There is not any, for it is more fair and efficient than private run with it's endless paperwork to rip off the people with insurance.

Yes, they can be just as bureaucratic and wasteful as the privately run systems. The advantage is that the wealth transferred through them is passed more uniformly (e.g. not to large multinational corps) and the system is accessible to all people in the same way. Up here in Canada folks do get ticked that they can not jump to the front of the line if they are willing to pay the costs of the test or whatever (they still have the alternative to go to the USA). Another costly aspect of trying to run this sort of system next door to a privately run one in the USA is that the education of doctors is heavily subsidized up here, yet their freedom to practice where they want is not (yet) controlled. As a consequence the publics return on the investment in their education is low. After their education some of them opt out of Canada and go to the greener pastures of the USA with its higher malpractice and business costs. They also have the option of opting out of the public system, thereby not being able to see insured patients and practicing on their own in Canada (e.g. many plastic surgeons do this as cosmetic surgery is sually not covered by public insurance).

what are the economic advantages to government involvement in health care?