What forces influence the price in a market, wherein the consumer must choose a good before knowing the price, and afterwards only knows the price of the good purchased, not that of any competing goods? For instance, in the US healthcare industry, a patient wanting to assess the cost associated with treating a particular condition - historically, at least - cannot receive quotes for the treatment.
Yes, we can model this as a choice under uncertainty, where the distribution is over the set of prices the consumer may face. The consumer could have an expected budget with expected demands and you would need to allow the consumer to go into debt if prices are higher than their wealth (think a quasilinear environment).
The second part, where the consumer doesn't know the other competing firms can also be modeled with uninformed consumers vs informed consumers, where firms compete on price over those kinds of consumers. The informed choose better option, the uninformed just choose the same regardless.
This seems like a case where the individuals in this market have inelastic demand curve for the good or service in question, as the individuals who do demand the service will consume the good or service regardless of the price.
the change in price would seem to come from the changes in the supply side i.e. change in technology making the service cheaper, increase in substitutes or provision of government subsidises.
I am unable to address the specific question of the US healthcare system inability to provide a quote for a given medical service, but the above two paragraphs should provide a basic framework for thinking about this issue.