taw's blog

Sunday, February 16, 2014

Simple theoretical basis for Hyperbolic Discounting

Exponential Discounting is the only time-consistent discounting method and it's mathematically very neat, so all economics and psychology likes to pretend that's how humans operate, but in reality everybody universally uses something much closer to Hyperbolic Discounting.

I've talked about evolutionary side of this problem before. These two discounting systems are not even close, so we need some kind explanation why all humans and all animals so consistently use Hyperbolic Discounting. Today I want to present a very simple and intuitive model in which Hyperbolic Discounting is optimal choice.

Exponential Discounting is wrong

But first, I want to disabuse you from the notion that Exponential Discounting is correct, a notion you might believe from excessive exposure to economics.

First imagine how much less do you care about yourself in 10 years compared to yourself right now. Is it 10%? 50%? 90%? Doesn't matter, just pick a number. Let's say this number is 50%, but analogous reasoning applies to each choice.

Now does any of this sound even remotely true:
• You care about yourself at age 90 only 6% as much as yourself at age 50.
• You're indifferent between one person dying today, or a trillion people dying in 200 years from now.
• You believe that wishes of any person who lived a few hundred years ago count for more than wishes of all existing and future humans put together (to remain consistent, going into the past reverses discounting process, so if you care about the future less, you must care about the past equally more)
These are straightforward consequences of exponential discounting. Does any of that sound even remotely sane? I don't think so.

Now compare that with prediction of hyperbolic discounting model, same discount rate (50% in 10 years):
• You care about yourself at age 90 only 42% as much as yourself at age of 50 (if you're 30 today). The older you get, the bigger the gap becomes since at young age both of them are extreme remote, but older you get, the closer 50 seems, but 90 still feels very remote.
• 1 person dying today is as bad as 21 people dying 200 years from now.
• Hyperbolic Discounting as normally defined doesn't provide any numbers about the past, but the most reasonable way to expand it is to make it symmetrical, so you care about past people less than today's people, and the more distant they are the less you care - you might sort of care what your grandparents wanted even if they passed away, but who the hell cares about some Medieval peasants tens of generations ago?
If you put it that way, Exponential Discounting is pretty much Flat Earth level of crazy.

Theoretical basis for Hyperbolic Discounting

There's still one problem left - Exponential Discounting has neat theoretical basis of self-consistency, but it's much harder to explain Hyperbolic Discounting even if it's so intuitively correct, and agrees with all experiments.

So here are two entirely reasonable assumptions, which together generate hyperbolic discounting:
• Assumption 1: You care about future or past you and past or present reality in proportion to how similar they are to you and your current reality.
This might sounds like an unusual way to put it, but it is intuitively about right. The future you will be a somewhat different person, with different preferences, and somewhat alien to you. You're likely to care about similar selves far more than about very different selves - and the greater distance in time, the more different a person you'll be.

The same applies to discounting reality - you might like having certain things or maintaining certain relationships, but the more different reality becomes, the more likely it is they'll become less valuable, or won't persist.

To show that what you care about it is difference, not time, consider these - your attachment to your phone or your collection of Magic: the Gathering cards or your funny cat picture collection won't change from today to tomorrow. But what if a zombie apocalypse started overnight? Or if you had to move to another country without ability to return, or if won a lottery, or had a major accident, or became a Scientologist, or whatnot? Suddenly all of this will be so much less relevant because reality will be so different from reality today, even if very little time has passed.

Now I admit, this assumption is unusual, but it doesn't generate Hyperbolic Discounting on its own. If each aspect of your personality and of reality had about the same chance to change every day, you'd still get Exponential Discounting.
• Assumption 2: Different aspects of your personality and of reality chance at different pace.
OK, this is totally obvious, but changing same pace to different pace assumption moves us from Exponential Discounting to Hyperbolic Discounting. Now technically you need some very particular and difficult to defend ratios to get exactly Hyperbolic Discounting, but you get Hyperbolic Discounting-like by just about any sensible choice of paces.

Let's say an aspect of your personality can have a half-life of 1, 2, 4, 8, etc. years with chance of 1/2, 1/4, 1/8, 1/16 that particular aspect has such half-life. This makes you value your 1.7 year removed self at 50%, but you get to 25% valuation at 4.1 years, and 10% valuation only at 10.5 years. That's not exactly Hyperbolic Discounting (it would predict 25% valuation at 6.8 fears and 10% valuation at 15.3 years), but we just made up pretty much the distribution we could come up with, and its behaviour is already far closer to Hyperbolic than to Exponential Discounting.

For continuous distribution of such half-lives you can assign numbers that give precisely Hyperbolic Discounting. I'm not going to bore you with the math - I'll leave that as an exercise to the reader. (future me will post the math if there's an overwhelming demand)

In any case, nobody really believes humans have exactly Hyperbolic Discounting. In fact both pace of chance of your personality and your empirically measured discount rates will depend on your age and circumstances in fairly predictable matter - just as model presented here would suggest, there is no "one true discount rate". To get exact shapes of people's discount curves you'd need detailed surveys and actuarial tables. Hyperbolic Discounting is basically the mathematically simplest formula that is gives decent approximation of how everybody actually discounts the future, and lacking highly precise and individualized measurements we can just as well use its simple formula.

So now you know of at least one model why people aren't all wrong.