It was not supposed to play out this way. The bold Medicaid overhaul, part of a grand bargain with the state’s most politically powerful health care players, has been promoted as a national model for curbing costs and reversing the incentives for fraud. It transferred tens of thousands of recipients of long-term care from a system in which providers billed Medicaid for each service to managed care, in which a capped monthly rate must cover all services to a company’s enrollees.

With the largest Medicaid budget in the country, $54 billion, New York is trying not only to rein in runaway spending, but also to “rebalance” it, away from costly institutional care, like nursing homes and medical models known for overbilling, to inexpensive supports that keep people safely in their communities.

In that context, Jason Helgerson, the state’s Medicaid chief, defended the rapid expansion of social-model adult day care, saying that without a chance to socialize and connect with others, Medicaid clients would suffer a decline in health that would add costs. But when a reporter described some of the practices observed at centers, he expressed surprise and anger.

“The idea that people are bicycling home from managed long-term care is a complete misnomer,” Mr. Helgerson said. “The idea that they’re playing Ping-Pong — I guess they could be wheelchair-bound Ping-Pong players, but otherwise it’s fraud and they are not eligible.”

Beneficiaries are supposed to be impaired enough to need at least 120 days of help with tasks like walking, bathing or taking medication. But managed care companies, not government agencies, are now mainly in charge of determining eligibility, typically by using nurses to assess each potential member.

“It is being gamed,” said an executive at a managed care company, speaking on the condition of anonymity. “There are just plums in the payment system. And the state will choose to be blind about this until something happens, which is what they did with nursing homes.”