I like the Dutch system of health care vouchers. It gives everyone coverage, but is competitive because private companies compete for the voucher money.
It's like that in Israel - there are several private companies competing for customers, each has hospitals and clinics and whatnot. They aren't paid by the customers though, but by the portion of health taxes collected, relevant to their portion of the overall taxpaying population. If they want people subscribing to them, they have to provide good service, so there's healthy competition, and yet as a patient, I'm not paying any premiums, it's all in the tax. The only problem is, in Israel the taxes are insanely high (I was paying 56%) and could be much lower, but the system itself seems to work very well
Clearly it doesn't though - taxes are to high - likely due to corruption.