In Defense of Ambiguous Problems

Chris_Leong

Yesterday, I posted the Curious Prisoner Puzzle. The puzzle was structured to attempt to fool people into answering 1/2, which is possible, but certainly not unambiguously true. Any answer has to acknowledge the ambiguity, but at least in my opinion, the best way to reinterpret the problem to ensure that there is a single, unambiguous answer is as follows: Assume that you are told the statement when it is true and nothing is otherwise. Then, the answer is 1/3. Well done to all the commentators who quickly managed to figure all this out.

So why did I post an ambiguous problem?:

If the point of these puzzles is to teach a lesson, dodging an incorrect answer suffices. I actually couldn't see a way of making the problem unambiguous without giving the game away with regarding the answer not being 1/2
You can't assume that every problem you'll try to solve will turn out unambiguous (particularly if you study philosophy), so it's useful to be able to recognise when this is the case. If you always practise with unambiguous problems, then you are biasing yourself to assume problems are unambiguous, which is why people often struggle with this
This problem demonstrates the importance of counterfactuals in terms of probability. In fact, an ambiguous problem is probably the best way to demonstrate that such problems are ambiguous without defining the counterfactuals. In particular, it shines a light on problems like the classic, "What is the chance that two kids were both born on the same day of the week if at least one of them was born on a Tuesday?"
Further ambiguous problems can reveal the implicit assumptions made elsewhere. Suppose we have are four boxes: red, green, blue and white. One is picked at random to contain a prize, then the host peeks and tells you that it isn't in the white box. What is the chance that it is in the blue box? This problem is mathematically the same, yet we generally don't say that it is ambiguous even though the counterfactual hasn't been specified. That's probably because the different natural interpretations of the host's algorithm all give the same answer of 1/3, but this is an important realisation because it affects a massive proportion of probability problems:

The host randomly picks a color that doesn't contain the prize and tells you that it isn't in that box
The host randomly pick a color and tells you if that box contains or doesn't contain the prize.
The host always tells you if the white box contains or doesn't contain the prize.
If the white box doesn't contain the prize host tells you that the white box doesn't contain it. They further tell you that they were always going to tell you if this was the case and that they weren't going to say anything otherwise.
As per the previous, but with a random box

I see that someone posted in the other thread that they though the most obvious answer is 1/2, but why is this the case? I don't see any obvious intuitive argument for why 1/2 is a reasonable answer.

Edit: I guess the idea is to just not perform any update on the statement the guard makes but just use it to infer that "Vulcan Mountain" is equivalent to "Vulcan", and then answer based on the fact that the latter probability is 1/2.

I think people reason as follows:

They figure that there is a 50% chance of being on Vulcan.

They then look at the following statement: "If you are on Vulcan, then you are on the mountain" and redistribute all of the Vulcan probability to the Vulcan Mountain. The mistake is not realising that the constraint of 50% chance of being on Vulcan only applies at the start and isn't necessarily maintained. But this feels weird because the statement seems to be limiting its scope to Vulcan; or at least until you start looking at the counterfactual.

This. The phrasing "if you are on Vulcan, then you are on the mountain" *sounds* like it should be orthogonal to, and therefore gives no new information on and cannot affect the probability of, your being on Vulcan.

This is quite false, as can be shown easily by the statement "if you are on Vulcan, then false". But it is a line of reasoning I can see being tempting.

1/2 is a reasonable answer because it was decided by a coinflip, and no new information has been given. For some possible behaviors of the guard, this is true - say "the guard flipped a coin, and made a statement about vulcan or earth based on that coin". For other possible behaviors ("the guard equiprobably picked one of the three non-true options to eliminate"), information _WAS_ revealed, by the unequal planetary weighting of possible guard statements. For still different others ("the guard eliminates the vulcan desert if you are elsewhere, and says nothing if you are there"), the information revelation is also in the distribution of making a statement or not.

I have no clue how a prisoner can distinguish between these guard-information mechanisms. Intentionally ambiguous situations are ambiguous, I guess.

Side-question: , was there a lesson or point to the less-direct phrasing of "if you are on Vulcan, you're in the mountain"? It's exactly equivalent but harder to read (for me) than "you are not in the Vulcan desert". Was this part of your intent to induce 1/2 as the instinctive answer, or just an accident of storytell

It was intended to push you towards the 1/2 answer. If I wrote: you are not in the Vulcan desert" many people would immediately reason that I've eliminated 1/4 possibilities leaving 3 left for a 1/3 chance. With the current formulation, many people won't even realise that they are equivalent.