There is such a thing as government housing (or government run housing) and yours truly doesn’t know much about it at all. As is known, it can be any level of government and for rather exclusive purposes. What immediately comes to mind is something like an orphanage or county home for the elderly (aged). There are also “low income housing projects.” Those might be several city blocks of housing with streets that aren’t listed on city maps. The entire complex is just a singular blob.
In addition, there is also government subsidized housing, and that comes in at least two different forms. One kind is on an individual person or family and the other kind is on a building (if occupied). If a family, for example, is low income, whoever is in charge of it (it can be something like a grandparent) can go to some office to ask for a voucher. Vouchers are rather hard come by, but anyone who has one can take it to any landlord that will accept it. It can be for any place. Of course, there’s only so much money in it.
The other kind of subsidy is on a building by contract with private property owners and based on a tenant’s income. “X” amount is considered a fair rent for apartments like “Y.” If (usually) one third of a prospective tenant’s income is less than the fair rental amount, the government makes up the difference, provided the manager or owner fills out and turns in all the paperwork and the prospective tenant meets any other qualifications, such as the right number of people for the space or older than such and such an age.
Some things are never definite.