Since this a crazy hard problem that won't be solved in a comments section, how about playing with a completely silly idea instead:

> Outlaw rental housing - force everyone to own their residence

The people we're most interested in helping are those pushed out by rising rents. Rents can't rise if they don't exist. New migrants can't drive rent up if rent doesn't exist - to the contrary, they would be increasing property value for all residents.

I'm confident we can make this work, people.