Hyperbolic discounting

Given two similar rewards, humans show a preference for one that arrives sooner rather than later. Humans are said to discount the value of the later reward, by a factor that increases with the length of the delay. In behavioral economics, hyperbolic discounting is a particular mathematical model thought to approximate this discounting process; that is, it models how humans actually make such valuations. Hyperbolic discounting is sharply different in form from exponential discounting, a rational function used in finance used in the analysis of choice over time. Hyperbolic discounting has been observed in humans and animals.

In hyperbolic discounting, valuations fall very rapidly for small delay periods, but then fall slowly for longer delay periods. This contrasts with exponential discounting, in which valuation falls by a constant factor per unit delay, regardless of the total length of the delay. The standard experiment used to reveal a test subject's discounting curve is to ask: "Would you prefer A today or B tomorrow?" and then, "Would you prefer A in one year, or B in one year and one day?"

For example in studies of pigeons the pigeon is given two buttons: button A provides a small amount of food quickly while button B provides more seed but after a delay. The bird then experiments for a while and settles on preferring A or B. With humans the typical experiment might ask: 'Would you prefer a dollar today or three dollars (today vs. tomorrow) or (in one year vs. in one year and one day)?" Typically, subjects will take less money today versus tomorrow, but will gladly wait one extra day in a year in order to receive more money.

Subjects using hyperbolic discounting reveal a strong tendency to make choices that are inconsistent over time. In other words, they make choices today that their future self would prefer not to make, despite using the same reasoning. This dynamic inconsistency happens because hyperbolic discounts value future rewards much more than exponential discounting.

Observations

The phenomenon of hyperbolic discounting is implicit in Richard Herrnstein's "matching law," the discovery that most subjects allocate their time or effort between two non-exclusive, ongoing sources of reward (concurrent variable interval schedules) in direct proportion to the rate and size of rewards from the two sources, and in inverse proportion to their delays. That is, subjects' choices "match" these parameters.

After the report of this effect in the case of delay (Chung and Herrnstein, 1967), George Ainslie pointed out that in a single choice between a larger, later and a smaller, sooner reward, inverse proportionality to delay would be described by a plot of value by delay that had a hyperbolic shape, and that this shape should produce a reversal of preference from the larger, later to the smaller, sooner reward for no other reason but that the delays to the two rewards got shorter. He demonstrated the predicted reversal in pigeons (Ainslie, 1974).

A large number of subsequent experiments have confirmed that spontaneous preferences by both human and nonhuman subjects follow a hyperbolic curve rather than the conventional, "exponential" curve that would produce consistent choice over time (Green et al., 1994; Kirby, 1997). For instance, when offered the choice between $50 now and $100 a year from now, many people will choose the immediate $50. However, given the choice between $50 in five years or $100 in six years almost everyone will choose $100 in six years, even though that is the same choice seen at five years' greater distance.

Hyperbolic discounting has also been found to relate to real-world examples of self control. Indeed, a variety of studies have used measures of hyperbolic discounting to find that drug-dependent individuals discount delayed consequences more than matched nondependent controls, suggesting that extreme delay discounting is a fundamental behavioral process in drug dependence (e.g., Bickel & Johnson, 2003; Madden et al., 1997; Vuchinich & Simpson, 1998). Some evidence suggests pathological gamblers also discount delayed outcomes at higher rates than matched controls (e.g., Petry & Casarella, 1999). Whether high rates of hyperbolic discounting precede addictions or vice-versa is currently unknown, although some studies have reported that high-rate discounting rats are more likely to consume alcohol (e.g., Poulos et al., 1995) and cocaine (Perry et al., 2005) than lower-rate discounters. Likewise, some have suggested that high-rate hyperbolic discounting makes unpredictable (gambling) outcomes more satisfying (Madden et al., 2007).

The degree of discounting is vitally important in describing hyperbolic discounting, especially in the discounting of specific rewards such as money. The discounting of monetary rewards varies across age groups due to the varying discount rate. (Green, Frye, and Myerson, 1994). The rate depends on a variety of factors, including the species being observed, age, experience, and the amount of time needed to consume the reward (Lowenstein and Prelec, 1992; Raineri and Rachlin, 1993).

Mathematical model

Comparison of the discount factors of hyperbolic and exponential discounting. In both cases, $k=1$. Hyperbolic discounting is shown to over-value future assets compared to exponential discounting.

Hyperbolic discounting is mathematically described as:

$f_{H}(D)={\frac {1}{1+kD}}$

where $f(D)$ is the discount factor that multiplies the value of the reward, $D$ is the delay in the reward, and $k$ is a parameter governing the degree of discounting. This is compared with the formula for exponential discounting:

$f_{E}(D)=e^{-kD}$

Quasi-hyperbolic approximation

The "quasi-hyperbolic" discount function, which approximates the hyperbolic discount function above, is given (in discrete time) by

$f_{QH}(0)=1$, and $f_{QH}(D)=\beta *\delta ^{D}$,

where $\beta$ and $\delta$ are constants between 0 and 1; and again $D$ is the delay in the reward, and $f(D)$ is the discount factor. The condition $f(0)=1$ is stating that rewards taken at the present time are not discounted.

Quasi-hyperbolic time preferences are also referred to as "present-biased" or "beta-delta" preferences. They retain much of the analytical tractability of exponential discounting while capturing the key qualitative feature of discounting with true hyperbolas.

Explanations

Uncertain risks

Notice that whether discounting future gains is rational or not – and at what rate such gains should be discounted – depends greatly on circumstances. Many examples exist in the financial world, for example, where it is logically reasonable to assume that there is an implicit risk that the reward will not be available at the future date, and furthermore that this risk increases with time. Consider: Paying $50 for your dinner today or delaying payment for sixty years but paying $100,000. In this case the restaurateur would be reasonable to discount the promised future value as there is significant risk that it might not be paid (possibly due to your death, his death, etc).

Uncertainty of this type can be quantified with Bayesian analysis. For example, suppose that the probability for the reward to be available after time $t$ is, for known hazard rate $\lambda$

$P(R_{t}|\lambda )=\exp(-\lambda t)$

but the rate is unknown to the decision maker. If the prior probability distribution of $\lambda$ is

$p(\lambda )=\exp(-\lambda /k)/k$

then, the decision maker will expect that the probability of the reward after time $t$ is

$P(R_{t})=\int _{0}^{\infty }P(R_{t}|\lambda )p(\lambda )d\lambda ={\frac {1}{1+kt}}$

which is exactly the hyperbolic discount rate. Similar conclusions can be obtained from other plausible distributions for $\lambda$.

Applications

More recently these observations about discount functions have been used to study saving for retirement, borrowing on credit cards, and procrastination. However, hyperbolic discounting has been most frequently used to explain addiction.

See also

• Time value of money
• Time preference
• Intertemporal choice
• deferred gratification

References

General references

• Ainslie, G. W. (1975) Specious reward: A behavioral theory of impulsiveness and impulsive control. Psychological Bulletin, 82, 463-496.
• Ainslie, G. (1992) Picoeconomics: The Strategic Interaction of Successive Motivational States Within the Person. Cambridge. Cambridge University Press.
• Ainslie, G. (2001) Breakdown of Will Cambridge, Cambridge University Press, ISBN 978-0521596947
• Bickel, W. K., & Johnson, M. W. (2003). Delay discounting: A fundamental behavioral process of drug dependence. In G. Loewenstein, D. Read & R. F. Baumeister (Eds.), Time and Decision. New York: Russell Sage Foundation.
• Chung, S. H. and Herrnstein, R. J. (1967). Choice and delay of Reinforcement. Journal of the Experimental Analysis of Behavior, 10 67-64.
• Green, L., Fry, A. F., and Myerson, J. (1994). Discounting of delayed rewards: A life span comparison. Psychological Science, 5, 33-36.
• Kirby, K. N. (1997) Bidding on the future: Evidence against normative discounting of delayed rewards. Journal of Experimental Psychology: General 126, 54-70.
• Loewenstein, G. and Prelec, D. (1992). Choices Over Time New York, Russell Sage Foundation
• Madden, G. J., Petry, N. M., Bickel, W. K., and Badger, G. J. (1997). Impulsive and self-control choices in opiate-dependent patients and non-drug-using control participants: Drug and monetary rewards. Experimental and Clinical Psychopharmacology, 5, 256-262.
• Madden, G. J., Ewan, E. E., & Lagorio, C. H. (2007). Toward an animal model of gambling: Delay discounting and the allure of unpredictable outcomes. Journal of Gambling Studies, 23, 63-83.
• Perry, J. L., Larson, E. B., German, J. P., Madden, G. J., and Carroll, M. E. (2005). Impulsivity (delay discounting) as a predictor of acquisition of i.v. cocaine self-administration in female rats. Psychopharmacology, 178, 193-201.
• Petry, N. M., and Casarella, T. (1999). Excessive discounting of delayed rewards in substance abusers with gambling problems. Drug and Alcohol Dependence, 56, 25-32.
• Poulos, C. X., Le, A. D., and Parker, J. L. (1995). Impulsivity predicts individual susceptibility to high levels of alcohol self administration. Behavioral Pharmacology, 6, 810-814.
• Vuchinich, R. E., and Simpson, C. A. (1998). Hyperbolic temporal discounting in social drinkers and problem drinkers. Experimental and Clinical Psychopharmacology, 6, 292-305.
• Rachlin, H. (2000). The Science of Self-Control Cambridge;London: Harvard University Press
• Raineri,A., and Rachlin, H. (1993). The effect of temporal constraints on the value of money and other commodities. Journal of Behavioral Decision-Making, 6, 77-94.