Wednesday, October 17, 2018

Yet another infinite hat-guessing story

Suppose first a countably infinite line of blindfolded people standing on tiles numbered 0,1,2,…, with the ones on a tile whose number is divisible by 10 having a red hat, and the others having blue hats. Suppose you’re in the line, with no idea where, but apprised of the above. It seems you should reasonably think: “Probably my hat is blue.”

But then the blindfolded people are shuffled, without any changes of hats, so that now it is the tiles with numbers divisible by 10 that have the blue hatters and the others have the red hatters. Such mere shuffling shouldn’t change what you think. So after being informed of the shuffle, it seems you should still think: “Probably my hat is blue.” It is already puzzling, though, why the first arrangement defined the probabilities and not the second. (What does temporal order have to do with these probabilities?)

Now suppose you gather the nine people after you (in the tile order—even though you are blindfolded, I suppose you can tell which direction the tile number numbers increase) along with yourself into a group of ten. In any group of ten successive people on the line, there is exactly one blue hat and nine red hats. Yet each of the ten of you thinks: “Probably my hat is blue.” And by a reasonable closure, you each also think: “Probably the other nine all have red hats.” You talk about it. You argue about it. “No, I am probably the one with the blue hat!” “No, my hat is probably the blue one.” “No, you’re probably both wrong: It’s probably mine.” I submit there is no rational room for any resolution to the disagreement, and indeed no budging of probabilities, no matter how much you pool your data, no matter how completely you recognize your epistemic peerhood, no matter how you apply exactly the same reasonable principles of reasoning. For nothing you learn from the other people is evidentially relevant. This is paradoxical.


Martin Cooke said...

I would not expect to learn much from such people, so I would not call it 'paradoxical'. For a start, they were all blindfolded, and yet they were all talking about what colours things were. How, I wonder, could you hope to learn much from them? But I think that things began to go wrong here with your "It seems you should reasonably think". That is because you go on to show why that would not be something that you should reasonably think. So I would emphasize the "seems" in that quote. It does indeed seem that way, but you do go on to show why one should not reasonably think it. You go on to show how the mere thinking of such shuffling should indeed change what you think!

Incidentally, there is a nice paradox in this area, due to someone called Levy (so far as I can tell). That is what I would call 'paradoxical'. There is a link to an article of mine that described a version of it here. Still, we have disagreed on much in this area...

I wonder if you find your story paradoxical because you would think that such people would know that their hats were blue (if in fact they were blue)? As you know, I do not think that we should get to knowledge that way. Here is another way of putting my reason why, inspired by recent news reports: It is a fact that women tend not to falsely accuse men of rape. That means that an accused man is very probably guilty. If knowledge is based on probability, then that would justify a claim to know that he was guilty. And you can imagine a crime where almost no one falsely accuses (has falsely accused), with as low a probability of that as you like; I would just ask you to imagine that you are on the jury. (I think that such scenarios are the sort that gave rise to our concept of knowledge in the first place.)

Martin Cooke said...
This comment has been removed by the author.
Martin Cooke said...

(sorry, my mouse is randomly double-clicking things)

IanS said...

The people in your group are not equivalent. ‘You’ are special. (Suppose the setup were repeated from the beginning, with a different initial ordering. Your new group would still contain you, but the other members would probably be different.) So everyone in the group could agree these credences: 0.1 that your hat is red, 89/90 for each of the others.

These credences reflect your special role, and the known fact that there are nine red hatters and one blue hatter in your group. There are of course no strictly defensible credences, because there are no natural uniform distributions over initial orderings or shuffling algorithms.

Alexander R Pruss said...


Designating a special member is interesting, but I would think that there wouldn't be the agreement on credences you suggest. Suppose you're special and marked out ahead of time. But everyone else in your group would think: "I am special, too." I haven't figured out exactly what the credences would be, but I would think that everyone would assign the same credence to their having the blue hat as they do to your having it.

OK, but now suppose you know all the infinity of people ahead of time. I.e., suppose that *everyone* is special. So now it seems everyone in the group should be treated equally, and have credence 1/10 of a blue hat.

IanS said...

Suppose the setup was that only one person – ‘you’ – was to form a group. (That is how I had read the OP). Then yes, ‘you’ and the ‘others’ in your group would indeed both be special, but special differently. You would be special in advance. The others would become special only after the shuffling, by virtue of being your successors in the new sequence. So you and the others could reasonably have different credences. (Note that you were special at the beginning, when the blue hats were dense in the line, while the others became special after the shuffling, when red was dense.)

Change the setup. Suppose everyone gets to form their own group. You might think that this would make them equivalent. Not so. By comparing notes, they could all work out their positions in the shuffled line. (The one who is in no one else’s group is #0, the one who is in no one’s group except #0 is #1, etc.) So they would not have to guess.

To avoid this issue, suppose they are shuffled into a doubly infinite line (-∞, …, -1, 0, 1, …, +∞), with blue hats at multiples of 10, red elsewhere. Then they can work out their ordering on the line (as before), but not their crucial absolute positions. This is an interesting problem :-).

Martin Cooke said...

"Such mere shuffling shouldn’t change what you think."
But it is the shuffling of Hilbert's Hotel. It can do a lot. Why should it not change what you think?

Personally, I would originally think that my hat was probably blue, too. But then I would think, after the shuffling, that it must therefore be probably red, and I would also see a problem with mere shuffling changing the probability like that. In light of that problem I would come to wonder about that original "probably blue" and decide that I had not had enough cause to believe it was probably blue in the first place (surprisingly). So the mere shuffling would have changed what I thought. (I would come to believe that the probability should derive from the details of how I am given a place in that line, which I do not know.)

You have set your scenario up so that none of the ten people thinks like that. And then you say that it is paradoxical that they cannot learn from each other!

IanS said...

Alex : “It is already puzzling though, that the first arrangement defined the probabilities and not the second.” Why is this puzzling? Isn’t is obvious that that we base our intuitive credences on the densities when the hats were assigned? (The OP does not say so directly, but I take it as implicit that the people started hatless, were placed on the tiles, and then given their hats as in the first arrangement of the OP.)

We can (sort of) formalise this as follows. The people are numbered 0, 1, 2, … These numbers are in effect their names; they stick with them in everything that follows. The people are first lined up on the tiles in natural order. They are then blindfolded and given a random permutation (details to be given later, but note that the people know the relevant probabilities, but not the actual permutation). They are given hats, as in the OP, according the tile numbers (i.e. red at every multiple of 10, blue otherwise.) They are shuffled to the new order (blue at every multiple of 10, red otherwise, as in the OP), according the obvious deterministic rule: Nth old red position to Nth new red position, Nth old blue position to Nth new blue position. A special person S (‘you’ of the OP) in chosen arbitrarily (but independently of any of the above). The numbers of S and her nine successors in the new ordering are announced (with S first, then in succession order) to everyone. Everyone knows in advance that all this is going to happen. What credences should S and her successors have about their hats?

The initial random permutation is supposed to represent the peoples’ ignorance about their positions on the tiles. It is clearly necessary: without it, the people could infer their hat colours from their numbers. We would like it to be fairly chosen from all possible permutations, but that could be hard to arrange :-) So start with something simpler: apply a (fairly chosen) random permutation to people 0 to 9, a similar but independent random permutation people 10 to 19, etc. This suffices to ensure that each person’s (unconditional) probability of getting a red hat is 1/10 (strictly, not just intuitively).

With this setup, everyone can (in principle) calculate the probabilities of ‘blue hat’ for S and for her nine successors, conditional on the announced information – no intuition is needed. Without even doing the calculations, three things are clear: (a) everyone will agree; (b) the conditional probabilities must add up to 9; (c) the conditional probabilities will vary wildly with the actual outcomes.

In this particular setup, the answer is simple, at least for all reasonably large-numbered S: the person with the smallest number is blue, the others are red. (Think about it…). Everyone can agree on this. Of course, this simple result is an artifact of the particular permutation probabilities and shuffling rule. But it shows how this sort of setup cannot yield a paradox.

IanS said...

Correction: second last para above: “(b) the conditional probabilities must add up to 1 [not 9]”. [Because there is always exactly 1 blue hat in the group of 10.]

Using the above model, you can work out the unconditional ‘blue’ probabilities for any S and her nine successors in the new order. For S = 1000, these are, in order: 0.9, 0, 0, 0, 0, 0, 0, 0, 0.1, 0. (If I have done it right – the tricky bit is the position of the 0.1. For different S, it appears in different positions.) Again, such a simple result is an artifact of the particular model details. But it illustrates the key point: the successors’ probabilities are much less than the initial probabilities of 0.9. There is no mystery here: the successors of S in the new order are a systematically biased selection from the original people. That was the point of the density-altering shuffling.