This has been in my drafts folder for ages, but in light of Eliezer's post yesterday, I thought I'd see if I could get some comment on it:
A couple weeks ago, Vladimir Nesov stirred up the biggest hornet's nest I've ever seen on LW by introducing us to the Counterfactual Mugging scenario.
If you didn't read it the first time, please do -- I don't plan to attempt to summarize. Further, if you don't think you would give Omega the $100 in that situation, I'm afraid this article will mean next to nothing to you.
So, those still reading, you would give Omega the $100. You would do so because if someone told you about the problem now, you could do the expected utility calculation 0.5*U(-$100)+0.5*U(+$10000)>0. Ah, but where did the 0.5s come from in your calculation? Well, Omega told you he flipped a fair coin. Until he did, there existed a 0.5 probability of either outcome. Thus, for you, hearing about the problem, there is a 0.5 probability of your encountering the problem as stated, and a 0.5 probability of your encountering the corresponding situation, in which Omega either hands you $10000 or doesn't, based on his prediction. This is all very fine and rational.
So, new problem. Let's leave money out of it, and assume Omega hands you 1000 utilons in one case, and asks for them in the other -- exactly equal utility. What if there is an urn, and it contains either a red or a blue marble, and Omega looks, maybe gives you the utility if the marble is red, and asks for it if the marble is blue? What if you have devoted considerable time to determining whether the marble is red or blue, and your subjective probability has fluctuated over the course of you life? What if, unbeknownst to you, a rationalist community has been tracking evidence of the marble's color (including your own probability estimates), and running a prediction market, and Omega now shows you a plot of the prices over the past few years?
In short, what information do you use to calculate the probability you plug into the EU calculation?