Counterargument #1:
If you are god, then the universe allows for "gods" which can arbitrarily alter the state of the universe. Therefore, any utility gains I make have an unknown duration - it's entirely possible that an instant after you grant my utility, you'll take it away. Furthermore, if you are god, you're (a) flipping a coin and (b) requiring a donation, so I strongly suspect you are neither friendly nor omni-benevolent. Therefore, I have no reason to favour "god will help me for $1" over "god will hurt me for $1" - you could just as easily be trying to trap me, and punish anyone who irrationally sends you $1.
1b) I have no reason to select you as a likely god candidate, compared to the ~infinite number of people who exist across all of space-time and all Everett branches.
Counterargument #2:
There are finite many states of "N".
2a) Eventually the universe will succumb to heat death. Entropy means that we can't gain information from the coin flip without approaching this state. 2b) Even if you flip coins incredibly fast and in parallel, I will still eventually die, so we can only count the number of coin flips that happen before then.
Counterargument #3:
Assume a utility function which is finite but unbounded. It cannot handle infinity, and thus your mugging relies on an invalid input (infinite utility), and is discarded as malformed.
3b: Assume that my utility function fails in a universe as arbitrary as the one implied by you being god, since I would have witnessed a proof that state(t+1) does not naturally follow state(t)
Counterargument #4:
Carefully assign p(you are god) = 1/N, where N approaches infinity in such a way as to cancel out the infinite sum you are working with. This seems contrived, but my mind assigns p(you are god) = "bullshit, prove it", and this is about the closest I can come to expressing that mathematically ;)
Counterargument #5:
Assign probabilities by frequency of occurrence. There have been no instances of god yet, so p(god) = 0. Once god has been demonstrated, I can update off of this 0, unlike with Bayesian statistics. My utility function may very well be poorly designed, and I believe this can still allow for FAI research, etc.: social standing, an interest in writing code, peer pressure, etc. all provide motivations even if p(FAI) = 0. One could also assume that even where p(x) = 0, a different function rewards utility for investigating and trying to update even zero-probability events (in which case I'd get some utility from mailing you $1 to satisfy my curiosity, although I suspect not enough to overcome the cost of setting up a PayPal account and losing $1)
Counterargument 3(b) is the most convincing of these to me.
If my decision theory is predicated on some kind of continuity in states of the universe, and my decision is based on some discontinuity in the state of the universe, my decision theory can't handle this.
This is troubling, but to try to make it more formal: if I believe something like "all mathematically possible universes exist" then promising to "change universes to UN(N)" is a meaningless statement. Perhaps the wager should be rephrased as "increase the measure of universes of higher utility"?
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: