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VoiceOfRa comments on Probabilities Small Enough To Ignore: An attack on Pascal's Mugging - Less Wrong Discussion
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Comments (176)
The M_i's can themselves be lotteries. The idea is to group events into finite lotteries so that the M_i's are >= N.
There is no principled way to chose utility functions either, yet people seem to be fine with them.
My point is that if one takes the VNM theory seriously as justification for having a utility function, the same logic means it must be bounded.
The VNM axioms are the principled way. That's not to say that it's a way I agree with, but it is a principled way. The axioms are the principles, codifying an idea of what it means for a set of preferences to be rational. Preferences are assumed given, not chosen.
Boundedness does not follow from the VNM axioms. It follows from VNM plus an additional construction of infinite lotteries, plus additional axioms about infinite lotteries such as those we have been discussing. Basically, if utilities are unbounded, then there are St. Petersburg-style infinite lotteries with divergent utilities; if all infinite lotteries are required to have defined utilities, then utilities are bounded.
This is indeed a problem. Either utilities are bounded, or some infinite lotteries have no defined value. When probabilities are given by algorithmic probability, the situation is even worse: if utilities are unbounded then no expected utiilties are defined.
But the problem is not solved by saying, "utilities must be bounded then". Perhaps utilities must be bounded. Perhaps Solomonoff induction is the wrong way to go. Perhaps infinite lotteries should be excluded. (Finitists would go for that one.) Perhaps some more fundamental change to the conceptual structure of rational expectations in the face of uncertainty is called for.
They show that you must have a utility function, not what it should be.
Well the additional axiom is as intuitive as the VNM ones, and you need infinite lotteries if you are too model a world with infinite possibilities.
This amounts to rejecting completeness. Suppose omega offered to create a universe based on a Solomonoff prior, you'd have to way to evaluate this proposal.
Given your preferences, they do show what your utility function should be (up to affine transformation).
You need some, but not all of them.
By completeness I assume you mean assigning a finite utility to every lottery, including the infinite ones. Why not reject completeness? The St. Petersburg lottery is plainly one that cannot exist. I therefore see no need to assign it any utility.
Bounded utility does not solve Pascal's Mugging, it merely offers an uneasy compromise between being mugged by remote promises of large payoffs and passing up unremote possibilities of large payoffs.
I don't care. This is a question I see no need to have any answer to. But why invoke Omega? The Solomonoff prior is already put forward by some as a universal prior, and it is already known to have problems with unbounded utility. As far as I know this problem is still unsolved.
Assuming your preferences satisfy the axioms.
No, by completeness I mean that for any two lotteries you prefer one over the other.
So why not reject it in the finite case as well?
Care to assign a probability to that statement.
Actually, I would, but that's digressing from the subject of infinite lotteries. As I have been pointing out, infinite lotteries are outside the scope of the VNM axioms and need additional axioms to be defined. It seems no more reasonable to me to require completeness of the preference ordering over St. Petersburg lotteries than to require that all sequences of real numbers converge.
"True." At some point, probability always becomes subordinate to logic, which knows only 0 and 1. If you can come up with a system in which it's probabilities all the way down, write it up for a mathematics journal.
If you're going to cite this (which makes a valid point, but people usually repeat the password in place of understanding the idea), tell me what probability you assign to A conditional on A, to 1+1=2, and to an omnipotent God being able to make a weight so heavy he can't lift it.
Ok, so care to present an a priori pure logic argument for why St. Petersburg lottery-like situations can't exist.
FInite approximations to the St. Petersburg lottery have unbounded values. The sequence does not converge to a limit.
In contrast, a sequence of individual gambles with expectations 1, 1/2, 1/4, etc. does have a limit, and it is reasonable to allow the idealised infinite sequence of them a place in the set of lotteries.
You might as well ask why the sum of an infinite number of ones doesn't exist. There are ways of extending the real numbers with various sorts of infinite numbers, but they are extensions. The real numbers do not include them. The difficulty of devising an extension that allows for the convergence of all infinite sums is not an argument that the real numbers should be bounded.
They have unbounded expected values, that doesn't mean the St. Petersburg lottery can't exist, only that its expected value doesn't.