ven den Bos & McClure, 2013)	$f(t; k_1, k_2, w) = w e^{-k_1 t} + (1 - w) e^{-k_2 t}$
additive-utility (Killeen, 2009)	$f(t; k, s, a) = \left( 1 - \frac{k}{V_D^a}t^s\right)^\frac{1}{a}$	$V_D$ is the value of the delayed reward. $f(t; k, s, a) = 0$ for $t > \left(V_D^a / k\right)^{1/s}$.
power (Harvey, 1986, eq. 2)	$f(t; k) = \frac{1}{(1 + t)^k}$	In equation 2 of the reference, the discount function is described as $\frac{1}{t^k}$, but time begins at $t = 1$.
arithmetic (Doyle & Chen, 2010)	$f(t; k) = 1 - \frac{kt}{V_D}$	$V_D$ is the value of the delayed reward. $f(t; k) = 0$ for $kt > V_D$.
fixed-cost (Benhabib, Bisin, & Schotter, 2010)	$f(t; w) = e^{-kt} - \frac{w}{V_D}$	$V_D$ is the value of the delayed reward. $f(t; w) = 0$ for $\frac{w}{V_D} > e^{-kt}$.
absolute-stationarity (Blavatskyy, 2024, eq. 3)	$f(t; k, s) = \exp\left\{ -k\frac{ts}{ts + 1}\right\}$	The original paper uses $t$ rather than $ts$. However, a scale factor appears necessary to account for different time units.
relative-stationarity (Blavatskyy, 2024, eq. 7)	$f(t; k, s) = \left( \frac{ts + 1}{2ts + 1} \right)^k$	The original paper uses $t$ rather than $ts$. However, a scale factor appears necessary to account for different time units.
constant (Franck et al., 2015)	$f(t; k) = k$	Null model; participants can be excluded if this model provides the best fit (Franck et al., 2015)
nonlinear-time-power	$f(t; k) = \frac{1}{(1 + t^s)^k}$	Experimental extension of the power discount function along the lines of the nonlinear-time-hyperbolic and nonlinear-time-exponential functions.
nonlinear-time-arithmetic	$f(t; k) = 1 - \frac{kt^s}{V_D}$	Experimental extension of the arithmetic discount function along the lines of the nonlinear-time-hyperbolic and nonlinear-time-exponential functions.
scaled-hyperbolic	$f(t; k, w) = \frac{w}{1 + kt}$	Experimental extension of the hyperbolic discount function along the lines of the scaled-exponential function.

The names of these discount functions can be accessed using get_available_discount_functions():

print(get_available_discount_functions())
#>  [1] "hyperbolic"                 "nonlinear-time-hyperbolic" 
#>  [3] "exponential"                "nonlinear-time-exponential"
#>  [5] "absolute-stationarity"      "relative-stationarity"     
#>  [7] "power"                      "nonlinear-time-power"      
#>  [9] "arithmetic"                 "nonlinear-time-arithmetic" 
#> [11] "inverse-q-exponential"      "scaled-exponential"        
#> [13] "scaled-hyperbolic"          "fixed-cost"                
#> [15] "dual-systems-exponential"   "additive-utility"          
#> [17] "model-free"                 "constant"