Same in Switzerland. Prices for services are fixed, and mandatory basic insurance won't pay more. Doctors and private hospitals are free to do their own thing,

Same in Switzerland. Prices for services are fixed, and mandatory basic insurance won't pay more. Doctors and private hospitals are free to do their own thing, as long as their clients (or their additional insurance) are willing to pay the difference. I think only hospitals are in public hand, but they still have to go via the insurances for payment.