This post has been retracted because it is in error.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.
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 nonnegative too. Or if you don't assume that the result in the infinite case is the limit of finite results, then you have different problems, but then who says the strategy in the infinite case is the same as the limit of finite strategies?
To pick a concave function at random, let U(x£) = log10(x) utilons. And let my bank account contain 10£ at the beginning of the experiment.
U(10£) = EU(9£+A+B) = 1u, so I pay 1£ for options A+B.
Assume WLOG that I'm considering option A first. EU(y+B) = .5*U(y) + .5*U(y+1£). Set that equal to 1u and solve for y: y=9.51249£. Thus I'm indifferent to selling option A for 0.51249£.
After doing so, I am then indifferent to selling option B for 0.48751£.
So I'm back to exactly 10£. No money pump.
Upvoted.