Britain has socialized medicine, where the whole health care system is part of the state. Canada has single payer, where the government runs the insurance market but the rest of the health care system is privatized. The Netherlands has a regulated private market system, where the government sets rules and subsidies while letting the private sector do all the leg work.
The United States has all three. Veterans have socialized coverage through the Veteran's Administration. The young, poor, disabled, and elderly have single payer through SCHIP, Medicaid, and Medicare accordingly. The rest of us use the free market system to obtain coverage and health care, which will be governed by each state, as well as the US Secretary of Health and Human Services, as governed by the Obama health care law.
Why not simplify the whole thing and put everyone on a market system based off of the Obamacare exchanges? The government can run a bare-bones single payer system called Medicare (like it currently does) that would cover 100 percent of costs for those over 65 (like it currently does), those under 18 (like SCHIP does), and those at 133 percent or below the poverty line (like Medicaid does). Everyone else would receive subsidies to purchase health care on the exchanges based off of their wealth income (like Obama-care will). Employers could purchase tax-deductible vouchers for their employees that each of them could use to buy health care on the exchanges (like the employer tax-deduction now). Others would be able to buy into this Medicare plan if they so chose to.
All private insurance plans will be required to cover at minimum the same things that Medicare does. Based off additional coverage, co-pays, and deductibles, all private plans would be designated bronze, silver, or gold allowing for people to do easy comparison shopping on the exchange. If people chose to opt out of Medicare and choose a private insurance plan instead in order to obtain better coverage, then they would receive a voucher that would cover 100 percent of Medicare coverage.
Example 1: A retired 67 year old man is currently covered by Medicare. Under the new system, they could still be on Medicare. Or they could upgrade from Medicare to a Bronze-level plan provided by Blue Cross Blue Shield. If they upgraded, then they would receive a $4000 voucher from the government - which would be put towards the cost of the private $4,500 plan - so that the man would only have to pay $500 out of pocket in order to gain better coverage.
Example 2: A 34 year old working woman currently has health care through her employer and her employer receives tax deductions for providing coverage. Under the new plan, the government would give her one voucher based off her income level and her employer will purchase her another voucher in order receive tax deductions for it. She will be able to choose her level of coverage instead of her employer deciding her coverage for her.
Example 3: A 21 year old college student male currently receives coverage from his parents' insurance. Under the new plan, the government would cover 100 percent of a bare-bones plan based off his income. If he needed additional coverage, then he (or his parents) could upgrade to a private plan on the exchange. The purchaser would receive tax deductions for buying it.