Copies Count
Hal Finney
> Would you say that because you think running multiple identical copies of a
> given mind in parallel doesn't necessarily increase the absolute measure of
> those observer-moments (that would be my opinion)...
Here is an argument I wrote a couple of years ago on another list
that first made me think that copies count, that is, that having more
identical copies should be considered to increase measure. Previously I
was skeptical about it, I thought what mattered was whether something
or someone got instantiated at all, not how many there were. And of
course since then I not only believe that copies count, I have followed
my logic to the absurd sounding conclusion that size and slowness increase
measure as well.
Consider an experiment where we are simulating someone and can give
them either a good or bad experience. These are not replays, they are
new experiences which we can accurately anticipate will be pleasant
or unpleasant.
Suppose we are going to flip a biased quantum coin, one which has a 90%
chance of coming up heads. We will generate the good or bad experience
depending on the outcome of the coin flip. I claim that it is obvious
that it is better to give the good experience when we get the 90% outcome
and the bad experience when we get the 10% outcome. That's the assumption
I will start with.
Now consider Tegmark's level 1 of parallelism, the fact that in a
sufficiently large volume of space I can find a large number of copies
of me, in fact copies of the entire earth and our entire visible universe
(the "Hubble bubble"?). When I do my quantum coin flip, 90% of the copies
will see it come up heads and cause the good experience for the subject,
and 10% will see tails and cause the bad experience.
I will also assume that my knowledge of this fact about the physical
universe will not change my mind about the ethical value of my decision
to give the good experience for the 90% outcome.
Now the problem is this. There are really only two different programs
being run for our experimental subject, the guy in the simulation. One is
a good experience and one is bad. All my decision does is to change how
many copies of each of these two programs are run. In making my decision
about which experiences to assign to the two coin flip outcomes, I have
chosen that the copies of the good experience will outnumber copies of
the bad experience by 9 to 1.
But if I don't believe that the number of copies being run makes a
difference, then I haven't accomplished what I desired. The fact that
I am running more copies of the good program than the bad wouldn't make
any difference. Therefore there is no actual ethical value in what I
have done, I might have just as validly reversed the outcome of my coin
flips and it wouldn't have made any difference.
In this way I reach a contradiction between the belief that the number
of copies doesn't matter, the belief that the existence of distant
parallel copies of myself doesn't make much difference in what I should
do, and the idea that there is value in making people happy. Of these,
the most questionable seems to be the assumption that copies don't matter,
so this line of reasoning turns me away from that belief.
I can come up with similar contradictions from simpler cases like
our own observations of subjective probability. The fact that I do
experience a subjective 90% chance of seeing the quantum coin come
up heads corresponds very well with the fact that 90% of the copies
of me will see heads - but only if I assume that the multiplicity of
the copies matters. After the coin flip, in a certain voume of space
there are 90 copies of me that see heads and 10 copies that see tails.
But within the two groups all copies are identical (neglecting other
quantum events which would further split me). If the multiplicity
doesn't count, then there are really just two outcomes and I might
expect to subjectively experience equal probability for them.
This is a variant on an old argument against the MWI, but in that case I
always felt the answer was "measure", that some of the outcomes occured
in a quantum branch which had this intangible quality which made it count
more. In this case I can't invoke any such magic; all the copies of me
are running in the same universe and with equal quantum amplitude. I have
to resort to counting the instances separately and assuming that each one
makes its own independent contribution to my subjective experiences, in
order to gain correspondence with subjective probability. Therefore it
is most consistent to say that separate runs of identical programs do
"count", they do add to the measure of the subjective experience.
Hal Finney
Stathis Papaioannou
Here is another way of explaining this situation. When there are multiple
parallel copies of you, you have no way of knowing which copy you are,
although you definitely are one of the copies during any given moment, with
no telepathic links with the others or anything like that. If a proportion
of the copies are painlessly killed, you notice nothing, because your
successor OM will be provided by one of the copies still going (after all,
this is what happens in the case of teleportation). Similarly, if the number
of copies increases, you notice nothing, because during any given moment you
are definitely only one of the copies, even if you don't know which one.
However, if your quantum coin flip causes 90% of the copies to have bad
experiences, you *will* notice something: given that it is impossible to
know which particular copy you are at any moment, or which you will be the
next moment, then there is a 90% chance that you will be one of those who
has the bad experience. Similarly, if you multiply the number of copies
tenfold, and give all the "new" copies bad experiences, then even though the
"old" copies are left alone, you will still have a 90% chance of a bad
experience, because it is impossible to know which copy will provide your
next OM.
So, perhaps counterintuitively, you and all your copies are better off if
all but one is painlessly killed than if the total number is increased and a
proportion of the new copies given a bad experience.
This is what I was trying to show in my post "another puzzle". I think this
way of looking at it is simple, consistent, does not require any new
physical laws, and provides a reason to do good things rather than bad
things in the multiverse, as long as you don't make the terrible mistake of
assuming that the absolute measure of copies with good experiences is more
important than the relative measure.
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Hal Finney
> Here is another way of explaining this situation. When there are multiple
> parallel copies of you, you have no way of knowing which copy you are,
> although you definitely are one of the copies during any given moment, with
> no telepathic links with the others or anything like that. If a proportion
> of the copies are painlessly killed, you notice nothing, because your
> successor OM will be provided by one of the copies still going (after all,
> this is what happens in the case of teleportation). Similarly, if the number
> of copies increases, you notice nothing, because during any given moment you
> are definitely only one of the copies, even if you don't know which one.
> However, if your quantum coin flip causes 90% of the copies to have bad
> experiences, you *will* notice something: given that it is impossible to
> know which particular copy you are at any moment, or which you will be the
> next moment, then there is a 90% chance that you will be one of those who
> has the bad experience. Similarly, if you multiply the number of copies
> tenfold, and give all the "new" copies bad experiences, then even though the
> "old" copies are left alone, you will still have a 90% chance of a bad
> experience, because it is impossible to know which copy will provide your
> next OM.
I'm not sure I fully understand what you are saying, but it sounds like
you agree at least to some extent that "copies count". The number of
copies, even running in perfect synchrony, will affect the measure of
what that observer experiences, or as you would say, his subjective
probability.
So let me go back to Bruno's thought experiment and see if I understand
you. You will walk into a Star Trek transporter and be vaporized and
beamed to two places, Washington and Moscow, where you will have two
(independent) copies wake up. Actually they are uploads and running on
computers, but that doesn't matter (we'll assume). Bruno suggests that
you would have a 50-50 expectation of waking up in Washington or Moscow,
and I think you agree.
But suppose it turns out that the Moscow computer is a parallel
processor which, for safety, runs two copies of your program in perfect
synchrony, in case one crashes. Two synchronized copies in Moscow,
one in Washington.
Would you say in this case that you have a 2/3 expectation of waking up
in Moscow?
And to put it more sharply, suppose instead that in Washington you will
have 10 copies waking up, all independent and going on and living their
lives (to the extent that uploads can do so), sharing only the memory
of the moment you walked into the transporter. And in Moscow you will
have only one instance, but it will be run on a super-parallel computer
with 100 computing elements, all running that one copy in parallel and
synchronized.
So you have 10 independent copies in Washingon, and 100 copies that
are all kept in synchrony in Moscow. What do you expect then? A 90%
chance of waking up in Washington, because 9/10 of the versions of you
will be there? Or a 90% chance of waking up in Moscow, because 9/10 of
the copies of you will be there?
I think, based on what you wrote above, you will expect Moscow, and that
"copies count" in this case.
If you agree that copies count when it comes to spatial location, I
wonder if you might reconsider whether they could count when it comes
to temporal location. I still don't have a good understanding of this
situation either, it is counter-intuitive, but if you accept that the
number of copies, or as I would say, measure, does make a difference,
then it seems like it should apply to changes in time as well as space.
Hal Finney
Stathis Papaioannou
I agree that you will have a 90% chance of waking up in Moscow, given that
that is the *relative* measure of your successor OM when you walk into the
teleporter. This is the only thing that really matters with the copies, from
a selfish viewpoint: the relative measure of the next moment:
(a) If you are copied 100 times and 99% of the copies tortured, you will
certainly know this, as there is a 99% subjective probability that you will
be tortured.
(b) If you are copied 100 times and the copies allowed to diverge, then 99%
of the copies painlessly killed, that means you have a 99% chance of being
killed, because in two steps, (i) there is a 100% chance your next OM will
be one of the 100 copies; and (ii) there will be a 99% chance that you will
have become one of the copies that will be killed, and if you are, then
there will be 0% chance that you will have any "next moment".
(c) If you are copied 100 times and all the copies are kept running in
parallel, then 99% of the copies painlessly killed, you can't possibly know
that anything odd has happened at all, because there is a 100% chance that
your next OM will come from the one remaining copy. Similarly if there were
10^100 copies *kept running in perfect sync* and all but one terminated.
1/1=100/100=10^100/10^100=1.
--Stathis Papaioannou
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Hal Finney
> I agree that you will have a 90% chance of waking up in Moscow, given that
> that is the *relative* measure of your successor OM when you walk into the
> teleporter. This is the only thing that really matters with the copies, from
> a selfish viewpoint: the relative measure of the next moment:
So let me try an interesting variant on the experiment. I think someone
else proposed this recently, the idea of "retroactive causation".
I won't put that exact spin on it though.
Suppose you will again be simultaneously teleported to Washington
and Moscow. This time you will have just one copy waking up in each.
Then you will expect 50-50 odds. But suppose that after one hour,
the copy in Moscow gets switched to the parallel computer so it is
running with 10 times the measure; 10 copies. And suppose that you know
beforehand that during that high-measure time period (after one hour)
in Moscow you will experience some event E.
What is your subjective probability beforehand for experiencing E?
I think you agreed that if you had been woken up in Moscow on
the super-parallel computer that you would expect a 90% chance of
experiencing E. But now we have interposed a time delay, in which your
measure starts off at 1 in Moscow and then increases to 10. Does that
make a difference in how likely you are to experience E?
I am wondering if you think it makes sense that you would expect a 50%
probability of experiencing events which take place in Moscow while
your measure is 1, but a 90% probability of experiencing events like
E, which take place while your measure is 10? I'm not sure about this
myself, because I am skeptical about this continuity-of-identity idea.
But perhaps, in your framework, this would offer a solution to the
problem you keep asking, of some way to notice or detect when your
measure increases.
In that case we would say that you could notice when your measure
increases because it would increase your subjective probability of
experiencing events.
Perhaps we could even go back to the thought experiment where you have
alternating days of high measure and low measure. Think of multiple
lockstep copies being created on high measure days and destroyed on low
measure days. Suppose before beginning this procedure you flip a quantum
coin (in the MWI) and will only undergo it if the coin comes up heads.
Now, could you have a subjective anticipation of 50% of experiencing the
events you know will happen on low-measure days, but an anticipation of
90% of experiencing the events you know will happen on high-measure days?
Then that would be a tangible difference, and you would be justified in
pre-arranging your affairs so that pleasant events happen on the high
measure days and unpleasant ones happen on the low measure days.
It's an interesting concept in any case. I need to think about it more,
but I'd be interested to hear your views.
Hal Finney
Jesse Mazer
>
>Jesse Mazer writes:
> > Would you say that because you think running multiple identical copies
>of a
> > given mind in parallel doesn't necessarily increase the absolute measure
>of
> > those observer-moments (that would be my opinion)...
>
..
It's a bit hard for me to come up with a satisfactory answer to this
problem, because I don't start from the assumption of a physical universe at
all--like Bruno, I'm trying to start from a measure on observer-moments and
hope that somehow the appearance of a physical universe can be recovered
from the subjective probabilities experienced by observers (note that this
does not mean that all physical processes are a kind of dream in the minds
of high-level intelligent beings like ourselves; I think the mathematical
description of the set of all observer-moments in a theory of consciousness
would probably be something like a set of graphs describing the causal
relationships between events in that observer's 'brain', and even very
simple causal patterns, like those created by the random jostling of
molecules in a rock, could be fellow observer-moments, albeit very simple
ones, which we are observing 'from the outside' just like we can only
experience other people's minds from the outside). Without having any clear
idea of how to do this derivation of an apparent physical universe from the
conditional and absolute measure on observer-moments, I don't really know
what the appearance of multiple physical copies would correspond to in terms
of conditional or absolute measure on observer-moments.
But my speculation would be this: multiple copies only increase first-person
measure to the extent that, from a third-person perspective, they are likely
to create more *distinct* observers with a memory of the split going one way
than of the split going the other. So if we look at some very large region
where there are 90 copies of our local region of the universe where the
quantum coinflip went one way and only 10 copy of our local region where the
coinflip went the other way, then even if those 90 copies run in lockstep
for a while, chances are good that eventually random events in each region
will cause them to diverge. The end result would be that, after sufficient
time, in this very large region you'll have 100 distinct regions which all
share the same history up until that coinflip, with 90 having records of the
flip going one way and 10 having records of it going the other. And given my
views on splits having an "anticipatory" quality (so if you're split into
two and then one copy is scheduled to be split 999 times later, that will be
reflected in your subjective probabilities in the initial split), I
naturally tend to think that this means you'd have a 9 times greater
probability of experiencing the coinflip outcome that will later lead to 90
divergent copies, even if the 90 copies run in lockstep initially.
But like I said, it's difficult to make any definite claims about why we
experience quantum probabilities the way we do, especially since no one has
come up with a way to derive a simple frequentist notion of probability from
the universal wavefunction in the MWI.
Jesse
Hal Finney
> It's a bit hard for me to come up with a satisfactory answer to this
> problem, because I don't start from the assumption of a physical universe at
> all--like Bruno, I'm trying to start from a measure on observer-moments and
> hope that somehow the appearance of a physical universe can be recovered
I have a similar perspective. However I think it will turn out that the
simplest mathematical description of an observer-moment will involve a Big
Bang. That is, describe a universe, describe natural laws, and let the
OM evolve. This is the foundation for saying that the universe is real.
> But my speculation would be this: multiple copies only increase first-person
> measure to the extent that, from a third-person perspective, they are likely
> to create more *distinct* observers with a memory of the split going one way
> than of the split going the other. So if we look at some very large region
> where there are 90 copies of our local region of the universe where the
> quantum coinflip went one way and only 10 copy of our local region where the
> coinflip went the other way, then even if those 90 copies run in lockstep
> for a while, chances are good that eventually random events in each region
> will cause them to diverge. The end result would be that, after sufficient
> time, in this very large region you'll have 100 distinct regions which all
> share the same history up until that coinflip, with 90 having records of the
> flip going one way and 10 having records of it going the other.
I meant to imply that this kind of differentiation could not occur in
my thought experiment, because the subject involved was a simulation in
a computer. He is not interacting with the environment, he is running
a deterministic program. So I can give him a good or bad experience
and he will not split.
Do you think you would say, in that case, that when I flip my 90/10
quantum coin, there is no reason to give him the good experience with 90%
probability and the bad experience with 10% probability? It would be just
as good to reverse the odds?
Remember, I'm not running copies; I'm just running one program, and
flipping a biased coin to decide what to do.
> And given my
> views on splits having an "anticipatory" quality (so if you're split into
> two and then one copy is scheduled to be split 999 times later, that will be
> reflected in your subjective probabilities in the initial split), I
> naturally tend to think that this means you'd have a 9 times greater
> probability of experiencing the coinflip outcome that will later lead to 90
> divergent copies, even if the 90 copies run in lockstep initially.
This is an interesting idea; one thing that occurs to me though is the
problem of amnesia. (And let's face it, 99% of what we experience,
we don't remember.) If you're split into two and experience something
in the one copy that is forgotten before it gets split 999 times, does
that to-be-forgotten experience still gain the benefit of the 999-fold
multiplication?
In other words, is it by virtue of later memory of the event that you
expect the subjective probability to be magnified, or is it a more
metaphysical connection?
> But like I said, it's difficult to make any definite claims about why we
> experience quantum probabilities the way we do, especially since no one has
> come up with a way to derive a simple frequentist notion of probability from
> the universal wavefunction in the MWI.
That is indeed the case. I hope to write an essay soon about what I
think such a solution would look like (not to offer a solution, merely
to try to clarify what exactly I think it needs to explain).
Hal
Brent Meeker
>-----Original Message-----
>From: Jesse Mazer [mailto:laser...@hotmail.com]
>Sent: Tuesday, June 21, 2005 12:38 AM
>To: h...@finney.org; everyth...@eskimo.com
>Subject: RE: Copies Count
>
>
>Hal Finney wrote:
>
>>
>>Jesse Mazer writes:
>> > Would you say that because you think running multiple identical copies
>>of a
>> > given mind in parallel doesn't necessarily increase the absolute measure
>>of
>> > those observer-moments (that would be my opinion)...
>>
>
>...
There seems to be equivocation about what constitutes an "observer moment". Is
it a unit of conscious experience? Or is it more; something that includes
memories even when they aren't being remembered? The latter might provide some
connectivity between OMs. Without including memories though, OMs as units of
conscious experience are disjointed and sporadic. Consciousness seems to be
only a small part of out thinking. But including memories creates other
problems. Where are they encoded?...in observerless moments?...in the physics
of the brain?
Brent Meeker
Stathis Papaioannou
>So let me try an interesting variant on the experiment. I think someone
>else proposed this recently, the idea of "retroactive causation".
>I won't put that exact spin on it though.
>
>Suppose you will again be simultaneously teleported to Washington
>and Moscow. This time you will have just one copy waking up in each.
>Then you will expect 50-50 odds. But suppose that after one hour,
>the copy in Moscow gets switched to the parallel computer so it is
>running with 10 times the measure; 10 copies. And suppose that you know
>beforehand that during that high-measure time period (after one hour)
>in Moscow you will experience some event E.
>
>What is your subjective probability beforehand for experiencing E?
>I think you agreed that if you had been woken up in Moscow on
>the super-parallel computer that you would expect a 90% chance of
>experiencing E. But now we have interposed a time delay, in which your
>measure starts off at 1 in Moscow and then increases to 10. Does that
>make a difference in how likely you are to experience E?
Again, it's a two step process, each time considering the next moment.
First, 50% chance of waking up in either Moscow or Washington. Second, 100%
chance of experiencing E in Moscow or 0% chance of experiencing E in
Washington. The timing is crucial, or the probabilities are completely
different. Russell Standish realised this in his response to my green/red
light puzzle. To summarise, God places you in a room with a light changing
colour every 10 minutes, corresponding with a high measure state (10^100
copies of you, say green) and a low measure state (one copy of you, say
red), but you don't know which colour is which. In my original wording, I
said you don't remember how you got there and only after you notice the
light changing colour over several cycles do you see God's explanatory note.
Now, if you have to guess which colour corresponds with with which state,
you may as well toss a coin, because your experience is that you spend half
your time red and half green; or, to put it differently, when you anticipate
the next moment when the light is about to change colour, there is a 50%
chance you will be in the high measure state and a 50% chance you will be in
the low measure state, from the symmetry of the situation from your 1st
person perspective. But Russell's answer was that if you remembered what
colour the light was when you first arrived in the room, that would almost
certainly have been the high measure state. The reason this is so different
is that when you consider your next moment when God is about to put you in
the room, you have to take both possibilities into account simultaneously
rather than sequentially, and there are 10^100 times as many ways the light
could end up green as red. This is the error people make when they say that
you are more likely to find yourself living in a period of high measure
(when you are younger) than low measure (when you are millions of years
old), as an objection to QTI. It isn't valid to shuffle all the OM's from
all time periods and draw one at random, except when considering your
initial introduction into the world. Once you are already alive, you have to
pay attention to the special way our minds create continuity of
consciousness from moment to moment.
>I am wondering if you think it makes sense that you would expect a 50%
>probability of experiencing events which take place in Moscow while
>your measure is 1, but a 90% probability of experiencing events like
>E, which take place while your measure is 10? I'm not sure about this
>myself, because I am skeptical about this continuity-of-identity idea.
>But perhaps, in your framework, this would offer a solution to the
>problem you keep asking, of some way to notice or detect when your
>measure increases.
>
>In that case we would say that you could notice when your measure
>increases because it would increase your subjective probability of
>experiencing events.
I think the subjective probability stays the same, for the above reasons. I
consider my next moment: what are the possibilities? What is the relative
proportion of each possibility? It's probably easiest to visualize with a
tree diagram, or with the game I suggested in my post "objections to QTI".
You can't just mix up all the OM's from different time periods and hope to
make sense of it.
>Perhaps we could even go back to the thought experiment where you have
>alternating days of high measure and low measure. Think of multiple
>lockstep copies being created on high measure days and destroyed on low
>measure days. Suppose before beginning this procedure you flip a quantum
>coin (in the MWI) and will only undergo it if the coin comes up heads.
>Now, could you have a subjective anticipation of 50% of experiencing the
>events you know will happen on low-measure days, but an anticipation of
>90% of experiencing the events you know will happen on high-measure days?
>Then that would be a tangible difference, and you would be justified in
>pre-arranging your affairs so that pleasant events happen on the high
>measure days and unpleasant ones happen on the low measure days.
No, I think your expectations should be the same if we're talking about
consecutive days, for the above reasons.
--Stathis Papaioannou
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Hal Finney
> Hal Finney writes:
> >Suppose you will again be simultaneously teleported to Washington
> >and Moscow. This time you will have just one copy waking up in each.
> >Then you will expect 50-50 odds. But suppose that after one hour,
> >the copy in Moscow gets switched to the parallel computer so it is
> >running with 10 times the measure; 10 copies. And suppose that you know
> >beforehand that during that high-measure time period (after one hour)
> >in Moscow you will experience some event E.
>
> First, 50% chance of waking up in either Moscow or Washington. Second, 100%
> chance of experiencing E in Moscow or 0% chance of experiencing E in
> Washington. The timing is crucial, or the probabilities are completely
> different.
Doesn't this approach run into problems if we start reducing the time
interval before the extra copying in Moscow? From one hour, to one
second, to one millisecond? At what point does your phenomenological
expectation switch over from 90% Washington to 90% Moscow? And does
it do so discontinuously, or is there a point at which you are "just
barely" conscious enough in Moscow before the secondary duplication,
that perhaps the two probabilities balance?
I am doubtful that this approach works.
Jesse Mazer suggested backwards causation, that the secondary copying in
Moscow would influence the perceptual expectation of waking up in Moscow
even before it happens. So he would say 90% Moscow from the beginning.
However I think that has problems if we allow amnesia to occur in Moscow
before the amplification.
I have been enjoying these discussions but unfortunately I will have to
take leave, I am going on vacation with the family for a week so I will
have little chance to participate during that time. I'll look forward
to catching up when I return -
Hal Finney
Stathis Papaioannou
>Stathis Papaioannou writes:
> > Hal Finney writes:
> > >Suppose you will again be simultaneously teleported to Washington
> > >and Moscow. This time you will have just one copy waking up in each.
> > >Then you will expect 50-50 odds. But suppose that after one hour,
> > >the copy in Moscow gets switched to the parallel computer so it is
> > >running with 10 times the measure; 10 copies. And suppose that you
>know
> > >beforehand that during that high-measure time period (after one hour)
> > >in Moscow you will experience some event E.
> >
> > Again, it's a two step process, each time considering the next moment.
> > First, 50% chance of waking up in either Moscow or Washington. Second,
>100%
> > chance of experiencing E in Moscow or 0% chance of experiencing E in
> > Washington. The timing is crucial, or the probabilities are completely
> > different.
>
>Doesn't this approach run into problems if we start reducing the time
>interval before the extra copying in Moscow? From one hour, to one
>second, to one millisecond? At what point does your phenomenological
>expectation switch over from 90% Washington to 90% Moscow? And does
>it do so discontinuously, or is there a point at which you are "just
>barely" conscious enough in Moscow before the secondary duplication,
>that perhaps the two probabilities balance?
The time interval is the minimum time interval for you to experience a
conscious moment, which is also the minimum interval for two exact copies to
diverge so that they are no longer "identical". It is the same question as
how long you can be "alive" as the original in a teleportation thought
experiment before you mind being killed. I would say that if you walk out of
the teleportation booth and then someone comes along and shoots you a minute
later, that's bad, because you have had time to become a "different" person
since the teleportation and the teleported copy no longer provides
continuity of consciousness. It could be argued that there would only be a
minute of experience lost and maybe it doesn't matter, but I would agree
with your (HF) previous post on just this question that it *does* matter.
Similarly with whether there is a sharp or gradual transition: I think there
would be a sharp transition between the point where you wouldn't notice if
you got shot and the point where you would mind very much. I am not sure
exactly what the smallest possible conscious interval is, but it would
certainly be longer than a millisecond and shorter than a second.
>I am doubtful that this approach works.
>
>Jesse Mazer suggested backwards causation, that the secondary copying in
>Moscow would influence the perceptual expectation of waking up in Moscow
>even before it happens. So he would say 90% Moscow from the beginning.
>However I think that has problems if we allow amnesia to occur in Moscow
>before the amplification.
>
>I have been enjoying these discussions but unfortunately I will have to
>take leave, I am going on vacation with the family for a week so I will
>have little chance to participate during that time. I'll look forward
>to catching up when I return -
I wish I were unfortunate enough to have to take leave! I have been enjoying
these discussions too, and hope you have a good break.
--Stathis Papaioannou