Concave utility,transaction costs,and risk in measuring discounting of delayed rewards

Research has consistently found that the decline in the present values of delayed rewards as delay increases is better fit by hyperbolic than by exponential delay-discounting functions. However, concave utility, transaction costs, and risk each could produce hyperbolic-looking data, even when the underlying discounting function is exponential. In Experiments 1 (N = 45) and 2 (N = 103), participants placed bids indicating their present values of real future monetary rewards in computer-based 2nd-price auctions. Both experiments suggest that utility is not sufficiently concave to account for the superior fit of hyperbolic functions. Experiment 2 provided no evidence that the effects of transaction costs and risk are large enough to account for the superior fit of hyperbolic functions.     

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.     

Delay-discounting probabilistic rewards: Rates decrease as amounts increase

The independence of delay-discounting rate and monetary reward size was tested by offering subjects (N = 621) a series of choices between immediate rewards and larger, delayed rewards. In contrast to previous studies, in which hypothetical rewards have typically been employed, subjects in the present study were entered into a lottery in which they had a chance of actually receiving one of their choices. The delayed rewards were grouped into small ($30–$35), medium ($55–$65), and large amounts ($70–$85). Using a novel parameter estimation procedure, we estimated discounting rates for all three reward sizes for each subject on the basis of his/her pattern of choices. The data indicated that the discounting rate is a decreasing function of the size of the delayed reward (p < .0001), whether hyperbolic or exponential discounting functions are assumed. In addition, a reliable gender difference was found (p = .005), with males discounting at higher rates than females, on average.     

Social discounting and delay discounting

Social discounting was measured as the amount of money a participant was willing to forgo to give a fixed amount (usually $75) to another person. In the first experiment, amount forgone was a hyperbolic function of the social distance between the giver and receiver. In the second experiment, degree of social discounting was an increasing function of reward magnitude whereas degree of delay discounting was a decreasing function of reward magnitude. In the third experiment, the shape of the function relating delayed rewards to equally valued immediate rewards for another person was predicted from individual delay and social discount functions. All in all, the studies show that the social discount function, like delay and probability discount functions, is hyperbolic in form. Copyright © 2007 John Wiley & Sons, Ltd.     

DO ADJUSTING‐AMOUNT AND ADJUSTING‐DELAY PROCEDURES PRODUCE EQUIVALENT ESTIMATES OF SUBJECTIVE VALUE IN PIGEONS?

The current experiment examined whether adjusting-amount and adjusting-delay procedures provide equivalent measures of discounting. Pigeons' discounting on the two procedures was compared using a within-subject yoking technique in which the indifference point (number of pellets or time until reinforcement) obtained with one procedure determined the value of the corresponding variable in the yoked condition with the other procedure. Behavior on each procedure was well described by a hyperbolic discounting function. Results revealed no systematic differences in the degree of discounting as measured by the discounting rate parameter of the hyperbola in Experiment 1, which used 20-mg pellets. These results were replicated in Experiment 2 using smaller, 14-mg pellets, which potentially yield more precise measurement of indifference points on the adjusting-amount procedure. The finding that estimates of the k parameter in the hyperbolic discounting function obtained with one procedure did not differ systematically from estimates obtained from the same subjects with the other procedure represents strong support for the hypothesis that the same process underlies the discounting of delayed rewards on both adjusting-amount and adjusting-delay procedures.     

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.     

Tests of an indifference rule in variable‐delay and double‐reward choice procedures with humans

Four hundred and fifty participants were recruited from Amazon Mechanical Turk across 3 experiments to test the predictions of a hyperbolic discounting equation in accounting for human choices involving variable delays or multiple rewards (Mazur, 1984, 1986). In Experiment 1, participants made hypothetical choices between 2 monetary alternatives, 1 consisting of a fixed delay and another consisting of 2 delays of equal probability (i.e., a variable‐delay procedure). In Experiment 2, participants made hypothetical monetary choices between a single, immediate reward and 2 rewards, 1 immediate and 1 delayed (i.e., a double‐reward procedure). Experiment 3 also used a double‐reward procedure, but with 2 delayed rewards. Participants in all 3 experiments also completed a standard delay‐discounting task. Finally, 3 reward amounts were tested in each type of task ($100, $1000, and $5000). In the double‐reward conditions (Experiments 2 and 3), the results were in good qualitative and quantitative agreement with Mazur's model (1984, 1986). In contrast, when participants made choices involving variable delays (Experiment 1), there was relatively poor qualitative and quantitative agreement with this model. These results, along with our previous findings, suggest the structure of questions in hypothetical tasks with humans can be a strong determinant of the choice pattern.     

Effects of inflation on the subjective value of delayed and probabilistic rewards

In the years prior to 1994, there were very high rates of inflation in Poland, and the zloty depreciated relative to the U.S. dollar. However, the new zloty, introduced in 1995, was associated with greatly decreased rates of inflation and provided a more stable currency. We report a series of three experiments that take advantage of these changes to examine the effects of inflation on the subjective value of delayed and probabilistic rewards. Subjects were Polish citizens familiar with both zlotys and dollars. The first two experiments, conducted in 1994, used dollars and old zlotys, and the third experiment, conducted in 1996, used dollars and new zlotys. In all three experiments, the dollar and zloty rewards were of equivalent worth, according to the then current exchange rates. In Experiment 1, subjects chose between immediate and delayed rewards and, in Experiment 2, chose between certain and probabilistic rewards. The subjective value of a delayed reward was greater when its amount was specified in dollars than when it was specified in old zlotys. In contrast, the currency in which a reward was specified had no effect on the subjective value of probabilistic rewards. The results of these two experiments suggest a selective effect of inflation on decisions involving delayed rewards. This was verified in the third experiment, in which, using new zlotys, no differences in discounting were observed between the two currencies with either probabilistic or delayed rewards. Importantly, in all three experiments, the discounting of both delayed and probabilistic rewards was well described by the same simple mathematical model, suggesting that similar decision-making processes underlie both phenomena. However, the present results argue against a single-process theory in which the discounting of probabilistic rewards is derived from the discounting of delayed rewards.     

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.     

PREFERENCE REVERSALS DUE TO MYOPIC DISCOUNTING OF DELAYED REWARD

Abstract— A basic stationarity axiom of economic theory assumes stable preference between two deferred goods separated by a fixed time. To test this assumption, we offered subjects choices between delayed rewards, while manipulating the delays to those rewards. Preferences typically reversed with changes in delay, as predicted by hyperbolic discounting models of impulsiveness. Of 36 subjects, 34 reversed preference from a larger, later reward to a smaller, earlier reward as the delays to both rewards decreased. We conclude that the stationarity axiom is not appropriate in models of human choice.     

DISCOUNTING OF DELAYED FOOD REWARDS IN PIGEONS AND RATS: IS THERE A MAGNITUDE EFFECT?

Temporal discounting refers to the decrease in the present, subjective value of a reward as the time to its receipt increases. Results from humans have shown that a hyperbola-like function describes the form of the discounting function when choices involve hypothetical monetary rewards. In addition, magnitude effects have been reported in which smaller reward amounts are discounted more steeply than larger amounts. The present research examines the cross-species generality of these findings using real rewards, namely food pellets, with both pigeons and rats. As with humans, an adjusting amount procedure was used to estimate the amount of immediate reward judged equal in value to a delayed reward. Different amounts of delayed food rewards (ranging from 5 to 32 pellets in pigeons and from 5 to 20 pellets in rats) were studied at delays varying from 1 s to 32 s. A simple hyperbola, similar to the hyperbola-like mathematical function that describes the discounting of hypothetical monetary rewards in humans, described the discounting of food rewards in both pigeons and rats. These results extend the generality of the mathematical model of discounting. Rates of discounting delayed food rewards were higher for pigeons than for rats. Unlike humans, however, neither pigeons nor rats showed a reliable magnitude effect: Rate of discounting did not vary systematically as a function of the amount of the delayed reward.     

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.     

Temporal Patterning in Probabilistic Choice

Human subjects making repeated choices between probabilistic and near-certain monetary rewards tended to be less risk-averse the shorter the inter-trial interval. Subjects were also less risk-averse when the choice-outcome sequences were clustered in threes than when each choice-outcome sequence was separated from its neighbors by equal intertrial intervals. These results further extend the analogy between delay discounting and probability discounting. Probability discounting, like delay discounting, may be described by hyperbolic function relating discounted value to expected delay to the next reward.     

Temporal discounting predicts student responsiveness to exchange delays in a classroom token system

Typical assessments of temporal discounting involve presenting choices between hypothetical monetary outcomes. Participants choose between smaller immediate rewards and larger delayed rewards to determine how the passage of time affects the subjective value of reinforcement. Few studies, however, have compared such discounting to actual manipulations of reward delay. The present study examined the predictive validity of a temporal discounting procedure developed for use with children. Forty-six sixth-grade students completed a brief discounting assessment and were then exposed to a classwide intervention that involved both immediate and delayed reinforcement in a multiple baseline design across classrooms. The parameters derived from two hyperbolic models of discounting correlated significantly with actual on-task behavior under conditions of immediate and delayed exchange. Implications of temporal discounting assessments for behavioral assessment and treatment are discussed.     

DISCOUNTING OF DELAYED REWARDS:

Abstract— 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 selfcontrol Discounting of delayed rewards by all three age groups was well described by a single function with agesensitive parameters (all R²s>94) Thus, even though there are quantitative age differences in delay discounting, the existence of an age-invanant form of discount function suggests that the process of choosing between rewards of different amounts and delays is qualitatively similar across the life span     

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.     

A Measure of Delay Discounting Within the Academic Domain

It is important to better understand the decision‐making processes involved in student procrastination, in order to develop interventions that reduce this common problem. Students may procrastinate because studying produces delayed reinforcers; however, no task measuring delay discounting of academic outcomes currently exists. In Experiment 1, we developed and piloted a measure of academic discounting modeled on titrating‐amount tasks successfully used in the discounting literature. Participants made hypothetical choices between working for money (the smaller, sooner reinforcer) and working on an assignment that was due at various times (the larger, later reinforcer). Participants showed systematic decreases in the subjective value of the assignment as a function of delay, and the hyperbolic and hyperboloid models described the shape of this decrease in value well. In general, larger delayed rewards are discounted less steeply than smaller delayed rewards (the magnitude effect). In Experiment 2, we observed the magnitude effect in academic discounting: Participants discounted a “not important” assignment more steeply than an “important” assignment. In the hyperboloid model, this change was captured by an increase in the s parameter. Results provide support for the validity of the academic discounting task. Copyright © 2018 John Wiley & Sons, Ltd.     

Amount of reward has opposite effects on the discounting of delayed and probabilistic outcomes

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.     

The relation between temperament and rate of temporal discounting

The present study examines the relationship between several temperamental traits (sensation seeking, extraversion–introversion, and impulsivity) and rate of temporal discounting, that is the rate of decrease in the subjective value of a reward as the delay to its receipt increases. Participants made choices between hypothetical monetary rewards that could be obtained immediately or after a specified delay. Temporal discounting functions were calculated and differences in rate of discounting between different groups were evaluated. For all groups of participants, hyperbolic discounting functions described well the decreases in subjective value with delay (R² ranged from 0.700 to 0.985). The rate of discounting was equivalent in high and low sensation seekers. However, both extraverts and high impulsive individuals showed higher temporal discounting rates than introverts and low impulsive individuals. The relationship between temperament and temporal discounting may be related to differences in time perception and/or to differences in susceptibility to rewards in extraverts versus introverts and in high- versus low-impulsive individuals.