# The linear-scaling error

Say your first car got 10 mpg, and you replaced it with a 20 mpg car.  Now you’re ready to get another car.  How many mpg will your new car need to get, to be as much of an improvement over your last car (gas-wise), as that car was over your first car?

A recent Science article, summarized here, reports on this as an instance of a simple yet subtle bias: When given information, people assume that the effects relevant to them scale linearly with the measurement scale used. In this instance, it’s miles per gallon.

If this were wartime, and you were rationed 10 gallons per week, the measurement of interest to you in evaluating a car’s mileage might be the number of different places you could visit once a week with that car.  Then the relevant statistic would be (miles/gallon)2.  But since we aren’t rationing gas, a better measurement is gallons per mile, which can be translated into dollars and environmental impact per mile.

When people are given figures in miles per gallon, they usually think that the answer to the above question is 30 mpg.  "Sixty percent of participants ordered the pairs according to linear improvement and 1% according to actual improvement. A third strategy, proportional improvement, was used by 10% of participants."  (The proportional strategy says that the answer is 40 mpg.)

People get the right answer when you rephrase the question in units that scale linearly with the effect.  Try this:  Your first car could go 100 miles on 10 gallons of gas.  Your second car could go 100 miles on 5 gallons of gas.  Your third car needs to go 100 miles on… 0 gallons of gas.  So it needs to get infinite mpg, to match the improvement in going from 10 to 20 mpg.

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• http://blog.greenideas.com botogol

In Europe the unit of measurement is km per litre.
So perhaps some clever data analysis could tease out whether this choice of unit does, indeed, influence behaviour.

• ben

I’m slow, isn’t it still a linear bias to assume that the 3rd car needs to go 100 miles on 0 gallons? What if the improvement from the 1st to the 2nd was .5x? Then the third car needs to go 100 miles on 2.5 gallons. What am I missing?

• http://uncommon-priors.com Paul Gowder

Why is that the standard? Seems to me that “as much of an improvement” is ambiguous between the various senses that people pick.

• Doug S.

Your first car could go 100 miles on 10 gallons of gas.
Your second car could go 200 miles on 10 gallons of gas.
Your third car could go infinite miles on 10 gallons of gas?

/me shrugs

The question posed in this post seems to fail to confine the solution space to one correct answer.

• http://profile.typekey.com/sentience/ Eliezer Yudkowsky

To the extent that people have particular places they want to get to and routinely drive there, it is better in terms of the environment, or oil imports, etc., to upgrade some cars from 10mpg to 20mpg than to upgrade the same number of cars from 20mpg to 500mpg.

• billswift

Shouldn’t this post be titled “Stupid Math Tricks”?

• http://profile.typekey.com/aroneus/ Aron

How about the title be ‘Something we can all agree on’?

• Constant

I would title this, “Scale matters”, or, “Scale blindness,” since people tend not to realize that the choice of scale has such an effect, and so they tend to project on the basis of whatever scale they’re presented with.

• jmm

In Europe, Canada, and Japan, fuel efficiency is already reported either as liters/km, or liters/100 km.

How many mpg will your new car need to get, to be as much of an improvement over your last car (gas-wise), as that car was over your first car?

From the 1st to the 2nd, “Gas wise” you have a 50% reduction in usage. To get another “gas wise” 50% reduction your car needs to go 40 mpg.

The proportional strategy says that the answer is 40 mpg

The proportional strategy is correct.

• http://shagbark.livejournal.com Phil Goetz

Where I said “as much improvement”, say “reduces the amount of gas you use as much”. If going from 10mpg to 20mpg reduces the amount of gas you use by X gallons per year, you need to go from 20mpg to infinite mpg to again reduce the amount of gallons you use by X gallons per year (supposing you still go to the same places).

You’d still fail. The revision could still be read as percentage. You would need the question to read “reduce the number of gallons used on a journey as much”. You must specify the unit.

• http://shagbark.livejournal.com Phil Goetz

“You’d still fail. The revision could still be read as percentage. You would need the question to read “reduce the number of gallons used on a journey as much”. You must specify the unit.”

No; you just need to specify that you are asking about absolute, not relative, reduction.

Strange that most of the commentators have a bias to interpret “improvement” as “percent improvement”. As if, improving something by 50%, and then improving it by 50% again, were both the same amount of improvement. Is this a valid semantic interpretation, overexposure to advertisements, or just bad math?

• http://profile.typekey.com/aroneus/ Aron

I’d say it is semantics. “Equal Improvement” used without any other context could be measured in linear or logarithmic units.

That is, you can say

improved by 10%, or improved by 10 mpg. Both make sense. Both are improved. And since you can say A was improved by 10% over B, and C was improved by 10% over D, you can also say they were improved by the same amount.

It’s just that your interpretation is not useful for the real world.

To demonstrate, let’s go the other direction; You have \$10,000 and buy a 1000 shares of stock XYZ. It doubles in time [T] to 20 and you sell for \$20,000.

You buy 1000 shares of another stock for \$20 per share – \$20,000. How much would the 2nd stock have to improve in time [T} to be “just as great of an investment”.

I would submit that the stock has to double; and that this is the proper best interpretation of this question.

Based on what you put forth as the correct answer to the OP problem, the answer would be “the stock has to go to \$30”. Which is an interpretation, but a much weaker one.

However, If we specify Units in this example and say “what does the stock have to go to equal the number of dollars you made on the first investment” then of course your answer would be correct.

You need to overcome your Bias. ðŸ™‚

In fact, I would submit that you made a Linear scaling error in your OP when you submitted the answer was infinite. You still had a bias of linear over logarithmic answer, but used the linear scale of gallons rather than mpg. Unless a unit is specified, the most useful real world answer in the OP is 40; although infinite is also an answer, but a weaker one.

30 is not an acceptable answer because it the OP does specify “in gas”.

• John Maxwell

My first introduction to this type of problem was as follows: A car drives a one mile track at 30 miles per hour. How fast does it does it have to go around the track again to average 60 miles per hour? This formulation doesn’t have these interpretation problems.

• Sigivald

As Aron said, when you say “improve as much”, it is deeply unclear that you mean “save as many gallons of fuel per fixed distance traveled”, rather than “reduce consumption per unit traveled by the same proportion”.

I cannot speak for others, but I don’t travel a fixed distance at all times and care about how many gallons of fuel I use to do so. I travel varying distances, and concern myself with (roughly) cost per mile.

That being proportional to consumption per mile, if the first change reduced it by half, when someone asks “how can I get that same improvement again?” nobody is going to think “how can I make it cost nothing?” – the reasonable interpretation is “how can I halve the cost again?”.

(The worst part is that this confusion is masking the utility of gallons-per-mile/cost-per-mile as a more useful measure than miles-per-gallon.)

• Doug S.

(Usually, the easiest way to solve a problem like the one John Maxwell posed and get it right is to just grind through the algebra.)

Average speed = total distance / total time
Total distance = 2 miles
Total time = 1 mile / (30 miles/hour) + 1 mile / (X miles / hour)
60 miles/hour = 2 miles / [(1/30 hours) + (1/X miles / hour)]
[(1 / 30 hours) + (1/X / hours)] = 1/30 hours
1/X hours = 0 hours

In order to go around the track once more and end up with an average of 60 miles/hour over the entire trip, the car must go around the track in zero time, meaning the car must go infinitely fast.

So, yeah. d/dx(1/x) = -1/x^2, so the more fuel efficient a vehicle is, the bigger the improvement in fuel efficiency you need before you reduce fuel consumption by the same amount. This suggests that the most inefficient vehicles may account for a surprisingly large share of total fuel consumption.

And now back to playing World of Warcraft.

• Anonymous

Your first car could go 100 miles on 10 gallons of gas. Your second car could go 100 miles on 5 gallons of gas. Your third car needs to go 100 miles on… 0 gallons of gas.

I’ll take the fourth car.

• http://shagbark.livejournal.com Phil Goetz

The original article says, “They were asked to rank-order five pairs of old and new vehicles in order of “their benefit to the environment”.” It would not be reasonable to rank benefits to the environment by percent improvement.

So the bias exists, even if you find fault with how I restated it.

• http://goodmorningeconomics.wordpress.com jsalvati

And the benefit most people think about is probably “cost of gasoline” and since gasoline demand is relatively inelastic (and people know this) so the cost of gasoline is roughly proportional to gallons/mile

• Kyle

Nothing bothers me more than studies with ambiguous questions and smarmy conclusions about how people are irrational, biased, or uniformed. The question John Maxwell mentioned is a favorite of mine and avoids the ambiguity problems.

Plus, the standard is a weird way to define improvement. In order to match the improvement from 10 mpg to 30 mpg you need to generate gas by driving? Unless the question spells out that standard, it doesn’t make a lot of sense to assume that it’s the right one. What’s wrong with “saves 50% of gas” or “doubles fuel efficiency” as a measure of improvement (gas-wise)?

• http://shagbark.livejournal.com Phil Goetz

Most people believe that, when you go to a pot-luck dinner, you should more food when you expect more people to be there. (I’ve asked.) I wonder if that’s a related bias.

• http://shagbark.livejournal.com Phil Goetz

Er, you should bring more food.

• James

Actually, the more people at a potluck, the more you should bring, IF you expect your dish to be significantly more popular than average. If you expect your dish to be unpopular, you should bring less when there are more people. (Or more realistically, you should bring the same amount, to signal a sense of fairness, and expect to take more of it home again.)

• Felix

I thought you meant “eat more food”. And that’s not irrational, since you can get away with it more easily.

“So it needs to get infinite mpg, to match the improvement in going from 10 to 20 mpg”

Maybe people don’t give this answer because they have some kind of reality bias?

• Ben Jones

In Europe the unit of measurement is km per litre.

Botogol, jmm, in the UK we also talk about miles per gallon. Yes, the majority of the mainland is firmly metric, but we’re not the 51st state just yet. Perhaps you were referring to Yurrp?

• The Sheep Nazi

Where I said “as much improvement”, say “reduces the amount of gas you use as much”.

Not to flog a dead gas can here, but: your readers who got the right answer (40mpg, or two-and-a-half gallons) did just exactly that. They reduced the amount of gas you needed by half, both times. I think you’ve demonstrated something a little bit different from what you set out to: people actually do fine with geometric progressions for the first few terms. They might have no idea what thirty years at 6% compounds to, but they can handle 1, 1/2, 1/4 pretty well.

• http://www.cawtech.freeserve.co.uk Alan Crowe

The comments take miles per gallon versus gallons per mile as the only example where scale influences perception, but the issue is more general than that. The parameterization of tax rates provides a second example, with different underlying mathematics.

• billswift

Felix, thanks, “Maybe people don’t give this answer because they have some kind of reality bias?”

That’s basically what I was trying to say with my “Stupid Math Tricks” comment. When you are applying math to the real world, and you start getting zero or infinities for an answer, the more rational conclusion is that you’ve made some sort of mistake, not that everybody else is making one.

• Daniel

you said “improvement over your last car (gas-wise)” which is vague.
If you had said “same financial improvement to keep up the exact need for the car” or something of the sort I can see how it might be as you state. We need to look at the original article to see what was meant