Suppose I randomly pick a coin from all of Coinspace and flip it. What probability do you assign to the coin landing heads? Probably around 1/2.
Now suppose I do the same thing, but pick N coins and flip them all. The probability that they all come up heads is roughly 1/2^N.
Suppose I halt time to allow this experiment to continue as long as we want, then keep flipping coins randomly picked from Coinspace until I get a tail. What is the probability I will never get a tail? It should be the limit of 1/2^N as N goes to infinity, which is 0. Events with probability of 0 are allowed -- indeed, expected -- when you are dealing with infinite probability spaces such as this one.
It's also not true that we can't ever update if our prior probability for something is 0. It is just that we need infinite evidence, which is a scary way of saying that the probability of receiving said evidence is also 0. For instance, if you flip coins infinitely many times, and I observe all but the first 10 and never see "tails" (which has a probability of 0 of happening) then my belief that all the coins landed "heads" has gone up from 0 to 1/2^10 = 1/1024.
There are only countably many hypotheses that one can consider. In the coin flip context as you've constructed the probability space there are uncountably many possible results. If one presumes that there's a really a Turing computable (or even just explicitly definable in some axiomatic framework like ZFC) set of possibilities for the behavior of the coin, then there are only countably many each with a finite probability. Obviously, this in some respects makes the math much ickier, so for most purposes it is more helpful to assume that the coin is really ...
This post describes an infinite gamble that, under some reasonable assumptions, will motivate people who act to maximize an unbounded utility function to send me all their money. In other words, if you understand this post and it doesn't motivate you to send me all your money, then you have a bounded utility function, or perhaps even upon reflection you are not choosing your actions to maximize expected utility, or perhaps you found a flaw in this post.
Briefly, we do this with The St. Petersburg Paradox, converted to a mugging along the lines of Pascal's Mugging. I then tweaked it to extract all of the money instead of just a fixed sum.
I have always wondered if any actual payments have resulted from Pascal's Mugging, so I intend to track payments received for this variation. If anyone does have unbounded utility and wants to prove me wrong by sending money, send it with Paypal to tim at fungible dot com. Annotate the transfer with the phrase "St. Petersburg Mugging", and I'll edit this article periodically to say how much money I received. In order to avoid confusing the experiment, and to exercise my spite, I promise I will not spend the money on anything you will find especially valuable. SIAI would be better charity, if you want to do charity, but don't send that money to me.
Here's the hypothetical (that is, false) offer to persons with unbounded utility:
If I am lying and the offer is real, and I am a god, what utility will you receive from sending me a dollar? Well, the probability of me seeing N Tails followed by a Head is (1/2)**(N + 1), and your utility for the resulting universe is UTILITY(UN(N)) >= DUT * 2**N, so your expected utility if I see N tails is (1/2)**(N + 1) * UTILITY(UN(N)) >= (1/2)**(N + 1) * DUT * 2 ** N = DUT/2. There are infinitely many possible values for N, so your total expected utility is positive infinity * DUT/2, which is positive infinity.
I hope we agree that it is unlikely that I am a god, but it's consistent with what you have observed so far, so unless you were born with certain knowledge that I am not a god, you have to assign positive probability to it. Similarly, the probability that I'm lying and the above offer is real is also positive. The product of two positive numbers is positive. Combining this with the result from the previous paragraph, your expected utility from sending me a dollar is infinitely positive.
If you send me one dollar, there will probably be no result. Perhaps I am a god, and the above offer is real, but I didn't do anything beyond flipping the first coin because it came out Tails. In that case, nothing happens. Your expected utility for the next dollar is also infinitely positive, so you should send the next dollar too. By induction you should send me all your dollars.
If you don't send money because you have bounded utility, that's my desired outcome. If you do feel motivated to send me money, well, I suppose I lost the argument. Remember to send all of it, and remember that you can always send me more later.
As of 7 June 2011, nobody has sent me any money for this.
ETA: Some interesting issues keep coming up. I'll put them here to decrease the redundancy: