Post retracted: If you follow expected utility, expect to be money-pumped

byStuart_Armstrong10y29th Oct 200920 comments


This post has been retracted because it is in error.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.