Another way to turn it into a more competitive market is to introduce large competition in the form of not for profit housing co-ops where the coop buys to rent houses, renters have to buy into the coop, and all profits are put into either buying or building new houses.

The coops keep their rents below that of the competition to attract new members, and can even reduce the rents after the initial outlay has been payed off.

This would then create a number of large semi-altruistic competitors that would throw sand into the capitalistic machine and inhibit price gouging.