Housing demand isn't really that inelastic. People will move to less crowded cities if they become too expensive. Americans tend to have much more living space per capita than most of the world, so there is a lot of room to move in with roommates or family.
> all of your competitors collude to also raise their rates similarly.
Realpage doesn't control the prices for the majority of housing in the US, the majority of rental units in the US, or even the majority of multifamily rental units in the US. If they tried to facilitate collusion, their customers would simply lose money.
The vast majority of people die within 10 miles of where they were born. People want to live near their families, and for the most part do not want to move to another city, and won't even if their rent is difficult to afford.
> Realpage doesn't control the prices for the majority of housing in the US, the majority of rental units in the US, or even the majority of multifamily rental units in the US.
They don't need to. They just need to control the prices for particular cities, because market prices are heavily tied to regions and not to the entire US. Detroit's housing market crashing didn't affect SF's rent.
You could instead imagine a (still completely fictitious) figure that makes your argument look bad -- e.g., that the raise in rates causes a 0% drop in occupancy. Then the landlords/algorithm win by raising prices.
One reason occupancy rates might remain relatively stable even in the face of rising prices is if consumers have relatively few better options, because e.g. all of your competitors collude to also raise their rates similarly.