Know what you are optimising for

Seth Godin gave a math puzzle. I know! I’m shocked too! I’d have to plagiarise a bit, since the puzzle fills more than 50% of his article. I’ll take the minimum that still makes sense. Here it goes:

Let’s say your goal is to reduce gasoline consumption.

And let’s say there are only two kinds of cars in the world. Half of them are Suburbans that get 10 miles to the gallon and half are Priuses that get 50.

If we assume that all the cars drive the same number of miles, which would be a better investment:

• Get new tires for all the Suburbans and increase their mileage a bit to 13 miles per gallon.
• Replace all the Priuses and rewire them to get 100 miles per gallon (doubling their average!)

My first answer was the second scenario. It’s wrong. The first scenario is the better investment. 2 of Seth’s readers had provided their own explanations (see Charlie‘s and Nariman‘s explanations).

Charlie gave a concrete example with calculations. Nariman distilled the question into math symbols. *smile* Both explained the answer excellently. I’m going to borrow on Nariman’s math workings and continue from there. You might want to read both explanations first.

So, following up on Nariman’s math calculations, we have

Let m be number of miles driven by a car…
Let s be the gas consumption (in gallons) for Suburbans (= m/10)
Let p be the gas consumption (in gallons) for Priuses (= m/50)
Let T be the total consumption (in gallons) (= s + p = m/10 + m/50 = 6m/50 = 0.12/m)

Now, Charlie used a “magic number” to start, 1300 miles. We’ll use that. Without loss of generality, we’ll examine only 1 Suburban and 1 Prius (since we’re talking about 50% existence for each).

In the 1st scenario, the total gasoline consumption is
1300/13 + 1300/50
= 100 + 26
= 126 gallons

In the 2nd scenario, the total gasoline consumption is
1300/10 + 1300/100
= 130 + 13
= 143 gallons

So with simple numbers, it’s easy to see that the 1st scenario is better. But can we make the 2nd scenario better? We doubled the mileage of a Prius and it’s still not good enough. How much do we need to improve the mileage before it becomes comparable?

Let h be the mileage such that the 2nd scenario is comparable.
So for the 2nd scenario, it becomes
1300/10 + 1300/h
= 130 + 1300/h

Here’s where it gets interesting. Let h go to infinity. The expression
130 + 1300/h
goes to 130 (because 1300/h goes to zero),
which is still more than 126 (from the 1st scenario).

This means, even if the Prius can travel all the way to Alpha Centauri and back a gazillion times, and then run a bajillion laps on the circumference of the universe, all on just a drop of oil, the 1st scenario is still better!

The misdirection

I’m guessing your first answer is also that the 2nd scenario is better. The reason why it’s wrong is because we were optimising for the wrong thing. The very first statement is

Let’s say your goal is to reduce gasoline consumption.

We were supposed to minimise gasoline consumption. But when the question came up, the term “mileage” appeared and took centre stage. And subsequently wrangled our minds to forget about what we were trying to do, and coerced us to maximise mileage instead.

We were solving the wrong problem.

Parting thoughts

Improving something that’s fairly good (a 50 miles per gallon Prius) is harder than improving something that’s fairly terrible (a 10 miles per gallon Suburban). Individually speaking, you should go ahead and improve the Prius (it was a 100% improvement!). But taken together, you should be improving the weakest link. In this case, the Suburban.