Tony Miller spent years pushing the idea that health plans with special accounts for covering deductibles might transform patients into savvy shoppers.

Now Miller has moved on to a new idea for insurance. His latest startup is a Minneapolis-based company called Bind that is selling the concept of “on-demand” health insurance, which lets workers in employer plans buy coverage for certain ailments only if they need the services.

Workers pay for a core health plan that covers what most people need. Add-in coverage would cover a few dozen specific services like knee arthroscopy or bariatric surgery that people buy only as needed, for three months at a time.

Subscribers would pay premiums and copays, but no deductibles or “coinsurance,” Miller said.

“Consumers don’t like doing fuzzy math with health benefits,” Miller said. “And fuzzy math is deductibles and coinsurance, because you’re making them figure out a price that the market is not willing to expose.”

Miller was the co-founder and chief executive at St. Louis Park-based Definity Health, a company that helped pioneer what are called “consumer-directed health plans” (CDHPs). Launched in the late 1990s, Definity was at the forefront of a movement that eventually coupled high-deductible health plans with health savings accounts (HSAs).

In these arrangements, employers and employees can put money into an HSA that workers tap to pay for medical services before meeting their deductible. Unused dollars in the account are invested in a manner similar to a 401(k) retirement plan, so workers have incentives to conserve and grow the funds by getting the best deal when buying health care.

In 2004, Minnetonka-based UnitedHealth Group agreed to acquire Definity Health for $300 million. The share of workers covered by the consumer-directed health plans has kept growing, but problems have arisen with big deductibles that stop some people from getting care.

“What’s happened in the intervening 20 years is, the market is stretching the consumer on the CDHP rack,” Miller said, adding that deductibles in the beginning were relatively small.

Bind wants to use copays to guide consumers to high-value health care providers. The co-payment for treating an ear infection, for example, might depend on where patients seek care — $15 at a retail clinic, for example, vs. $100 at urgent care.

Copay ranges vary with the service being sought and depend on the benefit design established by employers. Patients use their mobile devices to see what’s covered and what it will cost at different locations.

Currently, Bind is not an insurance company, Miller said, but serves as a third-party administrator for health plans where employers take the financial risk for the cost of medical bills. The startup began in 2016 inside Miller’s Wayzata venture capital firm Lemhi Ventures.

Miller said he’s not yet disclosing outside investments in Bind, but the company expects to make announcements soon. It employs 85 people “and we’re growing,” he said.

Miller is well established in the Twin Cities as a serial entrepreneur in health care, said Carolyn Pare, executive director of the Minnesota Health Action Group, a Bloomington-based coalition of large employers.

Employers in a tight labor market might be hesitant to try new things with health benefits if workers are content, Pare said, but she anticipated both “early adopters” and “fast followers” with Bind. She lauded Miller for once again devising a structure that’s meant to let consumers drive improvements in health care.

Earlier this month, Pare led a panel discussion at her group’s annual leadership summit in Edina that featured Miller, who mentioned his company’s pioneering role in consumer-directed health plans before telling the audience: “I’m here to make amends for that.” Fellow panelist François de Brantes responded that CDHPs helped make it easier for people to buy relatively low-cost health care services while also setting the stage for broader changes.

“Alternative health plans like Bind are designed to get to that next layer of transforming the market by focusing on areas where the consumer is spending a tremendous amount of money on out-of-pocket expenses and doesn’t have the incentive to do the right thing or the information they need to determine the best path,” said de Brantes, a health care consultant, via e-mail.