Delay, Probability, and Social Discounting in A Public Goods Game

A human social discount function measures the value to a person of a reward to another person at a given social distance. Just as delay discounting is a hyperbolic function of delay, and probability discounting is a hyperbolic function of odds-against, social discounting is a hyperbolic function of social distance. Experiment 1 obtained individual social, delay, and probability discount functions for a hypothetical $75 reward; participants also indicated how much of an initial $100 endowment they would contribute to a common investment in a public good. Steepness of discounting correlated, across participants, among all three discount dimensions. However, only social and probability discounting were correlated with the public-good contribution; high public-good contributors were more altruistic and also less risk averse than low contributors. Experiment 2 obtained social discount functions with hypothetical $75 rewards and delay discount functions with hypothetical $1,000 rewards, as well as public-good contributions. The results replicated those of Experiment 1; steepness of the two forms of discounting correlated with each other across participants but only social discounting correlated with the public-good contribution. Most participants in Experiment 2 predicted that the average contribution would be lower than their own contribution.     

Within-subject comparison of real and hypothetical money rewards in delay discounting

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 $10 to $250. 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.     

The absence of numbers to express the amount may affect delay discounting with humans

Human delay discounting is usually studied with experimental protocols that use symbols to express delay and amount. In order to further understand discounting, we evaluated whether the absence of numbers to represent reward amounts affects discount rate in general, and whether the magnitude effect is generalized to nonsymbolic situations in particular. In Experiment 1, human participants were exposed to a delay‐discounting task in which rewards were presented using dots to represent monetary rewards (nonsymbolic); under this condition the magnitude effect did not occur. Nevertheless, the magnitude effect was observed when equivalent reward amounts were presented using numbers (symbolic). Moreover, in estimation tasks, magnitude increments produced underestimation of large amounts. In Experiment 2, participants were exposed only to the nonsymbolic discounting task and were required to estimate reward amounts in each trial. Consistent with Experiment 1, the absence of numbers representing reward amounts produced similar discount rates of small and large rewards. These results suggest that value of nonsymbolic rewards is a nonlinear function of amount and that value attribution depends on perceived difference between the immediate and the delayed nonsymbolic rewards.     

Notes on discounting

In general, if a variable can be expressed as a function of its own maximum value, that function may be called a discount function. Delay discounting and probability discounting are commonly studied in psychology, but memory, matching, and economic utility also may be viewed as discounting processes. When they are so viewed, the discount function obtained is hyperbolic in form. In some cases the effective discounting variable is proportional to the physical variable on which it is based. For example, in delay discounting, the physical variable, delay (D), may enter into the hyperbolic equation as kD. In many cases, however, the discounting data are not well described with a single-parameter discount function. A much better fit is obtained when the effective variable is a power function of the physical variable (kDS in the case of delay discounting). This power-function form fits the data of delay, probability, and memory discounting as well as other two-parameter discount functions and is consistent with both the generalized matching law and maximization of a constant-elasticity-of-substitution utility     

Social discounting of pain

Impatience can be formalized as a delay discount rate, describing how the subjective value of reward decreases as it is delayed. By analogy, selfishness can be formalized as a social discount rate, representing how the subjective value of rewarding another person decreases with increasing social distance. Delay and social discount rates for reward are correlated across individuals. However no previous work has examined whether this relationship also holds for aversive outcomes. Neither has previous work described a functional form for social discounting of pain in humans. This is a pertinent question, since preferences over aversive outcomes formally diverge from those for reward. We addressed this issue in an experiment in which healthy adult participants (N = 67) chose the timing and intensity of hypothetical pain for themselves and others. In keeping with previous studies, participants showed a strong preference for immediate over delayed pain. Participants showed greater concern for pain in close others than for their own pain, though this hyperaltruism was steeply discounted with increasing social distance. Impatience for pain and social discounting of pain were weakly correlated across individuals. Our results extend a link between impatience and selfishness to the aversive domain.     

Modeling the effect of reward amount on probability discounting

The present study with college students examined the effect of amount on the discounting of probabilistic monetary rewards. A hyperboloid function accurately described the discounting of hypothetical rewards ranging in amount from $20 to $10,000,000. The degree of discounting increased continuously with amount of probabilistic reward. This effect of amount was not due to changes in the rate parameter of the discounting function, but rather was due to increases in the exponent. These results stand in contrast to those observed with the discounting of delayed monetary rewards, in which the degree of discounting decreases with reward amount due to amount-dependent decreases in the rate parameter. Taken together, this pattern of results suggests that delay and probability discounting reflect different underlying mechanisms. That is, the fact that the exponent in the delay discounting function is independent of amount is consistent with a psychophysical scaling interpretation, whereas the finding that the exponent of the probability-discounting function is amount-dependent is inconsistent with such an interpretation. Instead, the present results are consistent with the idea that the probability-discounting function is itself the product of a value function and a weighting function. This idea was first suggested by Kahneman and Tversky (1979), although their prospect theory does not predict amount effects like those observed. The effect of amount on probability discounting was parsimoniously incorporated into our hyperboloid discounting function by assuming that the exponent was proportional to the amount raised to a power. The amount-dependent exponent of the probability-discounting function may be viewed as reflecting the effect of amount on the weighting of the probability with which the reward will be received.     

Age‐related differences in delay discounting: Immediate reward,reward magnitude,and social influence

Delay discounting refers to the tendency of individuals to subjectively devalue rewards that are to be received in the future, with high rates of delay discounting being associated with a variety of maladaptive life outcomes (e.g., unhealthy dietary and exercise behaviors). The current study explored the psychological and social processes involved in adult age‐related differences in delay discounting of monetary rewards. Younger adults exhibited higher levels of delay discounting than older adults. This increased level of patience in older adults was found whether smaller‐sooner rewards were to be received immediately or in the future. However, there was an interaction with reward magnitude, whereby younger adults exhibited higher levels of delay discounting for smaller reward magnitudes but not larger reward magnitudes. Social influence on delay discounting was investigated by having participants complete three phases of the delay‐discounting task: an individual precollaboration phase, a collaboration phase in age‐group‐matched dyads, and an individual postcollaboration phase. A convergence effect was observed in that dyad members' postcollaboration choices were significantly more similar compared to their baseline choices during the precollaboration phase. Moreover, levels of convergence were comparable between younger and older adults, suggesting age invariance in social influence on delay discounting. The current results demonstrate a degree of malleability in delay discounting that extends into older adulthood, making interventions targeting the construct a promising avenue for future research.     

Delay discounting of qualitatively different reinforcers in rats

Humans discount larger delayed rewards less steeply than smaller rewards, whereas no such magnitude effect has been observed in rats (and pigeons). It remains possible that rats' discounting is sensitive to differences in the quality of the delayed reinforcer even though it is not sensitive to amount. To evaluate this possibility, Experiment 1 examined discounting of qualitatively different food reinforcers: highly preferred versus nonpreferred food pellets. Similarly, Experiment 2 examined discounting of highly preferred versus nonpreferred liquid reinforcers. In both experiments, an adjusting-amount procedure was used to determine the amount of immediate reinforcer that was judged to be of equal subjective value to the delayed reinforcer. The amount and quality of the delayed reinforcer were varied across conditions. Discounting was well described by a hyperbolic function, but no systematic effects of the quantity or the quality of the delayed reinforcer were observed.     

Social discounting

The amount of money a person was willing to forgo in order to give 75 dollars to another person decreased as a hyperbolic function of the perceived social distance between them. Similar hyperbolic functions have previously been shown to describe both time and probability discounting.     

Now for Me,Later for Us? Effects of Group Context on Temporal Discounting

Delayed rewards are less valuable than immediate rewards. This well‐established finding has focused almost entirely on individual outcomes. However, are delayed rewards similarly discounted if they are shared by a group? The current article reports on three experiments exploring the effect of group context on delay discounting. Results indicate that discount rates of individual and group rewards were highly correlated, but that respondents were more willing to wait (decreased discounting) for shared outcomes than for individual outcomes. An explanatory model is proposed suggesting that decreased discount rates in group contexts may be due to the way the effects of both delay and social discounting are combined. That is, in a group context, a person values both a future reward (discounted by delay) and a present reward to another person (discounted by the social distance between them). The results are explained by a combined discount function containing a delay factor and a factor representing the social distance between the decision maker and group members. Practical implications of the fact that shared consequences can increase individual self‐control are also discussed. Copyright © 2011 John Wiley & Sons, Ltd.     

Delay discounting as impaired valuation: Delayed rewards in an animal obesity model

Obesity is a major public health problem, which, like many forms of addiction, is associated with an elevated tendency to choose smaller immediate rather than larger delayed rewards, a response pattern often referred to as excessive delay discounting. Although some accounts of delay discounting conceptualize this process as impulsivity (placing the emphasis on overvaluing the smaller immediate reward), others have conceptualized delay discounting as an executive function (placing the emphasis on delayed rewards failing to retain their value). The present experiments used a popular animal model of obesity that has been shown to discount delayed rewards at elevated rates (i.e., obese Zucker rats) to test two predictions that conceptualize delay discounting as executive function. In the first experiment, acquisition of lever pressing with delayed rewards was compared in obese versus lean Zucker rats. Contrary to predictions based on delay discounting as executive function, obese Zucker rats learned to press the lever more quickly than controls. In the second experiment, progressive ratio breakpoints (a measure of reward efficacy) with delayed rewards were compared in obese versus lean Zucker rats. Contrary to the notion that obese rats fail to value delayed rewards, the obese Zucker rats' breakpoints were (at least) as high as those of the lean Zucker rats.     

Physical and cognitive effort discounting of hypothetical monetary rewards

Effort discounting refers to the decrease in the subjective value of a reward as the effort required to obtain the reward increases. The main aims of this study were to ascertain whether the amount of the reward affects the steepness of the effort discounting process for hypothetical monetary rewards, to identify whether this steepness depends on the type of effort that is required, and to determine whether the steepness of different types of effort covary at the individual level. Two types of effort were studied under hypothetical choice situations: physical effort and cognitive effort. Both physical and cognitive effort discounting were well described by the hyperbolic model. Large rewards were discounted less steeply than small rewards for both types of effort. This finding agrees with the results of prior studies which have found that larger rewards have greater motivational power. In addition, the steepness of physical effort discounting was positively correlated with the steepness of cognitive effort discounting, which suggests that the effort discounting process is a trait‐like characteristic within an individual.     

Bidding for delayed rewards: Accumulation as delay discounting,delay discounting as regulation,demand functions as corollary

In the accumulation paradigm animals press one manipulandum to accumulate pellets or seconds of access to food, and then press another manipulandum, or run some distance, to collect it. The accumulation may be interpreted as delay discounting, with the animals adjusting the distal amount to compensate for its distance or delay. The amount accumulated before being collected is a linear function of the distance or time that the experimental paradigm stipulates for collecting it. That linear function follows from the Unit‐Price/Unit‐Amount axiom. The inverse of the linear compensation functions gives a delay‐discount hyperbola. The advantages of the accumulation paradigm and analytic framework for delay discounting studies are noted. Compensation functions are then derived from a behavioral regulation model, which generalizes them to contexts where the individual's budget for response cost becomes over‐taxed. In turn, such compensation/regulation models lead directly to representative demand functions. In sum, regulation models provide a theoretical grounding for demand functions, compensation functions, and delay discounting hyperbolas. The parameter that links them is the unit amount k, the slope of the compensation function and of the discount function, the setpoint for consumption in a regulation model, and the ideal quantity consumed at minimal price in a demand analysis.     

Evidence for a magnitude effect in temporal discounting with pigeons

A magnitude effect in human intertemporal choice is well established-larger rewards or outcomes are discounted over time at a lower rate than are smaller rewards. However, many recent studies have failed to find a corresponding effect in nonhuman animals. Here we report a magnitude effect in temporal discounting for pigeons' choices involving a tradeoff between reward delay and amount. Pigeons chose between a small reward (1-s access to food) after a 2-s delay, and a large reward (4.5-s access to food) after a 28-s delay. Across conditions, the delays to the small and large rewards were increased or decreased, respectively. Temporal discounting functions obtained through a value-estimation procedure showed clear evidence of a magnitude effect: The value of the large reward decreased more slowly with increasing delay than the value of the small reward. We linked this result to a nonlinear relationship between choice and the delays associated with the small and large rewards. The nonlinearity was contrary to the generalized matching law but was predicted by the contextual choice model. Our results confirm the existence of a magnitude effect in nonhuman temporal discounting, showing that this adaptation is not unique to humans.     

Delay discounting: I'm a k, you're a k

Delay discounting is the decline in the present value of a reward with delay to its receipt. Across a variety of species, populations, and reward types, value declines hyperbolically with delay. Value declines steeply with shorter delays, but more shallowly with longer delays. Quantitative modeling provides precise measures to characterize the form of the discount function. These measures may be regarded as higher-order dependent variables, intervening variables, or hypothetical constructs. I suggest the degree of delay discounting may be a personality trait. In the end, the ontological status of measures of delay discounting is irrelevant. Whatever delay discounting may be, its study has provided the field of behavior analysis and other areas measures with robust generality and predictive validity for a variety of significant human problems. Research on moderating the degree of delay discounting has the potential to produce substantial societal benefits.     

Reformative self‐control and discounting of reward value by delay or effort1

Abstract: The relation between reformative self‐control and discounting of reward value by delay or effort was examined. Thirty‐two Japanese undergraduates were selected based on their scores on a standardized pencil and paper test of self‐control, the Redressive–Reformative Self‐Control Scale ( Sugiwaka, 1995 ). The subjects indicated their preference (i) between a hypothetical 100 000 yen reward available with various delays and a certain reward of variable amount available immediately; or (ii) between a hypothetical 100 000 yen reward available with various efforts and a certain reward of variable amount available without effort. The discounting of delayed reward value was described well by a hyperbolic function proposed by Mazur (1987 ). The slopes of the delayed discounting curves were steeper for subjects who scored low on the Reformative Self‐Control Scale, which is a measure of reformative self‐control and a subscale of the Redressive–Reformative Self‐Control Scale. The discounting by effort was also described well by the hyperbolic function. The slopes of the effortful discounting curves did not correlate to those of the delayed discounting curves. The slopes were not systematically different across the groups of subjects with different scores on the reformative self‐control scale.     

Delay or probability discounting in a model of impulsive behavior: effect of alcohol

Little is known about the acute effects of drugs of abuse on impulsivity and self-control. In this study, impulsivity was assessed in humans using a computer task that measured delay and probability discounting. Discounting describes how much the value of a reward (or punisher) is decreased when its occurrence is either delayed or uncertain. Twenty-four healthy adult volunteers ingested a moderate dose of ethanol (0.5 or 0.8 g/kg ethanol: n = 12 at each dose) or placebo before completing the discounting task. In the task the participants were given a series of choices between a small, immediate, certain amount of money and $10 that was either delayed (0, 2, 30, 180, or 365 days) or probabilistic (i.e., certainty of receipt was 1.0, .9, .75, .5, or .25). The point at which each individual was indifferent between the smaller immediate or certain reward and the $10 delayed or probabilistic reward was identified using an adjusting-amount procedure. The results indicated that (a) delay and probability discounting were well described by a hyperbolic function; (b) delay and probability discounting were positively correlated within subjects; (c) delay and probability discounting were moderately correlated with personality measures of impulsivity; and (d) alcohol had no effect on discounting.     

Magnitude effect contributes to the domain specificity in delay discounting

Discounting is a useful framework for understanding temporal choices. A person who prefers $50 immediately over $100 in 1 month exhibits a higher discount rate than a person deciding to wait for the higher payoff. Although previous research shows that discount rates are domain-specific, we propose an alternative to the domain specificity account. We suggest that differences in discounting alternatives across various domains may result not so much from the domains' nature per se but from differences in perceived attractiveness of the discounted alternatives. We replicated that an illustrative study evidencing domain specificity in discounting (Experiment 1) showed that people's subjective values of the payoffs in domains discounted in this experiment were different (Experiment 2) and used a novel method to match the attractiveness of the available alternatives across domains (Experiment 3). Finally, Experiment 4 showed that when matching was applied, the domain effect disappeared. We conclude that a magnitude effect can, at least partially, explain domain specificity in delay discounting.     

Effects of reward magnitude frames on measures of delay discounting in a hypothetical money scenario

The current study analyzed the effects of three frames of reward magnitude—quantity, volume, and duration—on the rate at which college students discounted hypothetical, delayed monetary rewards. Hypothetical scenarios were presented using the fill-in-the-blank discounting questionnaire and participants made choices between immediate and delayed hypothetical monetary rewards. Scenarios framed the monetary choices as (a) quantity of dollar bills, (b) height (inches) of a stack of dollar bills, and (c) duration of time spent in a hypothetical cash machine to collect dollar bills. For each scenario, participants' subjective values were used to calculate the area under the curve (AuC). Framing resulted in a moderate effect size: The duration frame yielded significantly smaller AuC values compared to the quantity and volume frames. Thus, the framing of reward magnitude was a significant variable in controlling discounting rates for hypothetical, delayed monetary rewards. Subsequent investigations should be aware of the independent effects of the reward magnitude frames on delay discounting rates.