I believe most EU countries do this in some form, but Austria by far has the largest and most successful public housing scheme. It is very effective at controlling prices.
How so? It makes it cheap for the people inside the system but expensive for those that are outside of it. You don't solve a housing crisis by essentially subsidizing the cost.
No, the Austrian government builds a large amount of housing, which increases the overall supply of housing. People living in social housing do get lower rent prices, but having a massive builder in the market that keeps pumping out new inventory helps to make it a buyer’s market and keep prices low. Simple supply and demand.
In Amsterdam 57% of the rental stock is rent-controlled housing (capped at below €800/month). But that doesn't help at all when you don't keep building more: the average time on the waiting list for a new allocation is 13 years, and rarely move out. And if they do move out physically, it's popular to not move out administratively and illegally sublet it for a lot more than that they're paying (that's the story I've been told by locals).
