Medicare Advantage is the private alternative to the traditional insurance program for seniors. It covers 11.7 million Americans. Some argue that it's less expensive than the public programs. Others say it's more expensive. It would be very hard for both of these things to be true.

A bit of background is helpful here: In 1997, the Balanced Budget Act aimed to open up the traditionally-public insurance program to private plans. That plan got a tepid response from insurers as it offered them low rate increases, pegged to overall economic growth.

"Since the passage of the Balanced Budget Act (BBA), progress toward these goals [of increasing competition in Medicare] has been halting," the independent Medicare Payment Advisory Commission concluded in a 2002 report. "The availability of plan options has not increased; most beneficiaries in rural areas still cannot enroll."

That's when Congress passed another law, the Medicare Modernization Act, which boosted the payments private insurers receive for covering Medicare enrollees.

Starting in 2004, Medicare Advantage began paying private plans a higher per-enrollee cost than the public program. It was true then and, as you can see here, has been pretty true ever since.

This chart is great to make the case that Medicare Advantage is more expensive than the government-run program: The government clearly pays them more to provide the same set of health-care benefits.

But that's not the case that James Capretta and Yuval Levin made this week. They argue that Medicare Advantage plans are less expensive, citing an Aug. 1 study in the Journal of the American Medical Association. That research looks at what would happen if we implemented the Ryan-Wyden Medicare reforms (which are pretty similar to Romney's proposals) right away.

Seniors would start getting a check to buy a public or private health plan. How much would the check be for? Excellent question! Medicare would comb through all the bids that private insurers submit, which shows how much they say it will cost to cover the average Medicare beneficiary's costs. They would then cut each senior a check equal to the second-lowest bid.

In 2009, the second-lowest bidder said it could cover the standard set of Medicare benefits for significantly less than the public program -- $717 less annually, to be exact. And that's why Capretta and Levin argue that the private plans are actually cheaper.

"Beneficiaries could get their comprehensive Medicare benefits for no additional premium if they selected the less expensive private plans," they write.

How, exactly, can private Medicare plans be less expensive for seniors but more expensive for the government? David Cutler, one of the study authors, helpfully points out that it has to do with the difference between the cost of health benefits, and how much we pay for them. Right now, the federal government pays private plans are well beyond the actual cost of providing benefits.

Here's why: Each year, the federal government sets a "benchmark" for how much it will pay health plans for each senior they enroll. The benchmark is routinely higher than the per-senior cost of traditional Medicare. If a health insurance company undercuts the benchmark and bids lower -- as about 90 percent of health plans do -- they net the difference as a rebate.

"Medicare Advantage plans bid less than traditional Medicare, but they are paid more," Cutler writes.

But since Medicare Advantage plans are bidding lower, does that mean they are indeed more cost-effective? It could. Or it could not -- as Cutler points out, Medicare covers a number of non-medicine programs that private plans do not. It covers most graduate medical education in the United States and sends supplemental payments to hospitals that see more uninsured patients. Those extra costs get folded in when analysts estimate Medicare's per-senior costs.

Cutler also points out the possibility of risk selection, that Medicare Advantage plans have kept their costs down by wooing the less costly seniors who have fewer medical problems. There is some evidence of Medicare Advantage plans offering gym memberships as a benefit -- and subsequently wooing healthier members, the ones who would actually work out.

A lot, in short, hinges on the definition of "expensive." Medicare Advantage plans are more expensive for the federal government than the traditional public plan. But when you look at how much the private plans say their benefits actually cost -- rather than what they get paid to deliver them -- Medicare Advantage comes out as the less expensive option.