Split the Difference

Here’s a problem for you:

My roommate and I manage our expenses like this: both people take care of their bills individually and at the end of the month, we split the difference.

This month, we did the exercise and he owed me 1500. But after a few days, we realised I had actually paid 9400 that was incorrectly marked as paid by him. So how much does the other person now owe me?

If you wish, take a moment to think about this before we continue. Done? Okay.

So, I can see two ways of thinking about this:

First, earlier I was owed 1500, but then since I also paid 9400, I am now owed 9400 and the original difference i.e. 1500.

Second, we can do a step-by-step calculation:

Now, the question is why is the difference you get $\frac{X - Y}{2}$?

If A spent X and B spent Y, total expenses will be $X + Y$ and when this is split equally, you get $\frac{X + Y}{2}$.

But A has already paid X and B has already paid Y. So what they get back is $\frac{X + Y}{2} - X$ or $\frac{X + Y}{2} - Y$ respectively.

If you simplify this, you get $\frac{\lvert X - Y \rvert}{2}$

And so, you can work out that the amount owed by the friend would be the same as calculated earlier.

There is also a third way, which like one of my friends who is more of a exam-taker than problem-solver would use.

He’d go like: the problem has two numbers mentioned, 1500 and 9400. So first look at the options of the question and rule out any option which is not a simple arithmetic operation of these and then, based on pure vibes (he would argue intuition), you choose one that is the addition of these two numbers because that is what “seems right”.

For now, if we disregard going by the vibes, how and why would someone chose one of the approaches above over the other?

When I was asked this question, I first thought of the first approach mentioned above. But then I was not sure, so I wrote down the problem to work it out.

As I was writing it down, I initially wrote the difference to be $X - Y$ and not $\frac{X - Y}{2}$. But it did not seem right.

So I thought about how would one actually calculate the share of each person and I understood how the $\frac{\lvert X - Y \rvert}{2}$ comes into the picture.

But now, after all this I had two questions. First, I understood how split the difference works in the context only by working it out. I was not originally clear about why splitting the difference actually works. But why do some people immediately know or identify this?

Possible reasons are: They are better at pattern recognition. They could have experienced situations like this in the past and they then internalised this idea that you split the difference. This could also have happened because they had to solve this problem many time before and they saw the pattern.

Or maybe this became clear to them when they zoomed out and were not looking at the details but the overall picture?

The second question I had was: where else have I seen this situation of choosing one approach over the other while reasoning things out?

When I’m learning something new, I usually start by going step-by-step and then once I have understood the idea, I may just apply it without having to think through it the next time I see a similar pattern.

But once that pattern is repeated enough, we may not always remember how that understanding developed in the first place. Like in a recent discussion with a colleague, I intuitively knew how a graph would change for a certain change of conditions, but I could not recall the reasoning behind that intuition because I had forgotten it. After the discussion, I worked it out though, but I know I might forget it again and that’s alright.

This situation also plays out badly in a lot in academic talks where on asked a question some speakers, being very familiar with their domains will respond with something they have internalised. But if someone asks them a follow-up requiering them to work out their earlier response, they might not be able to readily work it out and demonstrate their thinking. But they realise they are right, so they will usually provide a hand-waving argument instead of just saying “I can’t remember how to demonstrate it right now but I know that initial response works”.

In fact, some academics think that the hand-waving, throwing around jargon, and the obfuscation of your work in general is some sign of a genius. I hope this mindset is not widespread. I’m going to add this to the long list of problems I see in academia around me - something I’ll explore in more depth another time.