New York hospital patients would no longer wake from surgery with the bad surprise of a big bill from an anesthesiologist outside their insurance networks under legislation unveiled this week by the Cuomo administration.

Instead, they would not be responsible for bills normally not covered by their health insurance companies in both emergencies and in situations like scheduled surgeries where they weren’t told that the anesthesiologist, radiologist or other specialist involved wasn’t part of their insurer’s network. It would be left to the insurance company and doctor or hospital to resolve those billing disputes, with unresolved matters sent to an independent arbitrator.

“It’s the No. 1 complaint we get at the Department of Financial Services,” Superintendent Ben Lawsky said. His agency, which regulates New York insurers, has been dealing with the issue for two years, talking to stakeholders, he said. After briefing legislators, the bill was unveiled in supporting documents to Gov. Andrew Cuomo’s proposed budget.

Unreimbursed out-of-network bills leave patients thousands of dollars in debt and can drive them into bankruptcy, Lawsky said. Arbitration between insurers and doctors or hospitals, modeled after the process used by Major League Baseball, would require arbitrators consider certain factors and choose one side’s proposal or the other, an approach that’s hoped to brings both sides closer together, he said.

“Patients are complete collateral damage in the wars between carriers and providers,” said Elisabeth Benjamin of the Community Service Society, which receives complaint calls at its help line. “It just happens all the time. The patient has the least amount of bargaining power.”

In one case, a patient was stuck with a bill for a mystery anesthesiologist and that doctor’s helper for more than $20,000, Benjamin said. The problem for patients is huge, as the solution would be, she said.

Under the legislation, insurers would have to permit access to out-of-network providers if they don’t have someone within their provider network with appropriate training and experience to meet the patient’s needs. It would take immediately after enactment.