Product Code Database
In economics, hyperbolic discounting is a time- inconsistent model of delay discounting. It is one of the cornerstones of behavioral economics and its brain-basis is actively being studied by neuroeconomics researchers.
According to the discounted utility approach, intertemporal choices are no different from other choices, except that some consequences are delayed and hence must be anticipated and discounted (i.e., reweighted to take into account the delay).
Given two similar rewards, humans show a preference for one that arrives in a more prompt timeframe. Humans are said to discount the value of the later reward, by a factor that increases with the length of the delay. In the financial world, this process is normally modeled in the form of exponential discounting, a time- consistent model of discounting. Many psychological studies have since demonstrated deviations in instinctive preference from the constant discount rate assumed in exponential discounting. Hyperbolic discounting is an alternative mathematical model that agrees more closely with these findings.
According to hyperbolic discounting, valuations fall relatively rapidly for earlier delay periods (as in, from now to one week), but then fall more slowly for longer delay periods (for instance, more than a few days). For example, in an early study subjects said they would be indifferent between receiving $15 immediately or $30 after 3 months, $60 after 1 year, or $100 after 3 years. These indifferences reflect annual discount rates that declined from 277% to 139% to 63% as delays got longer. This contrasts with exponential discounting, in which valuation falls by a constant factor per unit delay and the discount rate stays the same.
The standard experiment used to reveal a test subject's hyperbolic discounting curve is to compare short-term preferences with long-term preferences. For instance: "Would you prefer a dollar today or three dollars tomorrow?" or "Would you prefer a dollar in one year or three dollars in one year and one day?" It has been claimed that a significant fraction of subjects will take the lesser amount today, but will gladly wait one extra day in a year in order to receive the higher amount instead. Individuals with such preferences are described as "present-biased".
The most important consequence of hyperbolic discounting is that it creates temporary preferences for small rewards that occur sooner over larger, later ones. Individuals using hyperbolic discounting reveal a strong tendency to make choices that are inconsistent over time – they make choices today that their future self would prefer not to have made, despite knowing the same information. This dynamic inconsistency happens because hyperbolas distort the relative value of options with a fixed difference in delays in proportion to how far the choice-maker is from those options.
After the report of this effect in the case of delay, 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 hyperbola, and that when the smaller, sooner reward is preferred, this preference can be reversed by increasing both rewards' delays by the same absolute amount. Ainslie's research showed that a substantial number of subjects reported that they would prefer $50 immediately rather than $100 in six months, but would NOT prefer $50 in 3 months rather than $100 in nine months, even though this was the same choice seen at 3 months' greater distance. More significantly, those subjects who said they preferred $50 in 3 months to $100 in 9 months said they would NOT prefer $50 in 12 months to $100 in 18 months—again, the same pair of options at a different distance—showing that the preference-reversal effect did not depend on the excitement of getting an immediate reward.Ainslie, George and Haendel, V. (1983) The motives of the will. in E. Gottheil, K. Druley, T. Skodola, H. Waxman (eds.), Etiology Aspects of Alcohol and Drug Abuse, Springfield, Ill.: Charles C. Thomas, pp. 119-140. Nor does it depend on human culture; the first preference reversal findings were in rats and pigeons.
Many subsequent experiments have confirmed that spontaneous preferences by both human and nonhuman subjects follow a hyperbola rather than the conventional, exponential curve that would produce consistent choice over time. 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. (2025). 9780871545497, Russell Sage Foundation. ISBN 9780871545497 Some evidence suggests pathological gamblers also discount delayed outcomes at higher rates than matched controls. Whether high rates of hyperbolic discounting precede addictions or vice versa is currently unknown, although some studies have reported that high-rate discounters are more likely to consume alcohol and cocaine than lower-rate discounters. Likewise, some have suggested that high-rate hyperbolic discounting makes unpredictable (gambling) outcomes more satisfying.
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. 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. (1992). 9780871545589, Russell Sage Foundation. ISBN 9780871545589
As becomes very large, the value of becomes much larger than the value of , with the effect that the value of becomes much smaller than the value of Therefore, the minimum value of (the number of dollars in the immediate choice) that suffices to be greater than that amount will be much smaller than the hyperbolic discounter thinks, with the result that they will perceive -values in the range from to inclusive as being too small and, as a result, irrationally turn those alternatives down when they are in fact the better investment.
where is the discount factor that multiplies the value of the reward, is the delay in the reward, and is a parameter governing the degree of discounting (for example, the interest rate). This is compared with the formula for exponential discounting:
where and are constants between 0 and 1; and is the delay in the reward, but now it takes only integer values. The condition states that rewards taken at the present time are not discounted.
Quasi-hyperbolic discounting retain much of the analytical tractability of exponential discounting while capturing the key qualitative feature of hyperbolic discounting.
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 λ,
but the rate is unknown to the decision maker. If the prior probability distribution of λ is
then the decision maker will expect that the probability of the reward after time t is
which is exactly the hyperbolic discount rate. Similar conclusions can be obtained from other plausible distributions for λ.
where V is the present value, P is the annual cash flow, D is the number of annual payments and k is the factor governing the discounting.
A study by Daniel Read introduces "subadditive discounting": the fact that discounting over a delay increases if the delay is divided into smaller intervals. This hypothesis may explain the main finding of many studies in support of hyperbolic discounting—the observation that impatience declines with time–while also accounting for observations not predicted by hyperbolic discounting. However, although these observations depart from exponential discounting, they do not entail preference reversal as time from the choice to the earlier reward increases.
Arousal of appetite or emotion does sometimes lead to preference reversal, and this has been the most widely accepted alternative to a simply hyperbolic function: hyperboloid or quasi-hyperbolic discounting fuses exponential curves with an arousal bump as a visceral reward becomes imminent. Such cases are obviously important, but still do not account for cases where either both or neither choice is made during arousal.
The most obvious objection to hyperbolic discounting is that many or most people learn to choose consistently over time in most situations. Similarly, a 2014 paper criticized the existing studies for mostly using data collected from university students and being too quick to conclude that the hyperbolic model of discounting is correct. Human experiments have frequently reported wide between-subject variations. If overcoming the tendency to temporary preference takes learning, the next obvious task for experimenters is to test theories of how and when this learning occurs (e.g. Ainslie, 2012).
|
|
