And yet, property is for sale everywhere, go check zillow. You might not like the price, I imagine anybody looking to buy property to rent out probably has similar feelings; ultimately though, if they are using this "price fixing" algorithm, then it should be easy enough to out compete them by just lowering prices.
That doesn't make any sense. There is a dramatic under-supply of housing, there will be no "out-competing". They're going to rent all of their apartments, the price-fixers and the non-price-fixers. They'd just be leaving money on the table, not "out-competing". They're not going to magically acquire a bunch of new non-existent housing to grow their market share. I implore you to move beyond econ 101 thinking.