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Does Hyperbolic Discounting Really Exist?

19 Post author: gwern 03 December 2011 03:07AM

“Beware of WEIRD psychological samples” because results derived from them may reflect the specific sample more than any kind of generalized truth. And LessWrong has generalized hyperbolic discounting out the wazoo. (See the tags akrasia and discounting.) Hyperbolic discounting is bad, of course, because among other things it leaves on vulnerable to preference reversals and inconsistencies and hence money-pumping.

But isn’t it odd that for a fundamental fact of human psychology, a huge bias we have spent a ton of collective time discussing and fighting, that it doesn’t seem to lead to much actual money-pumping? The obvious examples like the dieting or gambling industries are pretty small, all things considered. And online services like BeeMinder specifically devised on a hyperbolic discounting/picoeconomics basis are, as far as I know, useful but no dramatic breakthrough or silver bullet; again, not quite what one would expect. Like many other heuristics and biases, perhaps hyperbolic discounting isn’t so bad after all, in practice.

Ainslie mentions in Breakdown of Will somewhere that financial incentives can cause people to begin discounting exponentially. What if… hyperbolic discounting doesn’t really exist, in practice? If it may reflect a failure of self-control, a kind of teenager trait, one we find in younger (but not older) populations - like university students?

The following quotes are extracted from the paper “Discounting Behavior: A Reconsideration” (102 pages) by Steffen Andersen, Glenn W. Harrison, Morten Lau & E. Elisabet Rutström, January 2011:

The implied econometrics calls for structural estimation of the theoretical models, allowing for joint estimation of utility functions and discounting functions. Using data collected from a representative sample of 413 adult Danes in 2009, we draw striking conclusions. Assuming an exponential discounting model we estimate discount rates to be 5.6% on average: this is significantly lower than all previous estimates using controlled experiments. We also find no evidence to support quasi-hyperbolic discounting or “fixed cost” discounting, and only modest evidence to support other specifications of non-constant discounting. Furthermore, the evidence for non-constant discounting, while statistically significant, is not economically significant in terms of the size of the estimated discount rates. We undertake extensive robustness checks on these findings, including a detailed review of the previous, comparable literature.

…We do find evidence in favor of flexible Hyperbolic specifications and other nonstandard specifications, but with very modest variations in discount rates compared to those often assumed. We find that a significant portion of the Danish population uses Exponential discounting, even if it is not the single model that best explains observed behavior.

Given the contrary nature of our findings, in terms of the received empirical wisdom, section 6 contains a systematic cataloguing of the samples, experimental procedures, and econometric procedures of the alleged evidence for Quasi-Hyperbolic and non-constant discounting. We conclude that the evidence needed reconsideration. The one clear pattern to emerge from the received literature is that non-constant discounting occurs for some university student samples.

One major robustness check is therefore to see if the disappointing showing for the Quasi- Hyperbolic model is attributable to our population being the entire adult Danish population, rather than university students. Although it is apparent that the wider population is typically of greater interest, virtually all prior experimental evidence that we give credence to comes from convenience samples of university students. We find that there is indeed a difference in the elicited discount rates with (Danish) university students, and that they exhibit statistically significant evidence of declining discount rates. On the other hand, the size of the discount rates for shorter time horizons is much smaller than the received wisdom suggests.

…Coller and Williams [1999] were the first to demonstrate the effect of a front end delay; their estimates show a drop in elicited discount rates over money of just over 30 percentage points from an average 71% with no front end delay.11 Using the same experimental and econometric methods, and with all choices having a front end delay, Harrison, Lau and Williams [2002] estimated average discount rates over money of 28.1% for the adult Danish population. Andersen, Harrison, Lau and Rutström [2008a] were the first to demonstrate the effect of correcting for non-linear utility; their estimates show a drop in elicited discount rates of 15.1 percentage points from a discount rate over money of 25.2%. These results would lead us to expect discount rates around 10% with a front end delay, with a significantly higher rate when there is no front end delay.

…The Exponential discounting model indicates a discount rate of only 5.6%, where all discount rates will be presented on an annualized basis. The 95% confidence interval for this estimate is between 4.1% and 7.0%, so this indicates even lower discount rates than the 10.1% reported by Andersen, Harrison, Lau and Rutström [2008a] for the same population in 2003.25 For comparison, the Exponential discounting model assuming a linear utility function implies an 18.3% discount rate, with a 95% confidence interval between 15.5% and 21.2%, so this is also lower than the estimate for 2003 (25.2%, with a 95% confidence interval between 22.8% and 27.6%). We again conclude that correcting for the non-linearity of the utility function makes a significant quantitative difference to estimated discount rates.

The most striking finding from Table 1, for us, is that there is no Quasi-Hyperbolic discounting. The key parameter, $, is not statistically or economically significantly different from 1, and the parameter * is virtually identical to the estimate from the Exponential discounting model. The p-value on a test of the hypothesis that $=1 has value 0.55, although the 95% confidence interval for $ is enough to see that it is not significantly different from 1.

…The Weibull discounting model in panel F allows a very different pattern of non-constant discounting. Indeed, these parameter estimates do imply discount rates that vary slightly, from 6.7% for a 1 day horizon, to 6.0% for a 2 week horizon, and then down to 5.1% for a one year horizon. But the 95% confidence intervals on all of these is at least between 3% and 7%, and one cannot reject the Exponential discounting model hypothesis that s=1 (p-value of 0.73).

…The only demographic covariate to have any statistically significant impact on elicited discount rates is whether the individual is a female. Women have discount rates that are 6.6 percentage points lower than men, and the p-value on this estimated effect of 0.092. In turn, this derives from women being more risk averse: their RRA is 0.294 higher than men, with a p-value on this estimated effect of 0.026. Hence they have a more concave utility function and, by Jensen’s inequality applied to (0), have a lower implied discount rate. Looking at total effects instead of marginal effects, men on average have discount rates of 7.4% and women have discount rates of 3.6%, and the difference is statistically significant (p-value = 0.004).

…Our results were a surprise to us, and the robustness checks reported above did not lead us to qualify that reaction. We fully expected to see much more “hyperbolicky” behavior when we removed the front end delay, and particularly when that was interacted with not providing the implied interest rates of each choice. We were not wedded to one hyperbolicky specification or the other, and did not expect the exponential model to be completely overwhelmed by the alternatives, but we did expect to see much more non-constant discounting. We therefore examined the literature, and tried to draw some inferences about what might explain the apparent differences in results.

…[Literature survey] We ignored all hypothetical survey studies, on the grounds that the evidence is overwhelming that there can be huge and systematic hypothetical biases, and it is simply inefficient to repeat those arguments and waste time taking such evidence seriously. [Like prisoners doing a long sentence, and knowing the jokes and arguments of cellmates by heart, we would rather just point to surveys and evaluations of the evidence in Harrison [2006] and Harrison and Rutström [2008b].] We also focused on experiments, rather than econometric inferences from naturally occurring data, because those data are easier to interpret and have generated the conventional wisdom.36 We excluded studies that did not lend themselves to inferring a discount function.37 Finally, we excluded any study that used procedures that were patently not incentive- compatible or that involved deception.38

…One conclusion that we draw is that virtually all previous evidence of non-constant discounting comes from studies undertaken with students. We therefore conducted a conventional laboratory experiment, described below, using the same procedures as in our (artefactual) field experiment but with students recruited in Copenhagen…In order to determine if the evidence for non-constant discounting, such as it is, derives from the general focus on students samples, we replicated our field experiments with a student sample in Copenhagen recruited using standard methods.39 The experimental tasks were identical, to ensure comparability. Table 7 lists estimates from the student responses of the basic models in Table 1. The background risk attitudes of this sample were virtually identical to those of the adult Danish population.40 The results are clear: we obtain no evidence of quasi-hyperbolic discounting, no evidence of fixed-cost discounting, and no evidence of simple hyperbolic discounting. We do observe some non-constancy of some discount rates with the Weibull discounting specification, although the overall effect of the student sample is not statistically significant, as shown by the p- value of 0.18 on the null hypothesis that the specification is actually Exponential.41

Comments (12)

Comment author: Yvain 03 December 2011 01:58:48PM 6 points [-]

Your summary is that there might be a difference between students and adults here, but isn't the last paragraph of the quoted text saying that they considered this hypothesis, but that students were also non-hyperbolic when they tested them under the same protocol?

Comment author: gwern 04 December 2011 09:12:19AM *  1 point [-]

The students were more hyperbolic, but not much. That they advertise it that way just shows how much they wanted to find hyperbolic discounting somewhere, and makes their failure to that much more credible (unless it's all an elaborate con!).

Comment author: Hyena 03 December 2011 10:39:54AM 4 points [-]

So far as I recall, Ainslie's thesis is that the various "modules" of the brain have hyperbolic discount curves which are then composed to yield an exponential curve. Akrasia is what happens when particularly strong specific impulses spike above the exponential discount curve. Ainslie predicts what you actually see: lots of people making rational decisions punctuated by failures of "willpower" large and small.

I'm also unsure whether you're overstating LessWrong's obsession with akrasia. It's never felt over-generalized. The focus on it seems reasonable enough insofar as LeWers seem to be drawn heavily from students and techies, two groups for whom akrasia can be particularly destructive. So even if hyperbolic discounting is rarer (than I'm still not sure what), the expected negative value of akrasia may be particularly high for LeWers, leading to its perennial popularity.

Comment author: jsalvatier 05 December 2011 04:01:19PM 1 point [-]

This is really interesting and I find it surprising given that Glimcher's Neuroeconomics claims you can find specific neurons in monkey brains which fire hyperbolically. Looking forward to reading more on this.

Comment author: jacob_cannell 09 December 2011 10:29:11AM *  0 points [-]

Does "Glimcher's Neuroeconomics" specify exactly which hyperbolic function fits these monkey neurons? All the neural firing models I've seen are based on the sigmoid function, which at least is similar to some hyperbolics.

Also, just to point out, there isn't necessarily any association whatsoever between some complex human behaviour (such as discounting), and the firing dynamics of individual neurons.

Making that leap is similar to believing that the voltage activation functions of transistors in your latest intel processor is related to that function you keep seeing in your matlab program.

Comment author: jsalvatier 09 December 2011 06:38:45PM 0 points [-]

No it does not (sadly). He also claims that these firing rates/function predicts actual monkey discounting behavior quite closely.

Comment author: vi21maobk9vp 03 December 2011 08:55:57AM 0 points [-]

Preface: I am explaining why this argument doesn't work that well; it doesn't say anything about HD existing.

Well, there are some shady short-term high-interest loan operations in many cases. And it has to be shady, which limits the amount pumped.

Imagine you want to open hyperbolic-discounting operation for money pumping. What can you do? For it to be profitable you have to offer people to exchange little money now for more money later. Looks like banking, so you need to be in the banking business - or be a shady operation. Some part of what banks do fits the bill here, but they can't overdo it without huge and obvious colusion between all players: because short-term interbank loans are relatively cheap, if you try to profit too much there is a huge incentive to undercut you.

Also, with short-term loans (in money or any other form) it simply looks too suspicious...

Comment author: Unnamed 03 December 2011 06:54:08PM 1 point [-]

There are a lot of short-term high-interest loans, which take the form of credit card debt, payday loans, refund anticipation loans, installment plans, and so on. Refund anticipation loans are probably the closest to pure temporal discounting - that's where a tax preparer like H&R Block offers a customer who is due a $X refund from the government a choice between getting that $X later or getting $Y right now instead (in the form of a loan for $Y, which will be repaid by handing over the $X tax refund once it comes in). These examples at least suggest a high discount rate, although they don't definitively point to a hyperbolic shape.

There are a few reasons why these kinds of high-interest loans aren't more common. One is that it's hard to sell a short-term loan to someone with money in the bank, so they mostly target poor people. A second is the point that vi21maobk9vp makes: there's a somewhat competitive market so businesses need to compete by lowering their interest rate (a person who is willing to borrow at a 100% rate will still choose the loan with a 20% rate if they have that option). Also, governments try to put a stop to lending that they see as predatory; for instance, there has been a crackdown on refund anticipation loans over the past few years.

These examples only show high discount rates, not the preference reversal that is characteristic of hyperbolic discounting. But a business model that relies on getting your customers to change their minds (and explicitly reverse their prior decision) seems hard to pull off, and the profit comes from the customers' high-discounting decisions so the incentives aren't really there to try to induce a visible preference reversal.

Comment author: Khoth 03 December 2011 08:08:17PM 3 points [-]

Subscriptions to services that are cheap for an initial period then get expensive might be a case of preference reversal. People often sign up for them intending to re-evaluate after the introductory period, but when the time comes start procrastinating about it.

Comment author: Vladimir_Golovin 03 December 2011 06:47:03AM *  -1 points [-]

But isn’t it odd that for a fundamental fact of human psychology, a huge bias we have spent a ton of collective time discussing and fighting, that it doesn’t seem to lead to much actual money-pumping? The obvious examples like the dieting or gambling industries are pretty small, all things considered.

I wonder if consumer credits qualify as an example of hyperbolic discounting. Basically, people are paying more for the same TV in order to get it faster. That is, a TV now is more valuable to them than the same TV later.

Comment author: vi21maobk9vp 03 December 2011 08:48:26AM 2 points [-]

Seems to be more about misapplied non-linearity of money.

TV now should be higher utility than TV later (assuming TV has positive utility and you changing TVs more often than they break down completely): some of your viewing gets shfited from worse TV to a better one. Consumer credits seem to be long enough to come into the area where hyperbolic discounting does not overweight exponential discounting that much.

What gets stressed is that "Ah well, you pay just pocket change now, just pocket change later, just pocket change a few more times later..." And the full price of TV is a big chunk of cash, of course.

Comment author: [deleted] 13 June 2014 10:24:43AM *  -2 points [-]

I have personal experience of the following in thought experiments:

"Students, the night before an exam, often wish that the exam could be put off for one more day. If asked on that night, such students might agree to commit to paying, say, $10 on the day of the exam for it to be held the next day. Months before the exam is held, however, students generally do not care much about having the exam put off for one day. And, in fact, if the students were made the same offer at the beginning of the term, that is, they could have the exam put off for one day by committing during registration to pay $10 on the day of the exam, they probably would reject that offer. The choice is the same, although made at different points in time. Because the outcome would change depending on the point in time, the students would exhibit time inconsistency People display a consistent bias to believe that they will have more time in the future than they have today. More specifically, there is a persistent belief among people that they are “unusually busy in the immediate future, but will become less busy shortly.” However, this “time slack” is shown to be a bias. However busy you are this week is generally representative of how busy you are in future weeks. When people are estimating their time and when deciding if they will make a commitment, they anticipate more “time slack” in future weeks than the present week. Experiments by Zauberman and Lynch (2005) on this topic showed that people tend to discount investments of time more than money. The authors have nicknamed this the “Yes…Damn” effect because of the tendency to agree to do things in advance (e.g., travel to a conference), but when the time arrives you are very busy and it is inconvenient to attend.[4]""