Hyperbolic Discounting


    I have always been fascinated with the study of how people make decisions.  Its a complicated process that involves as much ‘gut instinct’ as rational evaluation, and is rife with systematic errors in judgement.  For a long list of common mistakes, check out Wikipedia’s list of cognitive biases, particularly the decision-making section.  The majority of them are things that most people are probably aware of, like picking a option just because other people picked it.  However, a couple of them are a bit surprising, especially the one I want to talk about today: the phenomenon of hyperbolic discounting.

    The core question is how people penalize various options for having a ‘delayed payoff’.  Would you rather have $30 now and $50 in five years?  Implicit in the decision-making process is that later payoffs aren’t worth as much.  You might need the money now more than later, or there’s a risk you won’t get the money later, due to death/bankrupcty of the source/etc.  The drop in value of a payoff due to the delay involved is called the ‘delay discount’.

     How might one rationally evaluate this discount?  Well, virtually every reason you can name for de-valuing later payoffs is due to some source of roughly constant risk.  Therefore, the value of an object should decay exponentially with time, at a rate determined by the amount of risk.  However, this is not what people usually do!  Studies have shown that people discount delayed payoffs hyperbolically; that is, roughly proportional to the inverse of the delay.  Specifically, a payoff of value P_0 becomes one of value frac{P_0}{1+cT} if it is delayed for time T, where c is a constant that determines roughly how ‘risky’ the delay is.

    This has two main consequences.  The first is that we tend to over-prefer options with more immediate payouts, which should be no surprise to anyone who has interacted with humans before.  The second is that we tend to over-value the further of two distant options, which is a bit weird.  For instance, many people might choose $1200 in forty years over $1000 in thirty years, even though inflation alone would make the two roughly equal.

    So why does this happen?  It should be noted that animals do this as well as people, so it seems pretty hard-wired.  I don’t think a good answer is known at this point, but I can think of a few curious options. 

    One is that people have a skewed perception of future time, and that in their minds ‘in thirty years’ and ‘in forty years’ aren’t as far apart as ‘now’ and ‘in ten years’.  I think something like a logarithmic measure on actual time might give the hyperbolic discounting model.  Can anyone think of a different kind of experiment that would test how people value future lengths of time?

    A second possibility is that we just don’t viscerally understand the exponential function.  It is certainly true that most people lack a good qualitative understanding of it (just ask an undergrad with a graphing calculator how many times e^x and 10x^2 intersect).  However, it is unclear to me how much people are actually trying to model constant risk when making these decisions, versus just going with what their gut tells them.  Though, gut instinct can manage a surprising amount of mathematical prowess, so who knows.

    Now, I just need to figure out how to use this to swindle people…

17 Responses to “Hyperbolic Discounting”

  1. Isabel Lugo Says:

    You could definitely use it to swindle people somehow — the problem is that it’ll take a long time to make money with such a scheme, because you’re taking advantage of the skewed way in which people see long times. So this is not a get-rich-quick scheme.

    And what do undergrads say about e^x and 10x^2, anyway? I’ve never asked.

  2. John Armstrong Says:

    Isabel: I’m pretty sure the “with a graphing calculator” comes into it. That is, they plug them into the calculator, see the exponential cut across through the parabola — crossing twice — and never stop to think that the exponential eventually rises faster than the quadratic — thus crossing again, far above the part of the graph near the origin.

  3. Ben Webster Says:

    Now, I just need to figure out how to use this to swindle people…

    That’s easy; start a credit card company.

  4. Peter Luthy Says:

    That is pretty funny, Greg. You’d think anyone who took 2 years of high school algebra would immediately correlate changes in money with exponential growth/decay just like interest or inflation since that the exponential function is hammered into their brains for weeks while they do interest computations. Then again I’ve seen enough scratch tickets in convenience stores to realize typical people aren’t doing any computations when dealing with money.

  5. Greg Muller Says:

    Yeah, I think this is gonna be too much work to make money off of; all the ways of capitalizing on immediate gratification are pretty well-exploited, and all the long term ways of swindling people take too long.

    The e^x versus 10x^2 thing was actually on my GRE math subject test (or maybe a practice test I took? its hard to remember), with a picture kindly provided. It would seem that I have a fondness for ways of tricking people mathematically, since I remembered it from then.

  6. Nate Berkowitz Says:

    Here’s a a good scam (if you own a store): advertise some product at an after-rebate price. Assume some percentage of people will make a purchase based on that price that they wouldn’t have made otherwise.

    When those customers realize that they won’t actually get the rebate for 6 to 8 weeks, the value of the rebate will shrink. Probably for many people it will drop below the minimum value threshold required to make filling out a form and walking to the mailbox worth their time.

    the “No Payments For 90 Days!” trick is similar, but it relies on cost discounting over time. I don’t know whether that’s also hyperbolic.

    of course describing your prices as “So low, manufacturers have hired men to kill us!!!” could also be described as hyperbolic discounting, but that’s something different…

  7. Greg Muller Says:

    Argh, I hate hidden rebates. Though, I think this is a function of my difficulty in executing basic tasks, like filling out a form, mailing a letter and depositing a check, rather than discounting the future payoff.

  8. Andrew Muller Says:

    If someone was interested into continuing to study this, they might consider replacing monetary value with something of constant worth, like snacks. Greg, for example, might consider five jelly beans now substantially more important then six jelly beans tomorrow if not just for the fact that a single jelly bean would fail to make a detectable difference in the levels of enjoyment. However, were I to offer him ten jelly beans any amount of time down the line in place of his immediate five, he would be more likely to scratch his chin and consider where he will be at that point in the future (for studies have shown his love of jelly beans does not diminish over time).
    If the above is true, then I believe the initial base of gratification has a much larger role in people’s thought process, and the time frame has less to do with the equation than what multiples of the original offered (if that makes any sense). To relate to the above example, were I to offer Greg three jelly beans today or four tomorrow, the value of the single jelly bean in question is greater as it represents a larger portion of the immediate offer. Likewise, were I to offer three beans now, or nine after a wait, the amount of time he would wait is not even relative as three times as many jelly beans is worth any wait.
    Maybe I’ll conduct some research at my dorm with candies.

  9. ANC Says:

    Somewhat trivial point, but exponential discounting does assume an infinite time horizon. Humans have finite lifetimes (the value I derive from consumption of a payoff of $100 delivered to me 200 years in the future is exactly 0, for example).

  10. Johan Richter Says:

    Actually, exponential discounting does not assume an infinite lifetime. If you have a constant probability of dying per time period this can be baked in the discount factor.

  11. Johan Richter Says:

    “Baked in” is not an English expression, is it?

    Make it: “this [the probability of dying] can be accounted for by making the discount factor larger. “

  12. John Armstrong Says:

    Sure it’s an English expressions. I’m sure everybody understood what you meant.

  13. Ulf Hamster Says:

    There is a problem with hyperbolic discouting. I you think dynamically, you will figure out it fits to the liquidity preference hypothesis (term structures); e.g. it would explain how firms might get forced into short term financing due to investors CONSISTENT preferences.

    the flipside is, that hyperbolic discounting implies a CONSISTENT behavior. Thus, it is not much different from Samuelsons’s UT. You just look at two different ways to discount ALWAYS like this.

    however, emprical evidence from psychology (see Thaler, 1981) observed that people are NOT CONSISTENT regarding WHAT KIND OF DISCOUTING they will use. That means there are people who might switch between different kinds of discouting depending on whatever factors.

    if i look at Shiller’s (1984) comment that finance isn’t free of ”fashion”, it is not possible to judge how consistent a certain believe in discounting something like this or that, will be. If you wanna know how big a crash will be, you have to guess how many people for what kind of investment were and will discount inconsistently.

  14. Steve in W MA Says:

    One thing that occurs to me is that in many of the research scenarios, the question of getting the money is hypothetical. The reason this is important is that most people probably have to know who the actor is who is going to give the sum of money in question. In the real world, we understand that future promises have a relatively high probability of not being delivered upon. So $100 in 5 years might have a value approaching zero compared to 50 dollars now or $100 in 5 days–because 0 to 5 days is probably the limit of viability for verbal agreements between humans, especially when the second party to the transaction doesn’t have a personal history with the questionnaire/experimental respondent.

    IF you supply an actor that is trusted, such as the US Treasury (ok, not necessarily so trusted these days as it used to be), which is the guarantor of the promise, then that may change the results because of perception that the monetary delivery will be backed up and reliable. Leaving the actor out of the scenario leaves the average questioned person with a big gap of knowledge about the actor’s reliability, so they will (without mentioning it or, in some cases, even being conscious or explicit about it) discount future rewards simply based upon a factor of “”I don’t trust future promises from unknown parties, so I’ll take the $50 now–because presumable in the now I can exert pressure to compel the reward if it isn’t presented, but after a year I won’t be able to.”

    To really do this experiment well one might need to have actual funds, get banking and identification information from recipients, disclose that funds are in escrow for the prizes, then do a series of other reward-based prizes that would involve a combination of actually handing cash to the recipients and transferring money into their accounts to establish credibility of the actor. Only then would offering them a future vs a present prize yield results that reduce or highly limit the distortion factor of credibility and deliverability.

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