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.
