adopy.tasks.dd

../../_images/delay-discounting-task.png

Delay discounting refers to the well-established finding that humans tend to discount the value of a future reward such that the discount progressively increases as a function of the receipt delay (Green & Myerson, 2004; Vincent, 2016). In a typical delay discounting (DD) task, the participant is asked to indicate his/her preference between two delayed options: a smaller-sooner (SS) option (e.g., 8 dollars now) and a larger-longer (LL) option (e.g., 50 dollars in 1 month).

References

Green, L. and Myerson, J. (2004). A discounting framework for choice with delayed and probabilistic rewards. Psychological Bulletin, 130, 769–792.

Vincent, B. T. (2016). Hierarchical Bayesian estimation and hypothesis testing for delay discounting tasks. Behavior Research Methods, 48, 1608–1620.

Task

class adopy.tasks.dd.TaskDD

Bases: adopy.base._task.Task

The Task class for the delay discounting task.

Design variables
  • t_ss (\(t_{SS}\)) - delay of a SS option

  • t_ll (\(t_{LL}\)) - delay of a LL option

  • r_ss (\(R_{SS}\)) - amount of reward of a SS option

  • r_ll (\(R_{LL}\)) - amount of reward of a LL option

Responses

0 (choosing a SS option) or 1 (choosing a LL option)

Examples

>>> from adopy.tasks.ddt import TaskDD
>>> task = TaskDD()
>>> task.designs
['t_ss', 't_ll', 'r_ss', 'r_ll']
>>> task.responses
[0, 1]

Model

class adopy.tasks.dd.ModelExp

Bases: adopy.base._model.Model

The exponential model for the delay discounting task (Samuelson, 1937).

\[\begin{split}\begin{align} D(t) &= e^{-rt} \\ V_{LL} &= R_{LL} \cdot D(t_{LL}) \\ V_{SS} &= R_{SS} \cdot D(t_{SS}) \\ P(LL\, over \, SS) &= \frac{1}{1 + \exp [-\tau (V_{LL} - V_{SS})]} \end{align}\end{split}\]
Model parameters
  • r (\(r\)) - discounting parameter (\(r > 0\))

  • tau (\(\tau\)) - inverse temperature (\(\tau > 0\))

References

Samuelson, P. A. (1937). A note on measurement of utility. The review of economic studies, 4 (2), 155–161.

Examples

>>> from adopy.tasks.ddt import ModelExp
>>> model = ModelExp()
>>> model.task
Task('DDT', designs=['t_ss', 't_ll', 'r_ss', 'r_ll'], responses=[0, 1])
>>> model.params
['r', 'tau']
compute(t_ss, t_ll, r_ss, r_ll, r, tau)

Compute the probability of choosing a certain response given values of design variables and model parameters.

class adopy.tasks.dd.ModelHyp

Bases: adopy.base._model.Model

The hyperbolic model for the delay discounting task (Mazur, 1987).

\[\begin{split}\begin{align} D(t) &= \frac{1}{1 + kt} \\ V_{LL} &= R_{LL} \cdot D(t_{LL}) \\ V_{SS} &= R_{SS} \cdot D(t_{SS}) \\ P(LL\, over \, SS) &= \frac{1}{1 + \exp [-\tau (V_{LL} - V_{SS})]} \end{align}\end{split}\]
Model parameters
  • k (\(k\)) - discounting parameter (\(k > 0\))

  • tau (\(\tau\)) - inverse temperature (\(\tau > 0\))

References

Mazur, J. E. (1987). An adjusting procedure for studying delayed reinforcement. Commons, ML.;Mazur, JE.; Nevin, JA, 55–73.

Examples

>>> from adopy.tasks.ddt import ModelHyp
>>> model = ModelHyp()
>>> model.task
Task('DDT', designs=['t_ss', 't_ll', 'r_ss', 'r_ll'], responses=[0, 1])
>>> model.params
['k', 'tau']
compute(t_ss, t_ll, r_ss, r_ll, k, tau)

Compute the probability of choosing a certain response given values of design variables and model parameters.

class adopy.tasks.dd.ModelHPB

Bases: adopy.base._model.Model

The hyperboloid model for the delay discounting task (Green & Myerson, 2004).

\[\begin{split}\begin{align} D(t) &= \frac{1}{(1 + kt)^s} \\ V_{LL} &= R_{LL} \cdot D(t_{LL}) \\ V_{SS} &= R_{SS} \cdot D(t_{SS}) \\ P(LL\, over \, SS) &= \frac{1}{1 + \exp [-\tau (V_{LL} - V_{SS})]} \end{align}\end{split}\]
Model parameters
  • k (\(k\)) - discounting parameter (\(k > 0\))

  • s (\(s\)) - scale parameter (\(s > 0\))

  • tau (\(\tau\)) - inverse temperature (\(\tau > 0\))

References

Green, L. and Myerson, J. (2004). A discounting framework for choice with delayed and probabilistic rewards. Psychological Bulletin, 130, 769–792.

Examples

>>> from adopy.tasks.ddt import ModelHPB
>>> model = ModelHPB()
>>> model.task
Task('DDT', designs=['t_ss', 't_ll', 'r_ss', 'r_ll'], responses=[0, 1])
>>> model.params
['k', 's', 'tau']
compute(t_ss, t_ll, r_ss, r_ll, k, s, tau)

Compute the probability of choosing a certain response given values of design variables and model parameters.

class adopy.tasks.dd.ModelCOS

Bases: adopy.base._model.Model

The constant sensitivity model for the delay discounting task (Ebert & Prelec, 2007).

\[\begin{split}\begin{align} D(t) &= \exp[-(rt)^s] \\ V_{LL} &= R_{LL} \cdot D(t_{LL}) \\ V_{SS} &= R_{SS} \cdot D(t_{SS}) \\ P(LL\, over \, SS) &= \frac{1}{1 + \exp [-\tau (V_{LL} - V_{SS})]} \end{align}\end{split}\]
Model parameters
  • r (\(r\)) - discounting parameter (\(r > 0\))

  • s (\(s\)) - scale parameter (\(s > 0\))

  • tau (\(\tau\)) - inverse temperature (\(\tau > 0\))

References

Ebert, J. E. and Prelec, D. (2007). The fragility of time: Time-insensitivity and valuation of thenear and far future. Management science, 53 (9), 1423–1438.

Examples

>>> from adopy.tasks.ddt import ModelCOS
>>> model = ModelCOS()
>>> model.task
Task('DDT', designs=['t_ss', 't_ll', 'r_ss', 'r_ll'], responses=[0, 1])
>>> model.params
['r', 's', 'tau']
compute(t_ss, t_ll, r_ss, r_ll, r, s, tau)

Compute the probability of choosing a certain response given values of design variables and model parameters.

class adopy.tasks.dd.ModelQH

Bases: adopy.base._model.Model

The quasi-hyperbolic model (or Beta-Delta model) for the delay discounting task (Laibson, 1997).

\[\begin{split}\begin{align} D(t) &= \begin{cases} 1 & \text{if } t = 0 \\ \beta \delta ^ t & \text{if } t > 0 \end{cases} \\ V_{LL} &= R_{LL} \cdot D(t_{LL}) \\ V_{SS} &= R_{SS} \cdot D(t_{SS}) \\ P(LL\, over \, SS) &= \frac{1}{1 + \exp [-\tau (V_{LL} - V_{SS})]} \end{align}\end{split}\]
Model parameters
  • beta (\(\beta\)) - constant rate (\(0 < \beta < 1\))

  • delta (\(\delta\)) - constant rate (\(0 < \delta < 1\))

  • tau (\(\tau\)) - inverse temperature (\(\tau > 0\))

References

Laibson, D. (1997). Golden eggs and hyperbolic discounting. The Quarterly Journal of Economics, 443–477

Examples

>>> from adopy.tasks.ddt import ModelQH
>>> model = ModelQH()
>>> model.task
Task('DDT', designs=['t_ss', 't_ll', 'r_ss', 'r_ll'], responses=[0, 1])
>>> model.params
['beta', 'delta', 'tau']
compute(t_ss, t_ll, r_ss, r_ll, beta, delta, tau)

Compute the probability of choosing a certain response given values of design variables and model parameters.

class adopy.tasks.dd.ModelDE

Bases: adopy.base._model.Model

The double exponential model for the delay discounting task (McClure et al., 2007).

\[\begin{split}\begin{align} D(t) &= \omega e^{-rt} + (1 - \omega) e^{-st} \\ V_{LL} &= R_{LL} \cdot D(t_{LL}) \\ V_{SS} &= R_{SS} \cdot D(t_{SS}) \\ P(LL\, over \, SS) &= \frac{1}{1 + \exp [-\tau (V_{LL} - V_{SS})]} \end{align}\end{split}\]
Model parameters
  • omega (\(r\)) - weight parameter (\(0 < \omega < 1\))

  • r (\(r\)) - discounting rate (\(r > 0\))

  • s (\(s\)) - discounting rate (\(s > 0\))

  • tau (\(\tau\)) - inverse temperature (\(\tau > 0\))

References

McClure, S. M., Ericson, K. M., Laibson, D. I., Loewenstein, G., and Cohen, J. D. (2007). Time discounting for primary rewards. Journal of neuroscience, 27 (21), 5796–5804.

Examples

>>> from adopy.tasks.ddt import ModelDE
>>> model = ModelDE()
>>> model.task
Task('DDT', designs=['t_ss', 't_ll', 'r_ss', 'r_ll'], responses=[0, 1])
>>> model.params
['omega', 'r', 's', 'tau']
compute(t_ss, t_ll, r_ss, r_ll, omega, r, s, tau)

Compute the probability of choosing a certain response given values of design variables and model parameters.

Engine

class adopy.tasks.dd.EngineDD(model, grid_design, grid_param)

Bases: adopy.base._engine.Engine

The Engine class for the delay discounting task. It can be only used for TaskDD.