## Hyperbolic Discounting

In economics, hyperbolic discounting is a time-inconsistent model of 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. This process is traditionally modeled in form of exponential discounting, a time-consistent model of discounting.

A large number of studies have since demonstrated that the constant discount rate assumed in exponential discounting is systematically being violated. Hyperbolic discounting is a particular mathematical model devised as an improvement over exponential discounting. 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 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?’ For certain range of offerings, 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.’

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 make, despite using the same reasoning. This dynamic inconsistency happens because the value of future rewards is much lower under hyperbolic discounting than under exponential discounting.

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, 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.

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. 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 discounting rats 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.

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 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.).

More recently these observations about discount functions have been used to study saving for retirement, borrowing on credit cards, and procrastination. It has frequently been used to explain addiction.  Hyperbolic discounting has also been offered as an explanation of the divergence between privacy attitudes and behavior.