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Elithrion comments on Pascal's Mugging as an epistemic problem - Less Wrong Discussion
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Why wait for the mugger to make his stupendous offer? Maybe he's going to give you this stupendous blessing anyway -- can you put a sufficiently low probability on that? Don't you have to give all your money to the next person you meet? But wait! Maybe instead he intends to inflict unbounded negative utility if you do that -- what must you do to be saved from that fate? Maybe the next rock you see is a superintelligent, superpowerful alien who, for its superunintelligible reasons requires you to -- well, you get the idea.
The difference between this and the standard Mugger scenario is that by making his offer, the mugger promotes to attention the hypothesis that he presents. However, for the usual Bayesian reasons, this must at the same time promote many other unlikely hypotheses, such as the mugger being an evil tempter. I don't see any reason to suppose that the mugger's claim promotes any of these hypotheses sufficiently to distinguish the two scenarios. If you're vulnerable to Pascal's Mugger, you've already been mugged by your own decision theory.
If your decision theory has you walking through the world obsessed with tiny possibilities of vast utility fluctuations, like a placid-seeming vacuum state seething with colossal energies, then your decision theory is wrong. I propose the following constraint on utility-based rational decision theories:
The Anti-Mugging Axiom: For events E and current knowledge X, let P(E|X) = probability of E given X, U(E|X) = utility of E given X. For every state of knowledge X, P(E|X) U(E|X) is bounded over all events E.
The quantifiers here are deliberately chosen. For each X there must be an upper bound, but no bound is placed on the amount of probability-weighted utility that one might discover.
Well, it's been two-and-a-quarter years since that post, but I'll comment anyway.
Isn't the anti-mugging axiom inadequate as stated? Basically, you're saying the expected utility is bounded, but bounded by what? If the bound is, for example, equivalent to 20 happy years of life, you're going to get mugged until you can barely keep from starving. If it's less than 20 happy years of life, you probably won't bother saving for retirement (assuming I'm interpreting this correctly).
Another way of looking at it, is that, let's say the bound is b, then U(E|X) < b/P(E|X) ∀ X, ∀ E. So an event you're sure will happen can have maximum utility b, but an event that you're much less confident about can have vastly higher maximum utilities. This seems unintuitive (which is not as much of an issue as the one stated above).
Perhaps a stronger version is necessary. How about this: P(E|X) U(E|X) should tend to zero as U(E|X) tends to infinity. Or to put that with more mathematical clarity:
For any sequence of hypothetical events E_i, i=0, 1, ..., if the sequence of utilities U(E_i|X) tends to infinity then the sequence of expectations P(E_i|X) U(E_i|X) must tend to zero.
Or perhaps an even stronger "uniform" version: For every e > 0 there exists a utility u such that for every event E with U(E|X) > u, its expected utility P(E|X) U(E|X) is less than e.
I called this an axiom, but it would be more accurate to call it a principle, something that any purported decision theory should satisfy as a theorem.
Hm, to be honest, I can't quite wrap my head around the first version. Specifically, we're choosing any sequence of events whatsoever, then if the utilities of the sequence tend to infinity (presumably equivalent to "increase without bound", or maybe "increase monotonically without bound"?), then the expected utilities have to tend to zero? I feel like there's not enough description of the early parts of the sequence. E.g. if it starts off as "going for a walk in nice weather, reading a mediocre book, kissing someone you like, inheriting a lot of money from a relative you don't know or care about as you expected to do, accomplishing something really impressive...", are we supposed to reduce probabilities on this part too? And if not, then do we start when we're threatened with 3^^^3 disutilons, or only if it's 3^^^^3 or more, or something?
I don't think the second version works without setting further restrictions either, although I'm not entirely sure. E.g. choose u = (3^^^^3)^2/e, then clearly u is monotonically decreasing in e, so by the time we get to e = 3^^^^3, we get (approximately) that "an event with utility around 3^^^^3 can have utility at most 3^^^^3" with no further restrictions (since all previous e-u pairs have higher u's, and therefore do not apply to this particular event), so that doesn't actually help us any.
Anyway, it took me something like 20 minutes to decide on that, which mostly suggests that it's been too long since I did actual math. I think the most reasonable and simple solution is to just have a bounded utility function (with the main question of interest being what sort of bound is best). There are definitely some alternative, more complicated, solutions, but we'd have to figure out in what (if any) ways they are actually superior.