Who are the relevant experts?

The Pig that Wants to be Eaten

#34: Don't Blame Me

In which an expert is not an expert, except when they are.

Who is ultimately responsible for a person's actions? Your instinctive answer is probably that a person themselves is generally responsible for their own actions. This is correct. End of blog post.

...not quite.

Because people don't operate in a vacuum. Our decisions are influenced by events and people around us, and sometimes these factors are out of our control.

In particular, sometimes we obtain advice from others regarding the best course of action. If we then follow that advice, to what degree can we abdicate our responsibility for any negative consequences? For of course, people are happy to attribute positive consequences to their own behaviour.

Unfortunately, it's not as clear-cut as it first appears. For instance, if my company computer stops working and I ask a computer technician friend what to do, it seems reasonable that his advice should be followed. However if this advice only damages the computer further and the company is unhappy, is it fair to pin the blame on the friend rather than myself?

In some sense it is - my friend is a 'relevant' expert, which means that I ought to trust their opinion on this matter over my own. But relevance is relative. My friend might be an expert on computers, but not on giving advice about computers. Or he might not even really be an expert at all, and I'm just mistaken. So how are we to know how to apportion blame?

The key thing is that me choosing to ask my friend for advice and then choosing to follow it are my actions - I can't blame these on my friend. I can only blame him for the advise itself. If there's good reason to believe he is a relevant expert, then the burden of blame shifts more towards him, but it doesn't absolve me of blame completely.

So there's a spectrum of degree of responsibility, based on the suitability of the received advice. But determining the relevance of an expert is not necessarily straight-forward. For example, consider the following examples:

• "I'm dumping you, because my financial advisor told me to! You're bad for my bank balance!"
• "I'm dumping you, because my psychologist told me to! You're bad for my mental health!"
• "I'm dumping you, because my palm reader told me to! You're bad for my life line!"

Different people can have different opinions as to the degree of relative responsibility that can be placed on the dumper in each scenario.

So what can we conclude? Some degree of responsibility must lie with the person who performs an action. But maybe if they were advised by another to act in a certain way, we can't lay all the blame on them.

Up next: A Turing test

To heap, or not to heap?

The Pig that Wants to be Eaten

#94: The Sorites Tax

In which a heap is not a heap, except when it is.

Does increasing the tax level by 0.01% leave anyone worse off financially? Of course it does - you will have less money in your next paycheck. But will you be substantially worse off? Unless you believe all taxation is theft, the answer is probably no - someone earning the NZ minimum hourly wage of $15.75/hour would pay an extra 5 cents/week in tax[note 1], which is unlikely to break the bank.

But if the government kept up the tax increases daily for 300 days, suddenly that person's paying an extra $19/week in tax. For some people, that really would have a substantial impact on their finances. So how can we reconcile this apparent incongruity?

That's easy, of course - many small changes can combine to make a large one. We see this in protest movements, in bird migration, in the pollution of the ocean with plastics - every individual member of a group performs a small action, but the sum total of these actions is something major.

The situation here is slightly different, however. And it's a tricky, but interesting difference.

This problem as stated above is a modern version of one called the Sorites paradox. In the original problem, the question concerns a pile of sand grains and the definition of the word 'heap', and rests of the following principles:

      1a) A single grain of sand is not a 'heap'.
      2a) Removing a single grain of a heap from a 'heap' will keep it being a heap [or alternatively,  adding a grain of sand to a 'non-heap' will keep it a 'non-heap'].

Both of these principles seem valid on the face of it - but their co-existence is contradictory! They partition the world of sand piles into two groups: 'heap' and 'non-heap', with no amount of adding or removing grains able to transfer a pile from one group into another. But this is clearly absurd.

There are ways around this, of course. The easiest way is to reject principle 1a) - to say that all piles of sand with 1 or more grain are to be considered 'heaps'. This avoids the contradiction.

However, this is rather disingenuous - it flies in the face of the regular definition of the word 'heap'. Furthermore, it is equivalent to asserting that a tax increase of 0.01% does have a substantial effect on finances. But unlike the heap problem, the tax one can be diluted. Is it fair to argue that a tax increase of 0.001% has a substantial effect? 0.0001%? One-one millionth of a percent? One-one billionth? To claim these all have a substantial effect is logically consistent - but linguistically unacceptable.

There is another way around this, and it rests in principle 2a) - specifically, what I've written between the square brackets. It follows logically, of course - unless you consider the adding or removal of grains to a pile to itself be part of the pile and thus its 'heap-ness'.

For example, it we disregard the section between the square brackets, we could imagine a pile A, which is not considered a heap. We add a grain of sand, making a new pile B, which we do consider a heap. Now we remove a grain of sand from pile B, making a new pile C. By principle 2a) we must consider this to be a heap too. So what is the difference (if any) between piles A and C?

Well, there is no intrinsic difference. They contain the same number of sand grains; although the exact grains used and the arrangement of them may differ, they are essentially identical. However, there is one key distinguishing property - the manner of their creation. Pile C was created from a heap - perhaps this is sufficient to distinguish it from pile A?

A casual observer viewing piles A and C on a table would not be able to distinguish between them. So should only one of them be a heap? Again, this seems linguistically unacceptable. The corresponding argument in the case of taxation, that a particular total tax rate increase would only be substantial if it were slightly less than another substantial one, but not if it were simply raised to that level, is clearly absurd. We would have to conclude that (for example) a 2% tax rise is not substantial if done on its own, but a rise to 2.1% and then a drop to 2% relative to the original would be substantial.

But perhaps a heap of sand is a very specific measurement! Perhaps 10,000 grains make a heap[note 2], and not a single grain less. This defies principle 2a) as well, but it does it in a more linguistically appealing way. Perhaps we've found a solution after all?

In the case of the heap, this solution is fine, as long as we modify principle 2a) as follows:

2b)  Removing a single grain from a 'heap' will keep it a heap, unless it is a heap of 10,000 grains, in which case removing a grain will turn it into a 'not-heap'.

This lets us determine that piles with 10,000+ grains of sand are heaps, while others are not.

The biggest problem with this is that the determination of the precise number of grains in a pile is unlikely to have widespread agreement. This makes the word 'heap' rather useless if its job is to communicate meaning between people. But if you don't mind that either, then you've solved the heap problem!

Okay, I lied. That's the smallest problem. The biggest problem is a lot more subtle, and a lot more intractable.

Let's stop talking about heaps and start talking about dominoes for a second. Suppose you set up a row of dominoes, such that the following rules apply:

      1c) The first domino will be knocked over.
      2c) A domino that is knocked over will also knock over the next domino in the row.

It's easy to see that under these conditions all the dominoes will be knocked over[note 3]. Of course, that rests entirely on the guarantee provided by rule 2 that a domino will always knock over the next domino in the row. If this fails even once, then we can't conclude that all dominoes will fall.

The problem with applying this to the tax scenario is that in the heap scenario, all grains of sand are equal. However, this is not the case with tax; different people pay different amounts of tax based on their income levels.

And the argument is as follows. Suppose you earn $Y and pay $X on tax. This leaves you with $(Y-X) to actually spend. Now if taxes are increased by 0.01%, then X will increase slightly, so you will have slightly less take-home income. But now you'll have just as much take-home income as someone who earns slightly less than you did before the tax was increased. Since for them the 0.01% tax increase was basically negligible, it will be for you as well, since you're now in that position. So another 0.01% tax increase won't affect you.

And now we see that the argument proceeds like the dominoes above - each increase in tax must be negligible, and (the key part) it was already negligible for someone else earlier. So there's no way to say that "10,000 grains is a heap" - a tax increase of 1% for you is just as negligible as a tax increase of 0.99% for someone earning a bit less than you [where negligibility is measured by the volume of reduction in take-home income]. But we agreed originally that a 0.01% tax increase is negligible for everyone - we've run into logical trouble!

Because we can run the same argument in reverse - if we keep increasing your tax by 0.01% a day, then eventually you'll be paying a substantial amount more in tax. But on whatever day that occurs, someone who earns slightly more than you will have hit that limit the previous day. And you can continue that chain of reasoning back to conclude that even on the first day the tax increase was substantial for someone, even though that's absurd.

So we have the paradox. The usual way we understand the concepts of 'heap', and 'substantial', are shown to lead to impossible conclusions in this scenario. But relaxing our definitions and allowing more vagueness doesn't allow us to handwave all our problems away - because the vaguer our terms get, the less useful they are as words at all! So we are left scratching our heads.

Next up: The blame game...

Note 1: My maths might not check out, but the point stands that the increase is negligible.

Note 2: Archimedes called the number 10,000 a 'myriad', which is a much cooler word than 'heap'.

Note 3: If there's at least one domino that isn't knocked over, then we'll consider the earliest domino that isn't knocked over (i.e. the domino closest to the first domino that isn't knocked over). In particular, this can't be the first domino (impossible under rule 1), so it must have had a domino before it in the row - and this domino must have been knocked over (since our domino of interest was the earliest in the row to not be) and so by rule 2 our domino would have been knocked over too. This means such a domino must not exist.