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What number is halfway between 1 and 9? (web.mit.edu)
189 points by aurelianito on Oct 8, 2012 | hide | past | favorite | 74 comments

According to the research discussed in this article, when the intensity of some sensory stimuli increases exponentially, our minds somehow unconsciously perceive the increase as linear. Our perception, in other words, is often logarithmic.

What I find most fascinating about this is that our conscious minds are not very good at thinking about exponential growth. My favorite example is that of the twin brothers who invest with annual gains of 12% each: one twin saves $2,000 from age 19 to 26 (a total of $16,000) and then stops saving; the other twin doesn't start saving until age 27 but then puts away $2,000 every year until age 65 (a total of $78,000). Do the math, and you will see that the savings of the second twin, growing at 12% every year, never catch up with those of the first one.

Our perception might often be logarithmic, and we might understand the math of exponentiation, but our brains are not very good at 'seeing' exponential growth!

Modern humans no more need to appreciate the economics of consistent 12% gains than they need to know how to fend off a polar bear, or how to date three supermodels on a modest budget--it just ain't gonna happen.

When I first learned about this investment problem (although, my version had 18-30 and 30-65), I worked out the math, and I think it held for as low as 2% interest. I could be remembering wrong though... Surely, someone's done a formal write up on this?

One way to simplify the problem is to figure out how much interest income the "early investor" is making at the moment the "late investor" kicks into gear. Obviously if Jane invests from age 16-20 and is now rolling $4,000/yr interest income back into the mix on her 21st birthday, John's yearly contribution of anything less than $4,000 ain't gonna cut it.

The Rule of 72 http://en.wikipedia.org/wiki/Rule_of_72 is a really handy tool the layperson can use to get a grip on how long it takes for compounding increases (which is most increases we deal with day to day--taxes, population, traffic) to result in doubling. It makes a much greater impact to say "this town's population will double in 10 years" than it does to talk about 7% growth rates.

> My favorite example is...

This exact thing caught me out a few weeks ago, it still boggles my mind. I started a pension recently and I was given examples of how much I would be worth if I started as soon as possible (my 21st birthday) or if I waited until I was 22, 23, 24... if my total monthly contributions are $2,000 and I start on my 21st birthday I will be worth hundreds of thousands of dollars more than if I started at 23, or 25, but my brain thinks that is insane and even though I know it's true, I still can't get over it. The difference between 21 and 23 is $160,000!

going even more on a tangent, it's really a great shame this sort of thing isn't taught in schools. It seems obvious now after being taught that savings can vary so much this way with so little time, but until I was explained it I had no idea. I figured waiting until 25 or 30 to start a pension made most sense!

Those estimates are often wrong though, because the idea of annual 12% returns is complete fiction. In fact the idea of any particular percentage annually is fiction. For instance, I only recently started a pension, however if you look at the stock market history (for simplicity), had I started in 2000, the net gain I could have expected was almost precisely zero, if not negative. If I could have found something that paid 12% from 2000 to 2012, sure, I would have missed out on a lot, but there in fact not only was no such beast (in the broad sense), there was also a lot of ways to experience negative gain.

This is one of my favorite visualizations ever: http://www.nytimes.com/interactive/2011/01/02/business/20110... Be sure to carefully read what it is actually visualizing; I've posted this a couple times and people often knee-jerk a reaction to it based on a complete misreading of the chart.

In the context of this discussion (and indeed as reflects how most people invest), wouldn't it be more relevant to see the average rate of return assuming regular (inflation or income growth adjusted) annual investments?

Exactly. Depending on volatility you could get a positive return investing a set amount in a stock regularly, even if the stock started and finished at the exact same price.

In addition some stocks pay dividends; reinvesting these can also improve rate of return beyond what would be expected from just looking at the stock price.

Interesting chart.

Strange, at one point it says "High inlfation led to negative returns". When money is worth less, then you get more money when you sell something. In this case your share of a company. So why would the returns of investing in the S&P be affected by inflation?

Because the nature of inflation is to devalue money. For example, let's say you put $10k under your bed to buy a car for your kid when they turn 18, and let's say inflation was at 2% for those 18 years.

Now the 18 years have speed buy and you go grab your very old stack of $10k from under the bed frame to celebrate your kid leaving the house. When you get to the car dealership, you find the equivalent car you could have purchased now costs $14,282. Your money has lost almost a third of it's value just by inflation.

Inflation affects the number that you use to value the worth of a company.

Say you buy one share at $10 of a company with a billion outstanding shares. That company is worth, on the market, $10 billion.

Let's say that next year, the company's share price is still $10. Let's also say that inflation in that period was 4%. The company is still worth $10 billion, but each dollar is worth 4% less. The company's total value has dropped.

Inflation didn't cause the total value to drop. The company's value dropped for other reasons (bad sales, company president went crazy, market jitters, whatever causes these things) and inflation simply moved the numbers a bit in the opposite direction.

In short: inflation doesn't affect your returns, but it affects the numbers you use to measure them.

You say "In short: inflation doesn't affect your returns". And thats what I said. But the linked article says the opposite: "High inlfation led to negative returns".

Inflation doesn't inflate everything at the same rate. In 1970, the DJIA was $809. 10 years later, it was $824. Inflation was more than 1%.

Starting early is only half the battle. The other half is the rate of return. If you start up a retirement account at age 17 with $2000 in it, but you just put it in a generic savings account, by age 37 you'll have . . . $2209.

People encouraging you to save often give unrealistically high rates of return. 12% is unrealistic for any investment vehicle I know of (though people sometimes say it for the stock market -- but they're crazy optimistic).

Here's a spreadsheet to show you the impact of both rate of return and time:


And of course, that doesn't account for the risk of the investment -- which, as a lot of people discovered in 2008, is a real, actual thing.

Indeed. These days, money in a savings account has a negative real rate of return.

That depends 100% on your timeframe. Start investing in March '09 and your money would have doubled in less than three and a half years.

"Savings account."

The contribution from your first year is earning interest for N years, so contributes (1 + R/100)^N (assuming fixed rate R, which is a fiction. That could be a lot of dosh...

You are not alone: www.youtube.com/watch?v=F-QA2rkpBSY

OTOH, sometimes our perception is multiplicative when it should be relative: charge me $100 extra on a house, I probably won't even notice; charge me $100 extra on a bar of chocolate, there's no way I'm going to buy it. But either way I lose $100!

You only buy a house once, you buy chocolate bars all the time. $100 on a chocolate bar every time compounds.

One-time coupon for $.50 off a chocolate bar: probably going to take the time to clip it. One-time coupon for $.50 off a laptop: probably won't bother.

You can find silly faults with individual examples all you want, but studies indicate that people have a tendency to think percentage-based about savings rather than absolute terms.

The $.50 off a chocolate bar may actually cost you money, since it will encourage you to buy the chocolate bar when you weren't planning on buying one at all. In fact worse than that, it may start a habbit of buying those chocolate bars.

>You can find silly faults with individual examples all you want, but studies indicate that people have a tendency to think percentage-based about savings rather than absolute terms.

Which sounds like the smart way to go about it.

It seems a little illogical in the chocolate vs laptop $.50 off voucher, but it too makes sense. The laptop voucher provides almost no value to you, considering you are about to spent $700 dollars on the thing anyway. $.50 could be just the gas to go to the store to buy it.

$.50 for the chocolate on the other hand, will be taken only by cash strapped people (or coupon maniacs).

I doubt people with $700 at the ready to buy a laptop would at the same time clip a $.50 coupon for a chocolate bar.

The $.50 is $.50 regardless of what the voucher is for. It provides exactly $.50 no matter what. Sure, maybe someone with $700 at the ready for a laptop won't be the kind to be clipping $.50 coupons for anything but the same tendency to consider percentages rather than absolute values remains regardless of actual numbers. People will pat themselves on the back for saving $10 on a $50 garment but won't bother going out of their way to save $100 on a $20000 car.

>People will pat themselves on the back for saving $10 on a $50 garment but won't bother going out of their way to save $100 on a $20000 car.

Which also makes sense. $100,000 to me is serious money. To Bill Gates it's small change. Money is a relative - percentage thing.

A $20,000 car is a purchase you make once in 5-10 years. Saving $100 means nothing in that context --you're already parting with 200 times the amount, so if the $100 amount meant something for you, you'd have gone for a cheaper, say $15000, car in the first place...

Savings of $10 (20%) on small items, like food or garments, on the other hand, pile up.

You spend more than $20,000 on small items per year than you do on cars...

I would clip it.

It's still a valid observation though. People don't drive to another store to save 10 dollars on a 500 dollar couch but they will drive all the way across town to save 10 cents a gallon on gas.

This isn't rational: they're clearly not buying the 100 gallons of gas for the gas-drive to make sense.

Give me 30 years and 4% interest on a loan to pay off the chocolate bar and I probably wouldn't notice either.

Dan Arielly's "Predictably Irrational" (http://www.amazon.com/Predictably-Irrational-Revised-Expande...) is a great book that covers such phenomenon.

When the intensity of some sensory stimuli increases exponentially, our minds somehow unconsciously perceive the increase as linear

By necessity, really. If you've ever spent much time with electronic sensors, you may discover as I did that seemingly half of the work is making nonlinear input linear, because nonlinear input is a mess to compute with.

SW and HW for digital imaging had/have to deal with this quite a bit.

I think your example illustrates the surprises of compound growth well, but with a realistic growth of 4.1% per year, twin A ends up with a pot of $88k and twin B ends up with $183k. So your example provides food for thought but probably shouldn't form the basis for a retirement strategy.

> What I find most fascinating about this is that our conscious minds are not very good at thinking about exponential growth.

I thought about it too, and come up with idea that some equivalent of Weber-Fechner law [1] might be working at it this level. So I have made a small write-up about this idea that you might find interesting [2].

[1] http://en.wikipedia.org/wiki/Weber%E2%80%93Fechner_law [2] http://xion.org.pl/2012/04/11/self-reinforcement-and-exponen...

What kind of investment brings 12% every year, for 40 years without the risk of losing all my money? With the investment options I have now (today), I'm better of consuming my money.

This is why the second twin should put in $5000 every year :D

on this topic - this is a video worth watching http://www.youtube.com/watch?v=F-QA2rkpBSY

1.8 (harmonic mean)

Seriously though, I thought we already knew that human perception is mostly logarithmic (for example, the eye performs well across 9 magnitudes of brightness – a linear system doesn't make sense). On the other hand, human perception of future value has generally found to be discounted hyperbolically, which is illogical (http://en.wikipedia.org/wiki/Hyperbolic_discounting).

I wonder is there is a correspondence with the harmonic mean but I can't get my head around it at the moment.

The reason why it is usually considered irrational/illogical is because hyperbolic discounting leads to dynamically inconsistent preferences. For example, under hyperbolic discounting I may prefer $100 today to $150 in one month, but prefer $150 in 61 months to $100 in 60 months. It's the same rate of discounting in both cases ($50 for an additional month of delay) but my preferences are not consistent.

Of course, there are other criteria for rationality besides coherence/consistency (as the wikipedia article alludes to).

But if you view the payoff as not certain, $100 now is worth a lot more than $150 in a month - because you are here now with $100. The $150 is worth its normal discount rate multiplied by the strength of the probability that you will be here in a month with $150. The discount rate here is entirely due differences in risk of nonpayment - which are marginal when comparing 60 months to 61 months, but massive when comparing now to an hour from now.

According to criteria that would label hyperbolic discounting irrational or illogical, debt collectors are equally confused for not accepting the promise of a $100,000 payoff 50 years from now rather than a settling of the debt today.

That's because the traditional economic definition of "rational" is an incomplete model of real world sanity that only works in incomplete models of the real world.

In the real world, you have to deal with the uncertain risk that the offer will be unfulfilled due to external factors, which always increases in time relative to present. You, the entity offering you the choice, or the entire environment both exist in, may not still exist in 60 months.

If the same offer is made 60 months from now, you should make a different choice, because the risk of non-existence between time of choice and time of reward has changed.

If the amount of hyperbolic discounting is genetically determined, then is likely that we are still adjusting to the massive drop in risk of death over the past several centuries.

"But pose the same question to small children, or people living in some traditional societies, and they're likely to answer 3."

Citation needed! I mean, if "half" is not understood as mathematically half, how can traditional societies even have an economy?

It's not "half," though, it's "halfway between." This implies a mean, but which mean to use is not explicitly expressed. Choosing the arithmetic mean gives 5, choosing the geometric mean gives 3.

Probably because there is a difference in perception between half of an amount and half of a number.

Radiolab's Numbers episode covered this fascinating topic back in 2009: http://www.radiolab.org/2009/nov/30/

Where is the study that showed "But pose the same question to small children, or people living in some traditional societies, and they're likely to answer 3."

Without a source on this, it's a difficult article to take the article seriously. It offers no support for its most prominent claim.

"Hi, quick question: Which number is halfway between 100 and 10,000?"

"1000. No, wait..."

(I might actually say that)

Interesting... this has been something I've postulated for several years now. My reasoning had most to do with our vision system. As we perceive distances, the distance between something 10 feet, 100 feet, 1000 feet away, etc, becomes less about knowing the absolute difference and more about relative distances; someone is more likely to determine that something is 110 feet away than they are to realize something is 1010... but realizing something is 1100 rather than 1000 feet away will feel about as off as suggesting something 110 feet away is really 100. Our vision measurements are further reinforced as we observe parallax scrolling as we are driven around the country in our youth.

I've also felt that teaching a number system that is linear means that it is harder to understand the difference between 100 and 1000000000. Even many adults find it difficult to understand the magnitude of numbers when you approach a billion or a trillion. I think this is largely because we are taught to think in a relatively small range of the number line and perceiving numbers outside that range, we try to relate it back to what we were taught in schools.

I'd love to see schools teach a logarithmical number line instead of the linear system kids are taught today.

Perhaps the fact that human brains seem hardwired to perceive relative differences by orders of magnitude is an argument for graphing logarithmically by default, instead of linearly.

Musical scales are exponential as well. (octaves are frequency doublings) There's also this experimental evidence about sensing loudness, weight, vibration, lightness, saltiness, thickness and several dozen other things. From: http://en.wikipedia.org/wiki/Stevens%27_power_law ...

The general form of the law is

ψ(I) = kI^a

where I is the magnitude of the physical stimulus, ψ(I) is the psychophysical function relating to the subjective magnitude of the sensation evoked by the stimulus, a is an exponent that depends on the type of stimulation and k is a proportionality constant that depends on the type of stimulation and the units used.

Radiolab did a good piece about this. http://www.radiolab.org/2009/nov/30/innate-numbers/

What does it mean when you can't answer the question properly and said "4.5"?

You're either a math major or an engineer. We can't do simple math anymore. Please give it to us in the form of an integral.

Also, there's the ambiguity of "inclusive or exclusive? And, which side?"

[1,9], (1,9], [1,9) or (1,9)?

Those ambiguities make no difference to the answer. Answering 4.5 just means you forgot that starting a count from 0 gives a different total than starting from 1.

If you're using the set of integers, then [1,9) is equivalent to [1,8]; and that would have a different answer from 5 for "what the center is"; so, it does affect the answer.

[1,9) is not equivalent to [1,8]. The center of [1,9) is infinitesimally close to the center of [1,9]. Its exactly equal when you take the limit.

... I was using that nomenclature for the short hand of "inclusive/exclusive". 1 to 9 exclusive is 1 to 8 in the integer set. Sorry for the lack of clarity. Perhaps now we can get to my original point?

4.5 is halfway between 0 and 9. Halfway between 1 and 9 is 1+(9-1)/2 = 5

Nu-uh. By that math, halfway between 1 and 2 is.... 1?!

It only equals 1 if you are using ints ;) I do that more often than I care to admit, I blame programming.

his formula is correct. you made a mistake in the order of operations.

For anyone who hikes or goes outdoor, this might be obvious.

If you notice how we perceive distances, it's logarithmic too. It's really hard to judge accurately the time to arrive over long distances. Like the child's example on the OP, people often perceive 1/4 of the distance as being "halfway there". It's some kind of psychological phenomenon.

I'm wondering if somehow our predisposition to think logarithmically is related to Benford's law - http://en.wikipedia.org/wiki/Benfords_law.

I think it is related to Benford's law in that they are both a manifestation of the fact that a log scale is the most natural (most uninformative, highest entropy) distribution (or prior) for things that have magnitude (having a parameter that can't go into the negatives such as height or length or volume as opposed to a positional parameter which can be negative).

I do believe this is a property of the universe and not just a psychological phenomenon. Our mind here is simply following the natural distribution and Benford's law is just the observed manifestation of this pattern.

This type of pattern is not odd if you are believer in the Bayesian interpretation of probability theory. The shape of Benford's distribution follows the shape of a maximum entropy, most uninformative distributions for a value of magnitude (as I mentioned, positions do not follow benford's law).

Take street lengths for example. Assuming that they follow a log prior simply means that for a length of street L, if you pick another random street, you are as likely to pick a street within the length range L/2 to L than L to 2*L. From the original street length, to get a street twice as short, you need to subtract much less than you would have to add to get a street twice as long. That is why this distribution is not linear or rather it is linear on the multiplication and division operation, not on additions or subtractions.

If instead you'd assume that a street x meter longer is as likely as a street x meter shorter you would end up with impossible probabilities. For example, for a street of 1 km, a 3 kilometer street would be as likely as a -1km street? Even if you'd assume probabilities were equal for all lengths between 0 and infinity that would mean you think there are as many streets measuring a tredecillion billion km long as there are street 5 km long. This is simply not how things are sized in the universe. Smaller things are in greater numbers. Log priors are one of these areas where the math predicts the universe logically and the universe is mirrored by the math beautifully iff you do your calculations properly (using Bayesianity). It seems evolution has made our psychology reflect this reality.

I don't think Benford's law has anything to do with human psychology, it's more of mathematical property that applies to random sample from certain distributions.

I suspect only in the sense that both are related to http://en.wikipedia.org/wiki/Power_law.

i wish it had described how people were asked. i guess it's through some graphical means. anyone know (original paper is paywalled)?

That question only seems to appear in the press release - it may be a reference to an earlier study, but press releases don't tend to cite sources. The paper in question is theoretical, although it gives an example with the intensity of various human and animal sounds. (I have access through my university)

One thing not mentioned in the paper is also music. Each octave increases exponentially (each is twice the frequency of the previous). However our ears perceive the difference between notes as linear (it is actually linear if you take the log of the frequency). One theory is due to the shape of the inner ear, where the individual hairs that hear sounds each octave apart are spaced lineally.

Did anyone permit for the possibility that some people may be wired to have an affinity for one and some for the other?

Could this be related to the recent post about software estimates?

Software estimates are just another unit of distance. That's a valid application of this general idea: Halfway between a day and a month is a week.

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