Why Pascal Won’t Get Mugged
Philosopher Nick Bostrom (2009) imagines that it would be quite easy to trick Blaise Pascal out of his money. In fact, he thinks a mugger could use the reasoning Pascal applies in his own “Wager argument” to trick him into giving up his wallet willingly. Bostrom’s fantastic dialogue culminates in the following manner:
Mugger: . . .Well, have I got good news for you! I have magical powers. I can give you any finite amount of money that you might ask for tonight. What’s more, I can give you any finite amount of Utility that I choose to promise you tonight.
Pascal: And I should believe you why?
Mugger: Trust me! OK, I realize this does not give you conclusive evidence, but surely it counts a least a little bit in favour of the truth of what I am asserting. Honestly, I really do have these powers.
Pascal: Your conduct tonight has not inspired me with confidence in your honesty.
Mugger: OK, OK, OK, OK. But isn’t possible that I am telling the truth?
Pascal: It is possible that you have the magic powers that you claim to have, but let me tell you, I give that a very, very low probability.
Mugger: That’s fine. But tell me, how low a probability exactly? Remember, you might think it all seems implausible, but we are all fallible, right? And you must admit, from what you’ve already seen and heard, that I am a rather atypical mugger. And look at my pale countenance, my dark eyes; and note that I’m dressed in black from top to toe. These are some of the telltale signs of an Operator of the Seventh Dimension. That’s where I come from and that’s where the magic work gets done.
Pascal: Gee . . . OK, don’t take this personally, but my credence that you have these magic powers whereof you speak is about one in a quadrillion.
Mugger: Wow, you are pretty confident in your own ability to tell a liar from an honest man! But no matter. Let me also ask you, what’s your probability that I not only have magic powers but that I will also use them to deliver on any promise – however extravagantly generous it may seem – that I might make to you tonight?
Pascal: Well, if you really were an Operator from the Seventh Dimension as you assert, then I suppose it’s not such a stretch to suppose that you might also be right in this additional claim. So, I’d say one in 10 quadrillion.
Mugger: Good. Now we will do some maths. Let us say that the 10 livres that you have in your wallet are worth to you the equivalent of one happy day. Let’s call this quantity of good 1 Util. So I ask you to give up 1 Util. In return, I could promise to perform the magic tomorrow that will give you an extra 10 quadrillion happy days, i.e. 10 quadrillion Utils. Since you say there is a 1 in 10 quadrillion probability that I will fulfil my promise, this would be a fair deal. The expected Utility for you would be zero. But I feel generous this evening, and I will make you a better deal: If you hand me your wallet, I will perform magic that will give you an extra 1,000 quadrillion happy days of life.
Pascal: I admit I see no flaw in your mathematics.
Mugger: This is my final offer. You’re not going to pass up a deal that we have just calculated will give you an expected Utility surplus of nearly 100 Utils, are you? That’s the best offer you are likely to see this year.
Pascal: Is this legitimate? You know, I’ve committed myself to trying to be a good Christian.
Mugger: Of course it’s legitimate! Think of it as foreign trade. Your currency is worth a lot in the Seventh Dimension. By agreeing to this transaction, you give a major boost to our economy. Oh, and did I mention the children? If only you could see the faces of the sweet little orphans who will be made so much better off if we get this influx of hard currency – and there are so many of them, so very, very, very many . . . .
Pascal: I must confess: I’ve been having doubts about the mathematics of infinity. Infinite values lead to many strange conclusions and paradoxes. You know the reasoning that has come to be known as ‘Pascal’s Wager’? Between you and me, some of the critiques I’ve seen have made me wonder whether I might not be somehow confused about infinities or about the existence of infinite values . . .
Mugger: I assure you, my powers are strictly finite. The offer before you does not involve infinite values in any way. But now I really must be off; I have an assignation in the Seventh Dimension that I’d rather not miss. Your wallet, please!
Pascal hands over his wallet.
Mugger: Pleasure doing business. The magic will be performed tomorrow, as agreed (Bostrom 2009, 444-445).1
I think Bostrom incorrectly characterizes how Pascal would respond to the “seventh-dimension” mugger of finite power. When he is asked how probable it would be that mugger possesses magical powers to give any finite sum of money, Pascal answers 1:1 quadrillion. But it seems to me that it is more likely that a finite being has the magical power to conjure up smaller sums of money or utility than they do to conjure up larger sums. So my Pascal would say something like: “We in the fourth-dimension have a magical ability too. We can calculate the probability that a seventh-dimension mugger will be able to produce any given finite sum of money to an amazing degree of accuracy. So if I ask you to produce finite sum n, I know the probability that you will be able to produce that sum is 1:1,000,000,000,000,000n Consequently, the likelihood that you will not make good on your promise always outweighs the potential reward for taking the risk. Sorry, you can’t have my wallet.” Interestingly enough, if the mugger were to claim omnipotence, and that he could give infinite utility to Pascal, then the risk and reward are balanced. One might just give a wallet to such a god. But Pascal might just as well take a risk on Christ instead, since Christ never threatened to take his wallet in a dark alley!
Bostrom’s analysis trades on Pascal evaluating the likelihood that a mugger could produce ANY given amount of money rather than evaluating the likelihood that the mugger could produce a PARTICULAR amount of money. We are to take the probability of producing 1 quadrillion dollars as equally likely as the probability of producing 10 quadrillion. But why should we buy into this? If magical power is analogous to any other finite physical power source, then the likelihood that a given quantity of some effect will be produced is directly proportionate to the quantity promised. Sure, we could pretend along with Bostrom that Pascal is some kind of a buffoon, but I don’t think this sheds much light on Pascal’s wager. It is just an uncharitable characterization of this genius of the 17th century.
1N. Bostrom (2009) “Pascal’s Mugging”, Analysis, Vol: 69 (3), pp. 443 -445.