When choosing between delayed or uncertain outcomes, individuals discount the value of such outcomes on the basis of the expected time to or the likelihood of their occurrence. In an integrative review of the expanding experimental literature on discounting, the authors show that although the same form of hyperbola-like function describes discounting of both delayed and probabilistic outcomes, a variety of recent findings are inconsistent with a single-process account. The authors also review studies that compare discounting in different populations and discuss the theoretical and practical implications of the findings. The present effort illustrates the value of studying choice involving both delayed and probabilistic outcomes within a general discounting framework that uses similar experimental procedures and a common analytical approach.Choice is relatively predictable when the alternatives differ on only one dimension. For example, if individuals are offered a choice between two rewards that differ only in amount, they generally choose the larger rather than the smaller reward. Similarly, if offered a choice between two rewards that differ only in delay, individuals tend to choose the reward available sooner rather than the one available later, and if offered a choice between two alternatives that differ only in probability, they tend to choose the more certain reward. These same general principles, and a complementary set of principles for negative outcomes (e.g., smaller punishments will be chosen over larger ones), apply both to humans and other animals. These behavioral tendencies obviously make both economic and evolutionary sense.Problems arise, however, when choice options differ on more than one dimension-for example, when an individual must choose between a smaller reward available sooner and a larger reward available later. It is also unclear, in general, which alternative an individual would (or should) select when choosing between a smaller, more certain reward and a larger, less certain one or between a less certain reward available sooner and a more certain, but more delayed, reward. These problems are only compounded when rewards differ on all three dimensions (i.e., amount, delay, and probability). The central issue is how individuals make trade-offs among their preferences on these dimensions (Keeney & Raiffa, 1993).This issue is not only of theoretical interest; it also has implications for everyday decision making, which often involves outcomes that differ on multiple dimensions. Examples of such decisions include deciding whether to purchase a less expensive item that can be enjoyed now or to save for a more expensive one; whether to choose a risky investment that potentially could pay off at a high rate or one that pays a low but guaranteed rate of return; and whether to buy

We describe a novel approach to the measurement of discounting based on calculating the area under the empirical discounting function. This approach avoids some of the problems associated with measures based on estimates of the parameters of theoretical discounting functions. The area measure may be easily calculated for both individual and group data collected using any of a variety of current delay and probability discounting procedures. The present approach is not intended as a substitute for theoretical discounting models. It is useful, however, to have a simple, univariate measure of discounting that is not tied to any specific theoretical framework.

In this study, children, young adults, and older adults chose between immediate and delayed hypothetical monetary rewards The amount of the delayed reward was held constant while its delay was varied All three age groups showed delay discounting, that is, the amount of an immediate reward judged to be of equal value to the delayed reward decreased as a function of delay The rate of discounting was highest for children and lowest for older adults, predicting a life-span developmental trend toward increased self-control Discounting of delayed rewards by all three age groups was well described by a single function with age-sensitive parameters (all R2s > 94) Thus, even though there are quantitative age differences in delay discounting, the existence of an age-invariant form of discount function suggests that the process of choosing between rewards of different amounts and delays is qualitatively similar across the life span

The present paper addresses the question of the form of the mathematical relation between the time until a delayed reward and its present value. Data are presented from an experiment in which subjects chose between hypothetical amounts of money available either immediately or after a delay (Green, Fry, & Myerson, 1994). Analyses of the behavior of individual young adults demonstrated that temporal discounting is better described by hyperbola-like functions than by exponential decay functions. For most individuals, the parameter that determines the rate of discounting varied inversely with amount. Raising the denominator of the discounting function to a power resulted in better descriptions of the data from most subjects. Two possible derivations of the temporal discounting function are proposed, a repeated choice model and an expected value model. These provide theoretical interpretations for amount-dependent discounting but amount-independent exponent parameters.Key words: temporal discounting, delayed rewards, behavioral economics, discounting function, choice, humans Humans and other animals typically will choose more immediate rewards over delayed rewards of equal magnitude. What is surprising, at least from certain perspectives, is that often they also will choose more immediate rewards over delayed rewards of larger magnitude. Many different accounts of the latter finding have been offered (e.g., lack of impulse control, poor ego strength, conflict between multiple selves), but the dominant account in the behavioral economic literature relies on the assumption that the value of a future reward decreases with increasing length of time to its receipt (Kagel, Battalio, & Green, 1995). This decrease in value as a function of delay is termed temporal discounting. The temporal discounting account posits that a smaller, more immediate reward may be chosen because the present (or subjective) value of the larger, more delayed reward is discounted; hence, its present value may be less than that of the more immediate reward.From the perspective of temporal discounting, the question of interest becomes the nature of the mathematical relation among amount, delay, and value. Economists and psychologists have typically employed two different approaches to determining this func-

The present, subjective value of a delayed reward is a decreasing function of the duration of the delay. This phenomenon is termed temporal discounting. To determine whether the amount of the reward influences the rate of temporal discounting, we had subjects choose between immediate and delayed hypothetical rewards of different amounts ($100, $2,000, $25,000, and $100,000 delayed rewards). As predicted by psychological models of the choice process, hyperbolic functions described the decrease in the subjective value of the delayed reward as the time until its receipt was increased (R2 S from .86 to .99). Moreover, hyperbolic functions consistently provided more accurate descriptions of the data than did exponential functions predicted by an economic model of discounted utility. Rate of discounting decreased in a negatively accelerated fashion as the amount of the delayed reward increased, leveling off by approximately $25,000. These findings are interpreted in the context of different psychological models of choice, and implications for procedures to enhance selfcontrol are discussed.