Eliezer's Coin Flipping Duplicates Paradox

From: Lee Corbin (lcorbin@rawbw.com)
Date: Sat Mar 08 2008 - 16:53:34 MST

Eliezer originally wrote

> If you flip a fair quantum coin, have your exoself generate 100
> separated isomorphic copies of you conditional on the coin coming up
> heads, then, when (all of) you are about to look at the coin, should
> your subjective anticipation of seeing "heads" be 1:1 or 100:1?

I wish to state more clearly the reasons why the answer should be
100:1, although let me hastily admit that this is trickier than I thought.

My new approach is to revert to the Many Worlds Interpretation
for clarity. I hope it makes understandable what I was saying
before. The answer that emerges should be the same as using
any of the other interpretations.

Let me break down the steps which discretely occur, to help
expose any invalid assumptions.

A. A fair coin is flipped and some process notes the outcome.
B. If the result is tails, nothing further happens.
C. If the result is heads, 100 separated isomorphic copies
     of a person are created.
D. Each copy in any universe only then examines the coin
     to see what turned up

What is the "subjective anticipation" that it will be heads?
(where I read that as "subjective probability").

If this all began in a Group of Identical Universes (a term often
used by Deutsch in "The Fabric of Reality"), then just after
the flip, the GIU splits into two, one universe called H and one
called T, each with a probability of .5 relative to the original GIU.

Next, in universe H the person's measure is multiplied by 100.

So how do we answer the question, "What is the person's
subjective probability of seeing an H now?". Well, each
different instance of the person has the same measure, since
in T no duplication occurred. That is, each of the 101 copies,
from the multiverse perspective, has the same salience, the
same measure. In other words, you get 100 times the
runtime in H as you do in T.

Therefore on this account, your subjective probability, sampled
evenly over the multiverse, is 100:1 that you will now see a "head".

The Evolutionary Test

Let's put this to the test I mentioned before, and check the result
by evolutionary logic.

You are furthermore required on each day that the coin is flipped
to undergo a 100/101 probability of dying (that is, the instance of
you perishing). This applies equally to all copies of you. I now
switch to the language of probability, though it would be possible,
but more cumbersome to stick with branching.

Consider the case now after five days have passed. We compute
that the expectation is that just one of you will still be alive, because
every day 100/101 are eliminated, whether or not they saw an H or a T.

What will this one remember? It's possible that he will remember
TTTTT, but that is very unlikely. That would only occur if each
day the 100/101 death toll struck only those who had received
"heads". The chances are (100/101)^5, which is close to .95,
that he would remember HHHHH.

And if this continues, then a "T" will crop up in a long sequence
of mostly H's about one time in one hundred and one.

Therefore, as before, the subjective probability is 100/101
that on each trial you'll see an H.


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