another paradox and a solution

From: Wei Dai <weidai.domain.name.hidden>
Date: Sat, 21 Feb 1998 18:15:19 -0800

Ok, here's another probability paradox. Suppose in the quantum suicide
experiment the assistant offers the experimenter a bet. The experimenter
gives the assistant $2 before he pulls the gun, and in return the
assistant will give $3 to the experimenter after pulling the gun, if the
result is a click instead of a bang. The assistant clearly has a positive
expected return from this bet. But if the experimenter believes she will
hear a click with 100% certainty, she also has a positive expected return.

These paradoxes show that the method of computing sensory probabilities
suggested by Tegmark is flawed. I think the right method involves removing
the restriction that

sum_over_X P(X|Y) = 1

where Y is current perception and X is future perception.
This sum should be greater than 1 if there is a posibility of copying
(cloning) and less than 1 if there is a possibility of death. In the
single-round quantum suicide experiment this sum would be 1/2. In an
experiment where an experimenter is copied with probability 1/2 the sum
should be 3/2. In general the sum should be the ratio between the measure
of one's continuation and the measure of oneself.

Let me give a specific example. Consider an experiment where at time 1 a
coin a flipped and the result observed by the experimenter, and at time 2
the experimenter the duplicated if the the coin landed heads. I suggest
the experimenter should have the following beliefs at time 0:

1. I will observe heads at time 1 with probability 1/2, and tails with
probability 1/2.

2. I will observe heads at time 2 with probability 1, and tails with
probability 1/2.

At time 1, if she observed heads:

3a. I will observe heads at time 2 with probability 2, and tails with
probability 0.

At time 1, if she observed tails:

3b. I will observe heads at time 2 with probability 0, and tails with
probability 1.
Received on Sat Feb 21 1998 - 18:16:04 PST