Thursday, June 13, 2013

Pascal's Wager and The St. Petersburg Lottery

Here's a fun bit of logic for you. The St. Petersburg Lottery was, for some time, an unsolved problem in math and economics. Imagine a lottery where the first day they flip a coin: heads, you win and get 1 cent. Tails, you get re-entered. Next round, heads, you win and get 2 cents; tails, again you get re-entered. And so on, with the pot doubling every time.

How much should you pay to buy into such a lottery? The chance that you will get to a certain point in the lottery (10 flips, 20 flips, wherever) gets smaller by 1/2 each round. But the prize also doubles each round! In theory (according to the old theory) we should be willing to pay an infinite amount to get into this racket. In practice we're not. How much are we willing to pay? The richer you are, the more you'll pay. What this is about is expected utility. For Bill Gates, paying a thousand bucks to enter hurts less than it hurts you. (I can say this because Carlos Slim doesn't read my blog.) The argument has even been made that this applies to high status/high self-esteem people who are willing to take risks in public and fail, because they lose less utility than others - it's just lost in the form of status rather than money.

Pascal's Wager was put by the mathematician and Catholic of the same name, who basically said you should believe just in case. Why? Because:
If you don't believe and you're right, fine.
If you don't believe and you're wrong, you're punished by going to Hell (so says a Catholic).
If you do believe and you're right, you're rewarded by going to Heaven.
If you do believe and you're wrong, you lead a less enjoyable life than you otherwise could have.

(This of course ignores that we can't voluntarily "choose" to believe in something. Try to choose to believe that the sky is red or your mother's name is Genghis. You can't. Which is a good thing.)

So if you do believe, the worst that happens to you is you waste some time, and you might get into Heaven. If you don't believe, the worst that happens is you go to Hell. (By the way, serious question: has anyone ever become a Christian because of Pascal's wager? Really. Ever?)

Of course the obvious problem is that in making this decision, you're assuming you know which god is the real one - that is to say, you must assume you know the nature of that god and how belief in that god affects your behavior - and that arriving at belief in said god in this manner (or at appearing to believe) will satisfy Him/Her/It/Them. Even ignoring that, the amount of reward by going to Heaven/punishment by going to Hell, and the chance of the chosen deity's existence, dramatically affect the outcome. So by believing, you will definitely restrain yourself to lead a less enjoyable life, and you have to compare that lost utility to the (possibly miniscule, as Pascal conceded) chance of the Catholic god's existence and (possibly large amount) that Heaven will feel good or Hell will hurt.

The key comparison here to the lottery above is that the utility lost will vary between individuals, based on their abilities to control their behavior. That is, in people making this calculation, those with less ability to control their behavior lose more utility - are hurt more - and rationally, are less likely to choose belief. Apparently, Yahweh created the simple-minded (and the rules of logic and game theory) in order to have a greater chance at punishing them for making (to them) rational choices. Not very nice, to punish the retarded and senile that He made that way in the first place.

Of course, we're already discussing a consequentialist ethics utility calculation that's supposed to result in your faking belief in a specific, omnipotent God that rules by virtue ethics. So go figure!

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