If you have a consistent utility function over outcomes, you cannot be money-pumped.
If your utility is convex in money and you follow independence, I can money pump you no matter what the situation, as L will always be worth more to you than £1.50. I will continue offering you that contract until you have no cash left, an event that is certain to eventually happen. So your statement is incorrect.
If your utility function is concave in money, it's a little harder, but I can use options. Contract A will give out £1 if a coin comes up head; contract B will give you £1 if that same coin comes up tails. I offer you cash for the possibility of buying these contracts from you for free (should you ever get your hands on them), as long as your capital is within £2 of your current amount. You should name a price less than 0.50 for these options, including a small utility profit for you; I take one option out on each of A and B. I then sell you A and B together, for £1 (since together they are exactly the same as a certain £1). I then exercise both my options and get A back, then B.
Of course, you would never do anything as stupid as accepting the contracts I've just described; but the fact remains that if your utility is not linear in money, you cannot put consistent prices on contracts and their combinations, so will end up losing if ever you blindly follow your utility function.
I will continue offering you that contract until you have no cash left, an event that is certain to eventually happen.
Only if you have an infinite bankroll. Otherwise, there is some tiny but nonzero chance that you lose all your money and the player makes a huge profit. And for the player with the convex utility function, the utility of that outcome is enough to make the whole ensemble of gambles worthwhile.
Then if you extend that to the infinite case by putting the limit outside the expected utility calculation, you will find that the limit is nonnega...
This post has been retracted because it is in error. Trying to shore it up just involved a variant of the St Petersburg Paradox and a small point on pricing contracts that is not enough to make a proper blog post.
I apologise.
Edit: Some people have asked that I keep the original up to illustrate the confusion I was under. I unfortunately don't have a copy, but I'll try and recreate the idea, and illustrate where I went wrong.
The original idea was that if I were to offer you a contract L that gained £1 with 50% probability or £2 with 50% probability, then if your utility function wasn't linear in money, you would generally value L at having a value other that £1.50. Then I could sell or buy large amounts of these contracts from you at your stated price, and use the law of large number to ensure that I valued each contract at £1.50, thus making a certain profit.
The first flaw consisted in the case where your utility is concave in cash ("risk averse"). In that case, I can't buy L from you unless you already have L. And each time I buy it from you, the mean quantity of cash you have goes down, but your utility goes up, since you do not like the uncertainty inherent in L. So I get richer, but you get more utility, and once you've sold all L's you have, I cannot make anything more out of you.
If your utility is convex in cash ("risk loving"), then I can sell you L forever, at more than £1.50. And your money will generally go down, as I drain it from you. However, though the median amount of cash you have goes down, your utility goes up, since you get a chance - however tiny - of huge amounts of cash, and the utility generated by this sum swamps the fact you are most likely ending up with nothing. If I could go on forever, then I can drain you entirely, as this is a biased random walk on a one-dimensional axis. But I would need infinite ressources to do this.
The major error was to reason like an investor, rather than a utility maximiser. Investors are very interested in putting prices on objects. And if you assign the wrong price to L while investing, someone will take advantage of you and arbitrage you. I might return to this in a subsequent post; but the issue is that even if your utility is concave or convex in money, you would put a price of £1.50 on L if L were an easily traded commodity with a lot of investors also pricing it at £1.50.