Discounting of Delayed Rewards: Models of Individual Choice

Abstract

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-

Effects of clomipramine on self-control choice in Lewis and Fischer 344 rats

“…This definition has led to the development of a quantitative model referred to as delay discounting (Mazur, 1987) that suggests that the value of a reinforcer is discounted or devalued as a result of its delay to presentation. This model has been supported by the findings from numerous experiments (e.g., Green et al, 1994;Madden et al, 1997;Mazur, 1987Mazur, , 1998Myerson and Green, 1995;Petry and Casarella, 1999). (1999) showed that substance abusers discounted delayed outcomes at greater rates than non-abusers.…”

A Discounting Framework for Choice With Delayed and Probabilistic Rewards.

“…The parameter s may represent the nonlinear scaling of amount and/or time and is generally equal to or less than 1.0 (Myerson & Green, 1995). Notice that in the special case when s equals 1.0, Equation 3 reduces to Equation 2.…”

“…As Myerson and Green (1995) showed, if one assumes that value is directly proportional to rate of reward (Rachlin, 1971), then discounting will have the form of Equation 2.…”

“…Note that the preceding equation is equivalent to Equation 2 with k = 1/m (for a more detailed derivation, see Myerson & Green, 1995).…”

“…Power functions have been shown repeatedly to provide a good description of the relation between perceived and physical magnitudes, although there may be systematic deviations with small stimulus magnitudes, and in some cases, the parameters of the power function may depend on how the physical magnitude is measured (Luce & Krumhansl, 1988). Myerson and Green (1995), therefore, assumed that the perceived rate would be equal to the perceived amount divided by the perceived delay, where the perceived magnitude of each variable (amount and delay) would be a power function of the actual magnitude. That is, the perceived amount equals cA r and the perceived delay equals a(D + m) q , and thus value, which depends on perceived rate, equals cA r /a(D + m) q .…”