What's the difference between Pascal's wager and Pascal's mugging?

I was discussing Pascal's mugging with a friend (a mugger comes up to Pascal, and says give me 5 dollars or I'll use my special powers to kill a billion people in another dimension), and there's something about it I didn't understand. In theory the difference between the two is that in Pascals' wager, because the utilities being discussed are all infinite, they all balance out (Christian God as opposed to God that rewards atheists, and so on), whereas in the mugging, you do the math for the probability of the event and its expected utility, and there's some number you get to where the utility is so high (say, 1 trillion people) where the probability isn't low enough to justify non-action (since the utility can grow at a much faster rate than the probability shrinks). But why would the balancing out idea not hold true when you're talking about finite (but unbounded) utilities? Why should I not be able to simply say that it's just as likely that the mugger will make a billion people very happy if I don't give him the money, and so choose not to give him the money?

• One would certainly wonder why someone who has the power to "kill a billion people in another dimension" would want 5 measly dollars. If someone had that power, why not just live in the other dimension and rule over them? I understand the point, but it's a rather silly presentation.
– user18800
Jan 9 '16 at 6:21

Pascal's wager is a special case of Pascal's mugging in which the mugger claims infinite power.

I'm not really clear on which formations of the wager and mugging you're discussing as they seem to differ for the originals that I am familiar with, but I will attempt to shed some light on your questions.

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there's some number you get to where the utility is so high (say, 1 trillion people) where the probability isn't low enough to justify non-action

To which Robin Hanson would argue:

Robin Hanson has suggested penalizing the prior probability of hypotheses which argue that we are in a surprisingly unique position to affect large numbers of other people who cannot symmetrically affect us. Since only one in 3^^^^3 people can be in a unique position to ordain the existence of at least 3^^^^3 other people who can't have a symmetrical effect on this one person, the prior probability would be penalized by a factor on the same order as the utility.

Source

Why should I not be able to simply say that it's just as likely that the mugger will make a billion people very happy if I don't give him the money, and so choose not to give him the money?

Or in the original formation:

Mugger: If you hand me your wallet, I will perform magic that will give you an extra 1,000 quadrillion happy days of life.

Well if there were actually people out there that could perform such magic, the probability of meeting one would scale with the power of their magic in theory. Then, the God of Pascal's wager is just a Pascal mugger (or blesser) with infinite power. By Robin Hanson's criteria, this infinite power confers an infinite improbability.

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Why should I not be able to simply say that it's just as likely that the mugger will make a billion people very happy if I don't give him the money, and so choose not to give him the money?

The probabilities being assigned are to the mugger being truthful, not being powerful.

The force of the "Pascal's Mugger" argument seems to be aimed at illuminating what is judged as the arbitrary nature of the alternatives. In the original wager, Pascal presumes that the alternatives are Christian belief or atheism.

"Pascal's Mugger" parodies this line of argument by inserting arbitrary alternatives and following the original reasoning.

Ultimately whether the arguments are distinguishable, and whether one is more persuasive than the other, depends on your judgment of the plausibility of the alternatives. Pascal's wager still remains effective for someone who feels roughly balanced between faith and atheism for some given definition of faith, but not for someone who finds the entire concept ludicrous.