This the reason that NY banned rental agents from using algorithmic pricing models.
Rental has a problem, made worse by algorithmic pricing, it is, simply, that in some places, housing on a free market would be much more expensive than people can afford. Which is a problem because people need a place to live, and kicking them out is not great. So in this case, government intervention makes sense: a ban on dynamic pricing is one thing, but also just capping rents, subsidized housing, banning short term rentals, etc... the usual "socialist" stuff. But this is, again, not a dynamic pricing problem (though it makes things worse in this case), it is a housing problem.
> they get taken for every penny that they are willing to spend
What is the problem with paying for what you are willing to pay? It is only a problem when you are paying for what you are not willing to pay. Housing is one of these cases, so is food, healthcare and other essentials, also taxes. These are special cases and they usually involve government intervention. And sure, in this case, algorithmic pricing can be a problem, because you leave the usual free market economy.
This is useful up to a point. One constraint is whether the goods are fungible. (Rental location is not.) Another is whether price differences even exist.
> What is the problem with paying for what you are willing to pay?
"Free" markets are a fiction. Transactions do not exist in a legal vacuum. (Except perhaps for cryptocurrencies, which exemplify why such laws exist.)
The fact is that some markets are dysfunctional. Cartels, monopolies, and monopsonies are generally seen as evidence of market failure.
There are sound reasons for regulators to limit price gouging. Just because retailers have pricing power does not mean they should be allowed to exploit it to the maximum degree possible. For example: company scrip has been outlawed for many years, even in the USA. Likewise, antitrust laws restrict -- or are intended to restrict -- the ability of retailers to collude in fixing prices at whatever the market will bear.
Consider a real-life example. Imagine you bought a diamond ring for your fiancee. You paid $10,000 for a tiny stone because a cartel has been restricting the supply for 100 years. Six months later, chemists make diamonds in a lab and break the monopoly. You can now buy the same stone for $500. Do you feel ripped off? Or do you feel that the $10,000 you paid represented market value, and put the humiliating loss down to poor timing?
("Same stone" is not quite accurate, because for the time being de Beers is still trying to market "natural" stones at the original price. They don't have much choice because their stockpile represents an enormous unrealized loss. Behind the scenes, you can bet that they are trying to get rid of them any way they can. In crypto parlance, they are left holding the bag.)
Now substitute "life saving drug" for diamonds, and "pharma bro" for de Beers. (And recall that the price gouging was legal, Shkreli was jailed for an unrelated fraud.)
Free market is indeed somewhat fictional, as it is far from free in practice. The stock market in particular may look unhinged, but it is heavily regulated, that's the reason why it works, and the reason it looks like the ideal of a free market is, ironically, because of regulations.
As for diamonds, these are luxury goods, being expensive is the whole point, so what if they are overpriced? If de Beers didn't have a history of things like using slave labor, I wouldn't have a problem with it. Would I have hated it, if my $10k diamond turned out $500, yes, no one likes losing money from a bad investment, but prices have to go down at some point. It also shows that competition can work, de Beers didn't go down because of regulation, it went down because it was beaten by technology and bad publicity.
The problem with Shkreli and the Epipen was because healthcare is not a normal market and completely unlike diamonds. Normally, one would expect the government to make sure that life-saving drugs stay affordable, and it is the case in many countries, the US being a notable exception. Instead, in the US, the government promotes this kind of behavior though a combination of its mostly private health insurance system, tight control on who gets to make and sell drugs, but no control on the price. EU countries (most of the developed world in fact) are much more sane in that regard. Dynamic pricing is allowed as a general rule, luxury goods are still overpriced, but essentials like health and housing are regulated to prevent pathological cases like the Epipen thing.