ata comments on The Presumptuous Philosopher's Presumptuous Friend - Less Wrong

3 Post author: PlaidX 05 October 2009 05:26AM

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Comment author: ata 05 October 2009 07:37:53PM *  7 points [-]

I don't think this requires anthropic reasoning.

Here is a variation on the story:

One day, you and the presumptuous philosopher are walking along, arguing about the size of the universe, when suddenly Omega jumps out from behind a bush and knocks you both out with a crowbar. While you're unconscious, she builds a hotel with 1,000,001 rooms. Then she makes a million copies of both of you, sticks them all in rooms, and destroys the originals.

You wake up in a hotel room, in bed with the presumptuous philosopher, with a note on the table from Omega, explaining what she's done.

"Which room are we in, I wonder?" you ask.

"Any of them is equally likely," says the presumptuous philosopher. "Because it's bloody obvious and all that. Million to one odds for any given room."

"Rubbish!" you scream. "Rubbish and poppycock! We have a 50% chance of being in room 870,199, and a 50% chance of being in one of the other rooms."

After the presumptuous philosopher stands in baffled silence for a moment, he says, "In that case, I'll make a deal with you. We'll go outside and check, and if we're in room 870,199 I'll give you ten bucks. If we're in one of the other rooms, I'll just smile smugly."

"Hah!" you say. "You just lost an expected five bucks, sucker!"

You run out of the room to find yourself surrounded by throngs of yourselves and smug looking presumptuous philosophers; you turn around and look at your door, labeled 129,070.

If I'm not mistaken (am I?), this version of the story is exactly isomorphic to PlaidX's original version; the only difference is that it's easier to see why the friend is wrong before you get to the end.

To anyone who agrees with the friend in the original story -- that the most reasonable estimate is that there is an even chance of being in either hotel -- would you disagree that this version is isomorphic to the original?

Comment author: PlaidX 05 October 2009 07:53:13PM 2 points [-]

I thought of this, but then, in the other direction, is the problem non-isomorphic to the original presumptuous philosopher problem? If so, why?

Is it because I used hotels instead of universes? Is it because the existence of both hotels has probability 100% instead of probability 50%? Is it some other thing?

Comment author: Nubulous 06 October 2009 07:15:49AM 0 points [-]

The most obvious difference is that the original problem involved the smaller or the larger set of people whereas this one uses the smaller and the larger.

Comment author: PlaidX 06 October 2009 08:52:10AM 0 points [-]

Ah, so the difference isn't that I used hotels instead of universes, it's that I used hotels instead of POSSIBLE hotels. In other words, your likelihood of being in a hotel depends on the number of "you"s in the hotel, but your likelihood of being in a possible hotel does not, is that what you're saying?

Unless the number of "you"s is zero. Then it clearly does depend on the number. Isn't this just packing and unpacking?

Comment author: Nubulous 06 October 2009 01:08:52PM 0 points [-]

You're reading a little more into what I said than was actually there. I was just remarking on the change of dependence between the parts of the problem, without having thought through what the consequences would be.

Now that I have thought it through, I agree with the presumptuous philosopher in this case. However I don't agree with him about the size of the universe. The difference being that in the hotel case we want a subjective probability, whereas in the universe case we want an objective one. Subjectively, there's a very high probability of finding yourself in a big universe/hotel. But subjective probabilities are over subjective universes, and there are very very many subjective large universes for the one objective large universe, so a very high subjective probability of finding yourself in a large universe doesn't imply a large objective probability of being found in one.

Comment author: PlaidX 06 October 2009 10:35:54PM 0 points [-]

I don't understand what you mean by subjective and objective probabilities. Would you still agree with the philosopher in my problem if omega flipped a coin (or looked at binary digit 5000 of pi) and then built the small hotel OR the big hotel?

Comment author: Nubulous 08 October 2009 12:43:05AM 0 points [-]

I don't know what I meant either. I remember it making perfect sense at the time, but that was after 35 hours without sleep, so.....

The answer to the second part is no, I would expect a 50:50 chance in that case.
In case you were thinking of this as a counterexample, I also expect a 50:50 chance in all the cases there from B onwards. The claim that the probabilities are unchanged by the coin toss is wrong, since the coin toss changes the number of participants, and we already accepted that the number of participants was a factor in the probability when we assigned the 99% probability in the first place.

Comment author: PlaidX 08 October 2009 03:08:37AM *  3 points [-]

So, if omega picks a number from 1 to 3, and depending on the result makes:

A. a hotel with a million rooms

B. a hotel with one room

C. a pile of flaming tires

you'd say that a person has a 50% chance of finding themselves in situation A or B, but a 0% chance of being in C?

Why does the number of people only matter when the number of people is zero? Doesn't that strike you as suspicious?

Comment author: Nubulous 10 October 2009 11:39:22PM 0 points [-]

When we speak of a subjective probability in a person-multiplying experiment such as this, we (or at least, I) mean "The outcome ratio experienced by a person who was randomly chosen from the resulting population of the experiment, then was used as the seed for an identical experiment, then was randomly chosen from the resulting population, then was used as the seed.... and so forth, ad infinitum".

I'm not confident that we can speak of having probabilities in problems which can't in theory be cast in this form.

In other words, the probability is along a path. When you look at the problem this way, it throws some light on why there are two different arguable values for the probability. If you look back along the path, ("what ratio will our person have experienced") the answer in your experiment is 1000000:1. If you look forward along the path, ("what ratio will our person experience") the answer is 1:1 (in the flaming-tires case there's no path, so there's no probability).

Comment author: PlaidX 11 October 2009 04:27:29AM 0 points [-]

But again I must ask, on the going-forward basis, why is the number of people in each world irrelevant? I grant you that the WORLD splits into even thirds, but the people in it don't, they split 1000000 / 1 / 0. Where are you getting 1 / 1 / 0?

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