• # Hyperbolic Discounting

Hyperbolic discounting is a behavioral bias which describes the tendency for people to increasingly choose a smaller-sooner reward over a larger-later reward as the delay occurs sooner rather than later in time. Full disclaimer: I am as guilty of it (if not more) than anybody else. Exhibit A is my book consumption: I never read as many books as since the time I gained to ability to 1-click download them and, to be fully transparent, I never compare the price of the Kindle version to that of the free-delivery (in one to three days) paperback. However, when ordering a book that’s offered exclusively in paper form, I watch the difference between the prices of delivery options like a hawk: unless I’m faced with some objective deadline, nine times out of ten I’ll pick the cheaper three-day delivery over the more onerous overnight one. Therefore, while my discount rate from zero to one is unquantifiably large, the one between one and three is a finite, reasonable, number.

And that is the definition of hyperbolic discounting: when offered a larger reward in exchange for waiting a set amount of time, people act less impulsively (i.e., choose to wait) as the rewards happen further in the future. Put another way, people avoid waiting more as the wait nears the present time.

The notion of discounting future rewards relative to immediate pleasure has a long history. People generally want rewards sooner rather than later. Thus, options that delay a reward appear less attractive and people discount them. Classical economics assumes that people discount a future reward by a fixed percentage for each unit of time they must wait. If the discount rate is 10% per year, a person should equally like $100 now and$110 a year from now. As well, the same person should also equally like $100 in a year and$110 in two years. According to this view (called exponential discounting), the amount people discount a future reward depends only on the length of the wait and a discount rate that is constant across different wait times.

Although exponential discounting has been widely used in economics, a large body of evidence suggests that it does not explain people’s choices. People choose as if they discount future rewards at a greater rate when the delay occurs sooner in time. To illustrate, many people prefer $100 now to$110 in a day, but very few people prefer $100 in 30 days to$110 in 31 days. It appears people would rather wait 1 day for $10 if the wait happens a month from now. However, they prefer the opposite if they must wait right now. More generally, the rate at which people discount future rewards declines as the length of the delay increases. This phenomenon has been termed hyperbolic discounting by psychologist Richard Herrnstein. There are several reasons why people might rationally choose a smaller reward now over a larger reward later. They may like the sure thing, their preferences could change, or they may have an urgent need such as hunger or paying the rent. Even so, people still seem to show inconsistencies in their choices over time. When choosing between$100 or $110 a day later as in the earlier example, people believe that in a month they will want to wait a day for an extra$10. Yet after a month passes, many of these people will reverse their preferences and now choose the immediate $100 rather than wait a day for an additional$10. In sum, even when facing the same exact choice, people act impulsively in the short-term but exhibit greater patience in the long- term.

The amount that people discount future rewards has been mathematically represented in several ways. The classical economic view of exponential discounting reduces a future reward by a factor of \frac{1}{(1 + k)^t} where k is the constant discount rate per time unit and t is the length of the delay. The amount a future reward is discounted depends only on the length of the delay, given a constant discount rate. Alternatively, hyperbolic discounting reduces a future reward by a factor of \frac{1}{(1 + k)^{\beta/\alpha}} where \alpha and \beta are greater than zero. The term “hyperbolic” is used because this formula is the generalized function for a hyperbola. With hyperbolic discounting, the discount rate decreases as the delay occurs further in the future. Thus, the amount a future reward is discounted depends on the length of the delay and when the delay occurs. Thus, hyperbolic discounting will generally discount future rewards more than exponential discounting for short delays, yet less than exponential discounting for long delays.

Two simpler versions of hyperbolic discounting have also been proposed and widely used. First, Herrnstein has modeled some behaviors quite well by assuming that \alphaand \betaare equal. In this formulation, future rewards are discounted by a factor of \frac{1}{(1 + kt)}. Second, economist David Laibson has accounted for several phenomena using a particularly simple form of “quasi-hyperbolic” discounting. Here, future rewards are discounted by a factor of \beta k^t for any t > 0 where \beta. This implies that people discount future rewards by a constant factor to reflect the presence of a delay. As well, they also discount by an exponential factor that grows at a constant rate with the length of the delay. Although not truly hyperbolic, this simpler formulation still captures many of the basic aspects of hyperbolic discounting such as greater impulsivity in the short-term.

While, as disclosed above, I am guilty of hyperbolic discounting in my personal life, I successfully manage to disassociate myself from the impulse in my professional one. What’s more, I manage to lever my awareness of the bias when dealing with others. The best example is in negotiations. My negotiating style is principles-based, with offers grounded on solid arguments and expecting similarly reasoned counteroffers. However, I have, on occasion, had the opportunity to leverage the other side’s hyperbolic discounting to my advantage. In fact, I have often closed deals by offering a deal-closing concession which might have seemed trivial with proper perspective but rewarded the other side with the handshake’s instant gratification. We don’t need a Ph.D. in psychology to recognize that a handshake (especially if it results from the other party’s concession) activates a neural reward, nor economists to recognize that often concessions with little value to one party have great value to the other. However, the hyperbolic discounting concept empowers a negotiator with the insight that a trivial concession leading to a handshake today has utility to the other side orders of magnitude larger than one left for tomorrow or another future date.

I also use hyperbolic discounting to get business for my company: it is as easy as offering to deliver our solutions now in exchange for payment sometime in the future. As long as the client is creditworthy, the lure of “enjoy now – pay later” works for both parties: my company gets the business at a premium rate that includes an implied interest rate while the client gets their solution immediately. The neuroscience behind it is that as soon as the client realizes they don’t have to pay right away, they don’t think about the price. Instead, the hyperbolic power of the availability dominates their thinking – “Wow! I can have it now!” overwhelms “Wow! That’s expensive!”.

More generally, and outside of the strict realm of my personal experience, I dare generalize to state that hyperbolic discounting introduces a fundamental paradigm change in economic theory: preferences with hyperbolic discounting, unlike those with exponential discounting, lack time-consistency. When an agent’s preferences are time-inconsistent, their preference ordering changes over time. Dynamic choice problems are therefore not determined by the solution of a simple maximization problem and require the agent to form their own expectations regarding their own decisions in the future. In fact, this can be assimilated to the psychological equivalent to a term structure of interest rates which, in the absence of tradable derivatives, can be arbitraged over and over again.

The takeaway is that, as rational as we may see ourselves, we all are equipped with a similar brain which is wired to maximize the rate of reward. In fact, neuroscience researchers empirically proved that the durations of saccades of varying amplitudes can be accurately predicted by a model in which motor commands maximize expected maximizing reward. So, while I continued downloading books at home even after I became aware of the robustness of the science behind hyperbolic discounting, I have forced myself to fight this behavioral bias at work whenever its lure could disadvantage me. But, even more importantly, I incorporated this knowledge to my “reason toolkit” to be levered as one more datum in the rational decision-making process.

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