Market behavior when consumer must choose product before knowing price? 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.