Re: self-sampling assumption is incorrect
Hi Wei Dai, About books. Concerning the provability logics I always mentionned the Boolos 1993 (or even his lovely lighter Boolos 1979), but I would like to mention also the book "Self-reference and modal logic" by Smorynski. The only problem is its very little caracters; I should go to the occulist! :0 But he has a nice chapter on the algebraic models of the provability logic: the so called diagonalisable algebras and fixed point algebras. As you know the Z logics I got are so weak that they loose (like G*) Kripke semantics or even Scott-Montague (sort of topological) semantics. So we need some algebraic move. Note that the G/G* story *begun* with those diagonalisable algebra through the work of Magari (Italy). But, perhaps more importantly at this stage I must recall the book "Mathematics of Modality" by Robert Goldblatt. It contains fundamental papers on which my "quantum" derivation relies. I mentionned it a lot some time ago. And now that I speak about Goldblatt, because of Tim May who dares to refer to algebra, category and topos! I want mention that Goldblatt did wrote an excellent introduction to Toposes: "Topoi". (One of the big problem in topos theory is which plural chose for the word "topos". There are two schools: topoi (like Goldblatt), and toposes (like Bar and Wells). :) Goldblatt book on topoi has been heavily attacked by "pure categorically minded algebraist like Johnstone for exemple, because there is a remnant smell of set theory in topoi. That is true, but that really help for an introduction. So, if you want to be introduced to the topos theory, Goldblatt Topoi, North Holland editor 19?(I will look at home) is perhaps the one. -Bruno PS I get your questions. I will think a little bit before answering. Thanks to Tim for Egan's exerp.
Re: self-sampling assumption is incorrect
On Tue, Jun 18, 2002 at 12:16:59AM -0700, Hal Finney wrote: > Let me start with this. If U(E1) > U(E2), then would a rational person > have to pick E1 over E2? What if he were someone who were contrary? > Or someone who preferred lesser utility? I think we can rule these > cases out by properly defining utility. With the proper definition it > will always be the case that if U(E1) > U(E2), he picks E1. Yes, this is part of the definition of utility. > Now consider a single-universe model. He can choose one of two > alternatives. In one alternative he is guaranteed to get E1, and in the > other alternative he has a 50-50 chance of getting E1 or E2. Is there > a rational way to prefer the second alternative? That is, can it be > better to have a chance of getting E2 rather than the certainty of E1? > I would like to rule this out for rational choosers, but I'm not 100% > sure. Some people seek risk, although a risk which has only a down side > still seems irrational. I agree with you here, because if he prefers the second alternative, he should not prefer E1 to E2. If faced with a choice between E1 and E2 he would do better to throw a mental coin and decide between them randomly. > There is an argument that there should be no differences, because the > information available in any sub-part of the multiverse is the same as in > the single universe case. In fact maybe we can never tell which theory > is correct, therefore the differences are entirely hypothetical. If > we accept this then what is irrational in the single universe case is > also irrational in the MWI. I disagree with you here. Although we have no direct sensory information about what happens in other branches of the multiverse, theory gives information about what happens in them, and that can be sufficient to change what we value in this branch. > Maybe you could expand on your argument about how diminishing utility > relates to evolutionary advantage across copies; I'm not sure what you > are getting at there. I see the reason higher quantities have less > marginal value to you as because of how they interact with each other > and with you; putting them all into separate universes would eliminate > the effects which I see as causing diminishing marginal value. You're right, the evolutionary advantage argument only applies to copies within a universe, not across universes (or non-interacting branches). The idea is that if you are content to have experiences similar to your copies, then the collection of your copies as a whole will contain less information (i.e. knowledge and skills) than the copies of someone who wants to have experiences different from his copies. So if your copies were to compete with his copies you would be at a disadvantage. P.S. I retract my claim that the self-sampling assumption is incorrect. I think I was just using it incorrectly. More on this in another post.
Re: self-sampling assumption is incorrect
After writing the following response, I realized that my argument against
the self sampling assumption doesn't really depend on E1 and E2 being
experiences. They can be any kind of events. Suppose they're prizes that
the copies can win for the original. E1 is a TV and E2 is a stereo. You'd
prefer a TV over a stereo but would rather have one TV and one stereo
instead of two TVs. Then my argument still works.
The issue of whether substitution effects can apply to experiences of
copies is of independent interest, so my original response still has a
point.
On Fri, Jun 14, 2002 at 07:42:27PM -0700, Hal Finney wrote:
> What about this variant on the experiment (the full experiment is below).
> Instead of B1 and B2 both getting E1, let B1 get E1 and B2 get E1'.
> E1' is another experience than E1 that is just about as good.
> U(E1) > U(E2) and U(E1') > U(E2). The idea is that this eliminates
> possible issues regarding whether two people (B1 & B2) who get exactly
> the same experience should count twice.
I think in that case it's still possible for U({E1,E1'}) < U({E1,E2}), if
for example E1 and E1' are very similar.
> It does seem that the SSA pretty much implies that if U(E1') > U(E2) then
> U({E1,E1'}) > U({E1,E2}). Is it really rational for this to be otherwise?
Yes, I believe it can be. If you believe otherwise you have to convince me
why it's impossible to value diversity of experience in your copies, or
why having that value would lead to absurd consequences.
We all know the law of diminishing marginal utility, which says that the
marginal utility of a good decreases as more of that good is consumed, and
the existence of substitution effects, where the marginal utility of one
good decreases when another similar good is consumed. I suggest there is
no reason to assume that the value of experiences of one's copies cannot
exhibit similar cross-dependencies. Actually I think the reason that
we have diminishing marginal utility and substitution effects, namely that
they provide an evolutionary advantage, also applies to the value of
experiences of copies.
> We know that rationality puts some constraints on the utility function.
> We can't have cyclicity in the utility preference graph, for example.
Our normative theories of rationality (i.e. decision theories) do put
constraints on preferences, but the history of decision theory has been
one of recognizing and removing unnecessary constraints, so that it can be
used by wider classes of people. The earliest decision theories for
example where stated in terms of maximizing expected money payoffs rather
than expected utility, which implicitly assumes that utility is a linear
function of money. Today, of course we recognize that utility can be any
function of money, even a decreasing one. Another example is the move from
objective probabilities to subjective probabilities.
> But in the case above, where U({X,Y}) means the utility of having two
> different independent experiences X and Y, maybe it does follow that
> U({X,Y}) and U({X,Z}) must compare the same as U(Y) and U(Z). You don't
> have any choice but to accept the equivalence. As Lewis Carroll wrote,
> "Then Logic would take you by the throat, and FORCE you to do it!"
> (http://www.mathacademy.com/pr/prime/articles/carroll/index.asp)
But remember that we choose the axioms. Logic doesn't tell use which
axioms to use.
Re: self-sampling assumption is incorrect
Hal Finney wrote:
>
> Wei writes:
> > Earlier (at http://www.lucifer.com/exi-lists/extropians/2612.html) I
> > argued that preference between the two choices is subjective (i.e. depends
> > on your utility function). I now realize this implies that the
> > self-sampling assumption (or SSA, the idea that you should reason as if
> > you were a random sample from the set of all observers, see
> > http://www.anthropic-principle.com/index.html for more details) cannot be
> > applied universally, because it implies that only choosing the two
> > identical experiences is rational.
> > ...
>
> What about this variant on the experiment (the full experiment is below).
> Instead of B1 and B2 both getting E1, let B1 get E1 and B2 get E1'.
> E1' is another experience than E1 that is just about as good.
> U(E1) > U(E2) and U(E1') > U(E2). The idea is that this eliminates
> possible issues regarding whether two people (B1 & B2) who get exactly
> the same experience should count twice.
>
> > Now which strategy I should choose
> > depends on whether U({E1,E1}) > U({E1,E2}), which can be independent of
> > whether U(E1) > U(E2).
>
> We can change this to whether U({E1,E1'}) > U({E1,E2}) in the modified
> form.
>
> It does seem that the SSA pretty much implies that if U(E1') > U(E2) then
> U({E1,E1'}) > U({E1,E2}). Is it really rational for this to be otherwise?
> We know that rationality puts some constraints on the utility function.
> We can't have cyclicity in the utility preference graph, for example.
>
> But in the case above, where U({X,Y}) means the utility of having two
> different independent experiences X and Y, maybe it does follow that
> U({X,Y}) and U({X,Z}) must compare the same as U(Y) and U(Z). You don't
> have any choice but to accept the equivalence. As Lewis Carroll wrote,
> "Then Logic would take you by the throat, and FORCE you to do it!"
> (http://www.mathacademy.com/pr/prime/articles/carroll/index.asp)
>
> Hal
>
I find it very hard to see how U({E1,E2}) is anything other than
p(E1)*U(E1)+p(E2)*U(E2) in this sort of experiment.
In the leadup to the discussion, Wei was suggesting that having two
different experiences may be better than repeating the same
experience. Surely this can only be true if you get to keep the first
experience when you experience the second, a situation that is false
in the current setup, since E2 is only experienced if you haven't
experienced E1.
Keep trying, but at this stage the argument against the SSA is not
compelling.
Cheers
A/Prof Russell Standish Director
High Performance Computing Support Unit, Phone 9385 6967, 8308 3119 (mobile)
UNSW SYDNEY 2052 Fax 9385 6965, 0425 253119 (")
Australia[EMAIL PROTECTED]
Room 2075, Red Centrehttp://parallel.hpc.unsw.edu.au/rks
International prefix +612, Interstate prefix 02
Re: self-sampling assumption is incorrect
Wei writes:
> Earlier (at http://www.lucifer.com/exi-lists/extropians/2612.html) I
> argued that preference between the two choices is subjective (i.e. depends
> on your utility function). I now realize this implies that the
> self-sampling assumption (or SSA, the idea that you should reason as if
> you were a random sample from the set of all observers, see
> http://www.anthropic-principle.com/index.html for more details) cannot be
> applied universally, because it implies that only choosing the two
> identical experiences is rational.
> ...
What about this variant on the experiment (the full experiment is below).
Instead of B1 and B2 both getting E1, let B1 get E1 and B2 get E1'.
E1' is another experience than E1 that is just about as good.
U(E1) > U(E2) and U(E1') > U(E2). The idea is that this eliminates
possible issues regarding whether two people (B1 & B2) who get exactly
the same experience should count twice.
> Now which strategy I should choose
> depends on whether U({E1,E1}) > U({E1,E2}), which can be independent of
> whether U(E1) > U(E2).
We can change this to whether U({E1,E1'}) > U({E1,E2}) in the modified
form.
It does seem that the SSA pretty much implies that if U(E1') > U(E2) then
U({E1,E1'}) > U({E1,E2}). Is it really rational for this to be otherwise?
We know that rationality puts some constraints on the utility function.
We can't have cyclicity in the utility preference graph, for example.
But in the case above, where U({X,Y}) means the utility of having two
different independent experiences X and Y, maybe it does follow that
U({X,Y}) and U({X,Z}) must compare the same as U(Y) and U(Z). You don't
have any choice but to accept the equivalence. As Lewis Carroll wrote,
"Then Logic would take you by the throat, and FORCE you to do it!"
(http://www.mathacademy.com/pr/prime/articles/carroll/index.asp)
Hal
> Here's a demonstration of this. Suppose you've agreed to participate in
> the following experiment. First you're copied. The original will observe
> while the copy (named A1) is told the following. A1 will be copied into
> A2, B1 and B2. All four will be run on seperate and identical computers.
> A1 and A2 will be shown a number equal to the millionth bit in the binary
> expansion of PI. B1 and B2 will both be shown a number equal to 1 minus
> that bit. All four will be asked to guess the millionth bit of PI. (Assume
> you have no idea what the millionth bit is.) If A1 guesses correctly, it
> will experience a very pleasant experience (call this experience E1). Same
> applies for B1 and B2, each of whom will also have E1 if he guesses
> correctly. If A2 guesses correctly however, he will experience a slightly
> less pleasant experience E2. If anyone guesses incorrectly, he's halted
> immediately. In any event all four copies are halted at the end of the
> experiment. (The setup can be changed so that the four runs are done
> sequentially instead of in parallel. I don't think that affects my
> argument at all.)
>
> Now put yourself in the position of A1 before he's been further copied,
> trying to devise a strategy for guessing the millionth bit of PI. Let's
> call that bit X and the number you'll be shown Y, and consider the two
> strategies A) guess Y, and B) guess 1-Y. It should be obvious at this
> point that if you prefer to have two identical very pleasant experiences
> you'll select strategy B, and if you prefer to have one very pleasant
> experience and one slightly less pleasant experience you'll select
> strategy A. However according to the SSA only strategy B is rational.
>
> Here's how I would analyze the situation given the SSA. After being shown
> Y, there's 1/4 probability that I'm A1, 1/4 probability that I'm A2, 1/4
> probability that I'm B1, and 1/4 probability that I'm B2. So if I guess Y,
> there's 1/4 probability that I cause a copy of me to experience E1 and 1/4
> probability that I cause a copy of me to experience E2, therefore my
> expected utility is U(A) = 1/4*U(E1) + 1/4*U(E2). If I guess 1-Y instead,
> there's 1/4 probability that I cause a copy of me to experience E1 and
> another 1/4 probability that I cause a copy of me to experience E1, so my
> expected utility is U(B) = 1/2*U(E1). Since U(E1) > U(E2), U(B) > U(A).
>
> Here's my proposed non-SSA way of analyzing the situation. After being
> shown Y, I consider myself to be A1, A2, B1, and B2 "simultaneously". If I
> guess 1-Y, there's probability of 1 that I cause two copies of me to
> experience E1 (call this {E1,E1}). If I guess Y, there's probability of 1
> that I cause one copy of me to experience E1 and one copy of me to
> experience E2 (call this {E1,E2}). Now which strategy I should choose
> depends on whether U({E1,E1}) > U({E1,E2}), which can be independent of
> whether U(E1) > U(E2).
>
> So my position is that rather than being a principle of correct reasoning,
> the status of the SSA should be reduced to that of an approximation useful
>
