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In your example, how much should you spend to choose A over B? Would you give up an unbounded amount of utility to do so?
This was helpful, thanks!
As I understand it, you are proposing modifying the example so that on some H1 through HN, choosing A gives you less utility than choosing B, but then thereafter choosing B is better, because there is some cost you pay which is the same in each world.
It seems like the math tells us that any price would be worth it, that we should give up an unbounded amount of utility to choose A over B. I agree that this seems like the wrong answer. So I don't think whatever I'm proposing solves this problem.
But that's a different problem than the one I'm considering. (In the problem I'm considering, choosing A is better in every possible world.) Can you think of a way they might be parallel--any way that the "I give up" which I just said above applies to the problem I'm considering too?
In response to
New Pascal's Mugging idea for potential solution
Let's be conservative and say the ratio is 1 in a billion.
Why?
Why not 1 in 10? Or 1 in 3^^^^^^^^3?
Choosing an arbitrary probability has good chances of leading us unknowingly into circular reasoning. I've seen too many cases of using for example Bayesian reasoning about something we have no information about, which went like "assuming the initial probability was x", getting some result after a lot of calculations, then defending the result to be accurate because the Bayesian rule was applied so it must be infallible.
It's arbitrary, but that's OK in this context. If I can establish that this works when the ratio is 1 in a billion, or lower, then that's something, even if it doesn't work when the ratio is 1 in 10.
Especially since the whole point is to figure out what happens when all these numbers go to extremes--when the scenarios are extremely improbable, when the payoffs are extremely huge, etc. The cases where the probabilities are 1 in 10 (or arguably even 1 in a billion) are irrelevant.
In response to
New Pascal's Mugging idea for potential solution
See https://arxiv.org/abs/0712.4318 , you need to formally reply to that.
Update: The conclusion of that article is that the expected utilities don't converge for any utility function that is bounded below by a computable, unbounded utility function. That might not actually be in conflict with the idea I'm grasping at here.
The idea I'm trying to get at here is that maybe even if EU doesn't converge in the sense of assigning a definite finite value to each action, maybe it nevertheless ranks each action as better or worse than the others, by a certain proportion.
Toy model:
The only hypotheses you consider are H1, H2, H3, ... etc. You assign 0.5 probability to H1, and each HN+1 has half the probability of the previous hypothesis, HN.
There are only two possible actions: A or B. H1 says that A gives you 2 utils and B gives you 1. Each HN+1 says that A gives you 10 times as many utils as it did under the previous hypothesis, HN, and moreover that B gives you 5 times as many utils as it did under the previous hypothesis, HN.
In this toy model, expected utilities do not converge, but rather diverge to infinity, for both A and B.
Yet clearly A is better than B...
I suppose one could argue that the expected utility of both A and B is infinite and thus that we don't have a good reason to prefer A to B. But that seems like a problem with our ability to handle infinity, rather than a problem with our utility function or hypothesis space.
It only takes less than 30 bits if your language supports the ^^^^ notation and that's not standard notation.
True. So maybe this only works in the long run, once we have more than 30 bits to work with.
In response to
New Pascal's Mugging idea for potential solution
"It takes less than 30 bits to specify 3^^^^3, no?"
That depends on the language you specify it in.
Yes, but I don't think that's relevant. Any use of complexity depends on the language you specify it in. If you object to what I've said here on those grounds, you have to throw out Solomonoff, Kolmogorov, etc.
In response to
New Pascal's Mugging idea for potential solution
See https://arxiv.org/abs/0712.4318 , you need to formally reply to that.
Yes, I've read it, but not at the level of detail where I can engage with it. Since it is costly for me to learn the math necessary to figure this out for good, I figured I'd put the basic idea up for discussion first just in case there was something obvious I overlooked.
Edit: OK, now I think I understand it well enough to say how it interacts with what I've been thinking. See my other comment .
In response to
The map of p-zombies
I disagree with your characterization of 0. You say that it is incompatible with physicalism, but that seems false. Indeed it seems to be a very mainstream physicalist view to say "I am a physical object--my brain. So a copy of me would have the same experiences, but it would not be me."
In response to
Experiment: Changing minds vs. preaching to the choir
How do I do step II? I can't seem to find the relevant debates. I found one debate with the same title as the minimum wage one I argued about, but I don't see my argument appearing there.
There is no way to raise a human safely if that human has the power to exponentially increase their own capabilities and survive independently of society.
Yep. "The melancholy of haruhi suzumiya" can be thought of as an example of something in the same reference class.
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The problem there, and the problem with Pascal's Mugging in general, is that outcomes with a tiny amount of probability dominate the decisions. A could be massively worse than B 99.99999% of the time, and still naive utility maximization says to pick B.
One way to fix it is to bound utility. But that has its own problems.
The problem with your solution is that it's not complete in the formal sense: you can only say some things are better than other things if they strictly dominate them, but if neither strictly dominates the other you can't say anything.
I would also claim that your solution doesn't satisfy framing invariants that all decision theories should arguably follow. For example, what about changing the order of the terms? Let us reframe utility as after probabilities, so we can move stuff around without changing numbers. E.g. if I say utility 5, p:.01, that really means you're getting utility 500 in that scenario, so it adds 5 total in expectation. Now, consider the following utilities:
1<2 p:.5
2<3 p:.5^2
3<4 p:.5^3
n<n+1 p:.5^n
...
etc. So if you're faced with choosing between something that gives you the left side or the right side, choose the right side.
But clearly re-arranging terms doesn't change the expected utility, since that's just the sum of all terms. So the above is equivalent to:
1>0 p:.5
2>0 p:.5^2
3>2 p:.5^3
4>3 p:.5^4
n>n-1 p:.5^n
So your solution is inconsistent if it satisfies the invariant of "moving around expected utility between outcomes doesn't change the best choice".
Again, thanks for this.
"The problem with your solution is that it's not complete in the formal sense: you can only say some things are better than other things if they strictly dominate them, but if neither strictly dominates the other you can't say anything."
As I said earlier, my solution is an argument that in every case there will be an action that strictly dominates all the others. (Or, weaker: that within the set of all hypotheses of probability less than some finite N, one action will strictly dominate all the others, and that this action will be the same action that is optimal in the most probable hypothesis.) I don't know if my argument is sound yet, but if it is, it avoids your objection, no?
I'd love to understand what you said about re-arranging terms, but I don't. Can you explain in more detail how you get from the first set of hypotheses/choices (which I understand) to the second?