Now it turns out that such experiments are going on. The most recent issue of Health Affairs includes this article, "Payers Test Reference Pricing and Centers of Excellence to Steer Patients to Low-Price and High-Quality Providers," by Robinson and Macpherson. Calpers, the California public employees system, pays $30,000 to insureds for a knee or hip replacement, with any excess over that the responsibility of the patient. Safeway, the grocery company, pays $1500 for a colonoscopy, after they observed almost 10-fold variation in colonoscopy prices.
Many pharmaceutical plans already use this reference pricing concept for drugs, giving a patient only the amount that a "reference"drug would cost.
The potential for savings here is quite large. As I pointed out in a class the other day, there are static and dynamic effects. The static effect is the one-time cost savings by having patients choose a lower cost provider. The dynamic effect arises when the entity losing business realizes that and lowers price, or when a supplier sees that their demand is now more elastic, and by offering lower prices they can attract business from new patients.
But note the free-rider problem inherent in this: Safeway's innovation will create lower prices for all buyers, to the extent that the dynamic effect of competition kicks in. Safeway pays all the cost of the innovation but accrues only part of the benefit.