To defend Medicare for a moment: it makes no economic sense to spend $10 in order to eliminate $5 worth of fraud. However, that math changes when everyone gets the idea that they can scam the system for small amounts of money with impunity, which is why insurance companies have fraud prevention systems.
A company in a free market will do its best to balance overzealous fraud prevention with preventing large losses due to fraud. A patient does not get any benefit from fraudulent payments nor for fraud prevention; both are essentially overhead. A company that strikes a good balance will either make a larger profit or will be able to pass along some of the cost savings to its customers (or both).
The problem comes in the superficial analysis of the efficiency of a company. If a company spends, say, $30 per customer to prevent $100 worth of fraud, and then charges $300 per month in premiums instead of $400 in premiums, it will look worse than a company that charges $370 per month for premiums and spends little to nothing on fraud prevention. In a superficial analysis, the former company spends 10% of premiums on “overhead” that does not “provide medical care,” which is substantially more than the second company spends. (Perversely, the higher premiums make the company’s percentage of overhead spending look even better: the larger denominator results in a smaller percentage.)
“Hey, Bridget, that would never happen in the real world. The insurance company would get creamed, because no one would spend an extra $70/month for the same product.”
Not in the free market, they wouldn’t, but I’ve just given you a simplified version of Medicare spending and the reason that Medicare famously “spends so little” on overhead.