Before we begin, see this similar article about a different mugging scenario, which demonstrated that all probability is conditional and that decision isn’t probability.
Here’s the serious version of Pascal’s Mugging (Eliezer Yudkowsky, or “Yud’s” from Less Wrong, makes little sense; paragraph breaks mine).
Blaise Pascal is accosted by a mugger who has forgotten his weapon. However, the mugger proposes a deal: the philosopher gives him his wallet, and in exchange the mugger will return twice the amount of money tomorrow. Pascal declines, pointing out that it is unlikely the deal will be honoured.
The mugger then continues naming higher rewards, pointing out that even if it is just one chance in 1000 that he will be honourable, it would make sense for Pascal to make a deal for a 2000 times return.
Pascal responds that the probability for that high return is even lower than one in 1000.
The mugger argues back that for any low probability of being able to pay back a large amount of money (or pure utility) there exists a finite amount that makes it rational to take the bet — and given human fallibility and philosophical scepticism a rational person must admit there is at least some non-zero chance that such a deal would be possible.
In one example, the mugger succeeds by promising Pascal 1,000 quadrillion happy days of life. Convinced by the argument, Pascal gives the mugger the wallet.
The previous article and the paragraph breaks should be a giveaway that the whole thing is silly. But if isn’t obvious, here’s the breakdown.
We’re trying to get inside Pascal’s head and form some list of evidence that is probative of the proposition, Y = “The mugger will another day give me X”, where X varies according to the deal. First, the mugger is always wrong: there is no probability for his promise. That probability only exists in Pascal’s head, deduced on the evidence Pascal decides to accept.
Some character comes up to you on the street and says “I’m a philosophical mugger. Gimme $100 and tomorrow I’ll give you $200”, and you’d form the evidence E= “This guy is a lunatic.” From which you’d deduce the probability of Y is 0; i.e. Pr(Y | E) = 0. Further, the higher the amount the mugger claims he’ll give you, the more you’d add to your evidence of the fellow’s insanity. “This guy is a lunatic. He has the power of granting 1,000 quadrillion happy days of life? Give me a break. And fetch me a straight-jacket.” From this, also you’d deduce the probability of Y as 0.
If you like, you can say the “weight” of the probability of Y has increased, in the sense each that addition to the evidence list would on its own lead to a 0 probability of Y (or close to that, depending on how you internally phrased each item of evidence). Learn more about probability “weight” in this fine book.
Anyway, why the interest in such a simple non-problem? Because of the mistaken belief that “events” have probabilities. The mugger insists Pr(Y) > 0, and therefore ups his offer so that whatever decision rule (see the first article about decision rules) Pascal uses, eventually the rule will say it’s optimal to hand over his cash.
This is true; it follows. But that is only because the false belief that events have probabilities. There is no such thing as “Pr(Y)”: it doesn’t exist. It is always wrong. Only Pr(Y | E) exists; so we must have evidence E.
Is Pascal’s Mugging salvageable? Suppose Pascal’s E is the proposition, “The mugger might pay me X”. But that’s logically equivalent to “The mugger might not pay me X”, which is equivalent to “The mugger might or might not pay me X”, which is a tautology. And which thus gives no information about Y.
No. The only way to save PM as a sensible problem is for Pascal himself to add evidence like, “I like the way this fellow looks; he has an honest face. People with honest faces keep their promises in bizarre situations like this sometimes, but not usually. He can keep his promise of giving me X if he really wanted to.” That’s a mouthful, but that’s how our minds work.
Still, there’s no reason to suppose anybody would think like that (except in fanciful, fantastic situations). If X is “Mugger will give me 1,000 quadrillion happy days of life”, nobody would believe it. I doubt anybody would believe it if the mugger promised a buck-fifty.