Crawl Across the Ocean

Thursday, August 20, 2009

25. Hyperbolic Discounting

Note: This post is the twenty-fifth in a series. Click here for the full listing of the series.

"The first beast that will appear
will entice us with money and fame.
If you listen long enough
you'll forget there's anything else.
Tie me to the mast
of this ship and of this band.
Tie me to the greater things
the people that I love."

The prototypical hyperbolic discounter was Odysseus. Ahead of time, he knew that he preferred being alive to hearing the song of the sirens, but he knew that once he heard the siren's song, his preferences would switch and he would be lured to his death by their singing. The only way to hear their song and not fall into their trap was to take preventative measures, tying himself to the mast and instructing his shipmates not to set him free.

Here is how Wikipedia defines 'hyperbolic discounting':

"In behavioral economics, hyperbolic discounting refers to the empirical finding that people generally prefer smaller, sooner payoffs to larger, later payoffs when the smaller payoffs would be imminent. However, when the same payoffs are both more distant in time, people tend to prefer the larger outcome, even though the time lag from the smaller to the larger would be the same as before."

As always, an example is helpful.

Say I offer you a choice between $50 now, and $100 one year from now. A lot of people will take the $50 now.

Now say I offer you $50 in five years or $100 in six years. A lot of the people who took the $50 now instead of $100 one year from now, will decide in this case to wait the extra year for the extra $50, choosing the $100 in year 6.

Even though the difference in the payout ($50) and the difference in time (one year) is the same, people have different preferences depending on how imminent the choices are.


It's generally accepted that given a choice between getting something sooner vs. later, sooner is better - after all, you could die before later comes! (plus, a bird in the hand is worth two in the bush). This process of putting less weight on things that are further into the future is known as 'discounting'.

Where there is disagreement is in the pattern of preferences - i.e. how people discount. It has been typical to model people's discounting using an exponential distribution.

In this model, the amount you discount for a certain size of time period always remains the same, regardless of how far into the future that time period is. So if I discount something that happens one year from now vs. something that happens now by 5%, then I will discount something that happens 10 years from now by 5% vs. something that happens 9 years from now (because in both cases, there is a one year time window between the two choices).

This type of discounting is so common (particularly in the world of finance) that it's sometimes a surprise to realize that there is any other way to do it.

However, experience with, and experiments on, people have demonstrated that many people actually employ a 'hyperbolic discounting' model (just ask the CEO of Money Mart!). The difference with the hyperbolic model is that you place a much higher discount rate on time periods that are near, and a lower discount rate on time periods that are far. One of the interesting consequences of this type of discounting is that ahead of time, when two potential events are at a distance, you may prefer one to another (going for a run, vs. eating a bag of potato chips, for example). However, once the events get closer in time, the short term payoff from the bag of chips may come to outweigh the longer term gain from going for a run. If you are self-aware enough to know your vulnerabilities, it may be possible to, like Odysseus, take preemptive measures (e.g. prayer - 'Lord, lead us not into temptation', or, more practically, not buying chips while at the grocery store).

Mathematically, the exponential function reads as: y = e-rt

(note that isn't just any old e, it's this e)

t represents time and as it gets bigger, the negative exponent gets larger, meaning the value of the function gets smaller - this is the mechanism through which the future payoffs (with a larger t value) are valued less than more current payoffs (with a small t value).

r represents the rate at which the future payouts shrink in importance - the higher the r value, the more you value the present vs. the future.

The hyperbolic function looks like this: y = 1 / (1 + rt)

t again represents time, this time in the denominator of a fraction, so again as t gets bigger the payout gets smaller.

It's easier if you see both curves plotted on a chart.

Note how the hyperbolic function drops off suddenly and then levels out, while the exponential function is much steadier. It is the sharp change in slope of the hyperbolic function which leads to the pattern where you do something in the short term (when the curve is steep, and you just need to do something now and damn the consequences) but come to regret it later once you are no longer on the steep part of the curve.

On the off chance that anyone actually made it to the bottom of the post, you might be wondering what this has to do with ethics. If so, thinklink back to the model of rational behaviour that David Gauthier sketched out where whatever a person decided to do must be rational, given that that was their preference. Hyperbolic discounting muddies the waters considerably since it means that not only will a person's preferences not be stable over time, but they will be predictably unstable. I know that there are actions I can take ahead of time to prevent myself from falling victim to temptation or short term thinking.

Furthermore, it seems that ethics often seem connected to this type of behaviour. What is patience but an attempt to place more weight on the future instead of the present? What is courage but an attempt to place more weight on the distant future instead of the very near future? What is procrastination but a failure to place more weight on the future than the present? Unlike Gauthier, we can't simply dismiss this running battle between the present and the future in any consideration of ethics.

Note: Lots more reading on the topic here.

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  • I'm leaving a comment because I notice that you're not being overwhelmed with comments on these posts... but I for one am really enjoying reading them. So I just wanted to say thanks for the ongoing series, it's fascinating reading :)

    By Blogger Renee, at 4:17 PM  

  • Thanks - even though my motivation for the series is primarily selfish, in that I'm trying to learn to satisfy my curiosity, comments (especially positive ones!) certainly help with the motivation to write more.

    By Blogger Declan, at 10:39 PM  

  • might be wondering what this has to do with ethics.

    Ethics? Damn, I had my cheque book out in the hope you were going to point out a a great arbitrage opportunity ;)

    By Blogger KevinG, at 1:04 PM  

  • Well, there's no doubt many people have made money off this phenomenon. The credit card companies make money off your inability to stop hyperbolic discounting, the counsellors and self-help programs and fitness clubs make money off your attempts to stop hyperbolic discounting and so on.

    I'm not sure I have any new suggestions. Markets may not be efficient in the sense that we don't have asset bubbles but they are fairly efficient in that if there is a way to part people from their money, someone will find it...

    By Blogger Declan, at 9:02 PM  

  • Indeed. I just figured that if it was traditional to model the discounting as exponential when hyperbolic may be a better fit then you should be able to work out some arbitrage.

    Anyway, as you note. It's a widely known phenomenon and is, IMO, a main reason climate change action will never be enough -- we won't even be here for the worst of the consequences.

    By Blogger KevinG, at 8:16 AM  

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