Despite the title, this is a more general and clearly solvable problem, I just can't research it because I have no idea what it's called.
Assume you are trying to buy something (such as a house) where there are many choices and gathering information about each choice is expensive enough to care about. Furthermore, assume that the process takes so long that you must decide whether to buy before you can evaluate the next house. Assume that you are capable of evaluating all the various dimensions each house can be rated on, and coming up with a single number that represents how good that house is for you (call it, HouseValue). Assume that you can express the cost of evaluating a house in terms of that HouseValue. Assume you don't know anything about the statistical distribution of the HouseValue number (I guess you would assume it's normal but you don't know the mean or the variance, but you are the experts and I'll take your word on this).
What is the formula for the optimal time to stop looking and buy the house you just evaluated? I'm pretty sure this is a well-known problem, or very close to a well known problem, but I don't know where to begin looking for it.
(I'm really guessing when I tag this, by the way).