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?

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

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.

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.

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.

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?

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.)

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.

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)?

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

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.

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.

(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 timeTotal distance = 2 milesTotal 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 hours1/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.

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.)

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.

## The linear-scaling error

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

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.

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.

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.

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?

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?

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.)

Er, you should bring more food.

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.

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)?

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

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.

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.

(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 timeTotal distance = 2 milesTotal 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 hours1/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.

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.)

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.