Under the current law "insurance companies" (a misnomer) have been dropping like flies. There are areas of the country with no insurance operators at all.
How do you inject competition into that scenario?
Or are you talking about provider competition?
Ah, the thread has wandered into something that seems to get little discussion, but is crucial: Effective ways to drastically reduce health care costs.
The root problems at hand are government interference in the market, and people's expectation of a high level of care that someone else pays for. Both end up driving up costs. Other factors include a relatively "tight" supply of doctors, "defensive medicine", and essentially no communication to patients /families of costs of services.