The combined effects of delay and probability in discounting (original) (raw)
Related papers
Journal of Experimental Psychology: Learning, Memory, and Cognition, 2015
The value of an outcome is affected both by the delay until its receipt (delay discounting) and by the likelihood of its receipt (probability discounting). Despite being well-described by the same hyperboloid function, delay and probability discounting involve fundamentally different processes, as revealed, for example, by the differential effects of reward amount. Previous research has focused on the discounting of delayed and probabilistic rewards separately, with little research examining more complex situations in which rewards are both delayed and probabilistic. In two experiments, participants made choices between smaller rewards that were both immediate and certain and larger rewards that were both delayed and probabilistic. Analyses revealed significant interactions between delay and probability factors inconsistent with an additive model. In contrast, a hyperboloid discounting model in which delay and probability were combined multiplicatively provided an excellent fit to the data. These results suggest that the hyperboloid is a good descriptor of decision making in complicated monetary choice situations like those people encounter in everyday life.
Within-Subject Comparison of Real and Hypothetical Money Rewards in Delay Discounting
Journal of the Experimental Analysis of Behavior, 2002
A within-subject design, using human participants, compared delay discounting functions for real and hypothetical money rewards. Both real and hypothetical rewards were studied across a range that included 10to10 to 10to250. For 5 of the 6 participants, no systematic difference in discount rate was observed in response to real and hypothetical choices, suggesting that hypothetical rewards may often serve as a valid proxy for real rewards in delay discounting research. By measuring discounting at an unprecedented range of real rewards, this study has also systematically replicated the robust finding in human delay discounting research that discount rates decrease with increasing magnitude of reward. A hyperbolic decay model described the data better than an exponential model.
A comparison of four models of delay discounting in humans
The present study compared four prominent models of delay discounting: a one-parameter exponential decay, a one-parameter hyperbola . An adjusting procedure for studying delayed reinforcement. In: Commons, M.L., Mazur, J.E., Nevin, J.A., Rachlin, H. (Eds.), Quantitative Analyses of Behavior: The Effect of Delay and of Intervening Events on Reinforcement Value, vol. 5. Erlbaum, Hillsdale, NJ, pp. 55-73], a two-parameter hyperboloid in which the denominator is raised to a power [Green, L., Myerson, J., 2004. A discounting framework for choice with delayed and probabilistic rewards. Psychol. Bull. 130, 769-792], and a two-parameter hyperbola in which delay is raised to a power [Rachlin, H., 2006. Notes on discounting. J. Exp. Anal. Behav. 85, 425-435].
Comparing hyperbolic, delay-amount sensitivity and present-bias models of delay discounting
Behavioural Processes, 2015
Delay discounting is a widely studied phenomenon due to its ubiquity in psychopathological disorders. Several methods are well established to quantify the extent to which a delayed commodity is devalued as a function of the delay to its receipt. The most frequently used method is to fit a hyperbolic function and use an index of the gradient of the function, k, or to calculate the area under the discounting curve. The manuscript examines the behavior of these quantification indices for three different datasets, as well as provides information about potential limitations in their use. The primary limitation examined is the lack of mechanistic specificity provided by either method. Alternative formulations that are thought to provide some mechanistic information are examined for the three separate datasets: two variants of a hyperboloid model (Rachlin 1989 Judgment, decision and choice. New York: W. H. Freeman) and the quasi-hyperbolic model (Laibson 1997 Q J Econ 112 443-477). Examination of the parameters of each formulation suggests that the parameters derived from the quasi-hyperbolic model allows groups and conditions within the three datasets to be reliably distinguished more readily than the hyperboloid models. However use of the quasi-hyperbolic model is complex and its limitations might offset its ability to discriminate within the datasets. MeSH attention deficit hyperactivity disorder; cigarette smoking; delay discounting; impulsive behavior; reward
Effects of Delay and Probability Combinations on Discounting in Humans
Behavioural Processes, 2016
To determine discount rates, researchers typically adjust the amount of an immediate or certain option relative to a delayed or uncertain option. Because this adjusting amount method can be relatively time consuming, researchers have developed more efficient procedures. One such procedure is a 5-trial adjusting delay procedure, which measures the delay at which an amount of money loses half of its value (e.g., 1000isvaluedat1000 is valued at 1000isvaluedat500 with a 10-year delay to its receipt). Experiment 1 (n = 212) used 5-trial adjusting delay or probability tasks to measure delay discounting of losses, probabilistic gains, and probabilistic losses. Experiment 2 (n = 98) assessed combined probabilistic and delayed alternatives. In both experiments, we compared results from 5-trial adjusting delay or probability tasks to traditional adjusting amount procedures. Results suggest both procedures produced similar rates of probability and delay discounting in six out of seven comparisons. A magnitude effect consistent with previous research was observed for probabilistic gains and losses, but not for delayed losses. Results also suggest that delay and probability interact to determine the value of money. Five-trial methods may allow researchers to assess discounting more efficiently as well as study more complex choice scenarios.
Behavioural Processes, 2013
Delay discounting (DD) and probability discounting (PD) refer to the reduction in the subjective value of outcomes as a function of delay and uncertainty, respectively. Elevated measures of discounting are associated with a variety of maladaptive behaviors, and confidence in the validity of these measures is imperative. The present research examined (1) the statistical equivalence of discounting measures when rewards were hypothetical or real, and (2) their 1-week reliability. While previous research has partially explored these issues using the low threshold of nonsignificant difference, the present study fully addressed this issue using the more-compelling threshold of statistical equivalence. DD and PD measures were collected from 28 healthy adults using real and hypothetical $50 rewards during each of two experimental sessions, one week apart. Analyses using area-under-the-curve measures revealed a general pattern of statistical equivalence, indicating equivalence of real/hypothetical conditions as well as 1-week reliability. Exceptions are identified and discussed.
Amount of reward has opposite effects on the discounting of delayed and probabilistic outcomes
Journal of Experimental Psychology: Learning, Memory, and Cognition, 1999
Previous research has shown that the value of large future rewards is discounted less steeply than is the value of small future rewards. These experiments extended this line of research to probabilistic rewards. Two experiments replicated the standard findings for delayed rewards but demonstrated that amount has an opposite effect on the discounting of probabilistic rewards. That is, large probabilistic amounts were discounted at the same or higher rates than small amounts. Although amount had opposite effects on the discounting of delayed and probabilistic rewards, nevertheless, the same form of mathematical function accurately described discounting of both types of reward. The findings suggest that fundamentally similar, but not identical, processes are involved in decision making regarding delayed and probabilistic rewards. The implications of these findings for impulsivity and self-control are discussed.
TOWARDS A GENERAL MODEL OF TEMPORAL DISCOUNTING
Psychological models of temporal discounting have now successfully displaced classical economic theory due to the simple fact that many common behavior patterns, such as impulsivity, were unexplainable with classic models. However, the now dominant hyperbolic model of discounting is itself becoming increasingly strained. Numerous factors have arisen that alter discount rates with no means to incorporate the different influences into standard hyperbolic models. Furthermore, disparate literatures are emerging that propose theoretical constructs that are seemingly independent of hyperbolic discounting. We argue that, although hyperbolic discounting provides an eminently useful quantitative measure of discounting, it fails as a descriptive psychological model of the cognitive processes that produce intertemporal preferences. Instead, we propose that recent contributions from cognitive neuroscience indicate a path for developing a general model of time discounting. New data suggest a means by which neuroscience-based theory may both integrate the diverse empirical data on time preferences and merge seemingly disparate theoretical models that impinge on time preferences. A tremendous variety of decisions faced by humans and animals require selecting between actions whose outcomes are realized at different times in the future. Moreover, it is quite commonly the case that more desirable outcomes can only be had at the expense of greater time or effort. As such, doing well in many behavioral contexts requires the ability to forego immediate temptations and to delay gratification. Given the ubiquity and importance of delaying gratification, there has been a longstanding interest in understanding and describing how humans and other animals respond to such decisions. The standard approach to investigating these phenomena is to present intertempo-ral choices-decisions, for example, between an immediate and a delayed reward-and to describe mathematically how choices are made. Both humans and other animals often prefer the immediate reward even when the delayed reward is larger. Mathematically, this can be summarized by asserting that the subjective value of a reward is discounted by an amount that depends on the delay until receipt (Ainslie, 1975; Rubinstein, 2003; Samuelson, 1937). From a research standpoint, the critical question is the nature of this delay discounting function. Fundamentally, this question can be satisfied by establishing a mathematical formulation of discounting that accounts for the diversity of behaviors involving choices over delays. Identifying such an equation seems likely to benefit from a close coupling between the mathematical function and the psychological and cognitive processes that underlie decision-making. Advances in understanding the delay discounting equation have been tremendous. In the next section we review these efforts that ultimately culminate in quasi-hyperbolic discounting models that capture behavior with impressive precision (published r 2 values commonly greater than 0.9). However, hyperbolic models are clearly limited in the range of discounting phenomena that they account for. In particular, the models capture behavior well in isolated contexts but are unequipped to account for how discounting varies across situations. These contextual effects can be profound. We believe this limitation arises from the fact that hyperbolic discounting models are distant from the basic cognitive processes that underlie decision-making. We review some highly cited models of discounting. However, our primary focus is on neuroscience, with the goal of proposing a brain-inspired model of discounting that (a) preserves the overall structure of hyperbolic discounting, but (b) is This work was supported by NIDA grant R03 032580 (SMM) and a Netherlands Organization for Scientific Research (NWO) Rubicon postdoctoral fellowship (WVDB).
Exponential Versus Hyperbolic Discounting of Delayed Outcomes: Risk and Waiting Time
Integrative and Comparative Biology, 1996
SYNOPSIS. Frequently, animals must choose between more immediate, smaller rewards and more delayed, but larger rewards. For example, they often must decide between accepting a smaller prey item versus continuing to search for a larger one, or between entering a leaner patch versus travelling to a richer patch that is further away. In both situations, choice of the more immediate, but smaller reward may be interpreted as implying that the value of the later reward is discounted; that is, the value of the later reward decreases as the delay to its receipt increases. This decrease in value may occur because of the increased risk involved in waiting for rewards, or because of the decreased rate of reward associated with increased waiting time. The present research attempts to determine the form of the relation between value and delay, and examines implications of this relation for mechanisms underlying risk-sensitive foraging. Two accounts of the relation between value and delay have been proposed to describe the decrease in value resulting from increases in delay: an exponential model and a hyperbolic model. Our research demonstrates that, of the two, a hyperbola-like discounting model consistently explains more of the variance in temporal discounting data at the group level and, importantly, at the individual level as well. We show mathematically that the hyperbolic model shares fundamental features with models of prey and patch choice. In addition, the present review highlights the implications of a psychological perspective for the behavioral biology of risksensitive foraging, as well as the implications of an ecological perspective for the behavioral psychology of risk-sensitive choice and decision-making.