Christopher Maloney, <dude.domain.name.hidden>, writes:
> I'd like to revive an old thread, that has been bothering me a lot
> lately. I hope you'll all agree that it's a fascinating puzzle.
> Wei Dai posed this way back, in February of last year:
> http://www.escribe.com/science/theory/index.html?mID=38.
> I've read the entire thread, and I don't think the question was
> ever resolved. Wei, Nick Bostrom, and Hal Finney were the main
> contributors.
>[...]
> But there's an alternative way of computing this probability. She
> believes that at time t1, the odds are 1/2 that she'll have seen
> heads. She also knows that if, at time t1, she saw heads, then the
> odds will be 1 that at all later times, she'll continue to remember
> seeing heads. Likewise, if she saw tails at time t1, she'll
> continue to remember seeing tails. So
>
> P(H,t3) = P(H,t1) = 1/2.
>
> So which is correct? I know which solution I prefer, but I'd like
> to get some feedback first.
I don't think the reasoning for this example is very solid. You have
to introduce this notion of "the probability that I will remember X at
time B, given that I remember it at earlier time A." Then you have to
assume that it is 1.
But there is no sound basis for this assumption or this methodology.
It is based on our common experience, but we have no experience with
copy machines of this type. I think we have to use mathematics rather
than experience to guide us in new situations. So this reasoning does
not seem convincing to me.
One thing you did not say was why the other answer, P(H,t3) = 2/3, was
problematical. I believe it had to do with decision theory. In classical
decision theory, a choice is evaluated by taking all possible outcomes,
multiplying the possible outcome times the utility of that outcome, and
summing the results. You then make the choice which maximizes this sum.
An example where this might seem to be a problem is the following.
You are offered a bet where you will pay $15 if the coin is tails, but
win $10 if it is heads. If P(H) is 2/3, you might want to take the bet.
The resolution is to look at when the payment is made. If it is after
you are duplicated, and each instance of your duplicates receives the
$10, then it is a good bet. If the payment is made before duplication
(but after the coin flip) then your head-observing duplicates don't
actually end up with $10 each, but rather $5. The duplicating machine
does not duplicate money, or any of your material possessions. In fact,
being duplicated in effect cuts your material possessions in half.
So the real choice is between $5 with probability 2/3 and -$15 with
probability 1/3, not a good bet.
An interesting variant though is to introduce nonmaterial rewards.
They used to have kissing booths at the fair. For a few dollars you
would get to kiss a beautiful girl. Suppose that is worth $10 to you.
Now the bet is offered; you pay $15 on tails but get a kiss on heads.
The payment will be made before duplication.
In this case I would say that it is a good bet. The memory of the kiss
gets duplicated with you, and each of your duplicates ends up with a
memory that is worth $10. So in this case you have P=2/3 of gaining
something worth $10 and P=1/3 of losing something worth $15, making
it favorable.
> Note that one need not bring MWI into this at all. The only big
> assumption is the existence of a copy machine. Instead of MWI, one
> can think of the identical experiment being carried out on an
> ensemble of, say, 100 hapless souls Albert, Bernard, Caroline, etc.
> At time t1, some number close to 50 will have seen heads. At time
> t2, there will be 150 people, 100 of whom remember seeing heads,
> and 50 of whom remember seeing tails. From a bird perspective,
> if you picked any person at random from this group, the chance that
> they'll have seen heads is 2/3.
One difference though is that the MWI duplicates all of your material
possessions, unlike the copy machine. That changes the answer I would
give, as in the kissing example above. So it is not clear to me that
copying presents issues involving subjective probabilities in the same
way that an MWI model does.
Hal
Received on Wed Aug 04 1999 - 09:59:41 PDT