This doesn't work with an unbounded utility function, for standard reasons:
1) The mixed strategy. If there is at least one lottery with infinite expected utility, then any combination of taking that lottery and other actions also has infinite expected utility. For example, in the traditional Pascal's Wager involving taking steps to believe in God, you could instead go around committing Christian sins: since there would be nonzero probability that this would lead to your 'wagering for God' anyway, it would also have infinite expected utility. See Alan Hajek's classic article "Waging War on Pascal's Wager."
Given the mixed strategy, taking and not taking your bet both have infinite expected utility, even if there are no other infinite expected utility lotteries.
2) To get a decision theory that actually would take infinite expected utility lotteries with high probability we would need to use something like the hyperreals, which would allow for differences in the expected utility of different probabilities of infinite payoff. But once we do that, the fact that your offer is so implausible penalizes it. We can instead keep our money and look for better opportunities, e.g. by acquiring info, developing our technology, etc. Conditional on there being any sources of infinite utility, it is far more likely that they will be better obtained by other routes than by succumbing to this trick. If nothing else, I could hold the money in case I encounter a more plausible Mugger (and your version is not the most plausible I have seen). Now if you demonstrated the ability to write your name on the Moon in asteroid craters, turn the Sun into cheese, etc, etc, taking your bet might win for an agent with an unbounded utility function.
Also see Nick Bostrom's infinitarian ethics paper.
As it happens I agree that human behavior and intuitions (as I weight them) in these situations are usually better summed up with a bounded utility function, which may include terms like the probability of attaining infinite welfare, or attaining a large portion of hyperreal expected welfare that one could, etc, than an unbounded utility function. I also agree that St Petersburg lotteries and the like do indicate our bounded preferences. The problem here is technical, in the construction of your example.
See Alan Hajek's classic article "Waging War on Pascal's Wager."
That article is paywalled. It was published in 2003. Hajek's entry about Pascal's Wager in the Stanford Encylopedia of Philosophy is free and was substantively revised (hopefully by Hajek) in 2008, so there's a good chance the latter contains all the good ideas in the former and is easier to get to. The latter does mention the idea that utilities should be bounded, and many other things potentially wrong with Pascal's wager. There's no neat list of four items that looks like a...
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: