I recall an article on HN quite a while ago about one of those property management companies/websites in the Bay Area. The article was detailing that, while the majority of people looking to rent out their properties were individuals and not the ultra wealthy, the app "encouraged" price-fixing, including things such as keeping a certain % of lots vacant on purpose in order to drive up prices artificially. I put encouraged in quotes because it turned out that a prerequisite of managing your properties through this application was actually complying with the price fixing, on an app-wide scale of all its users.

So despite it making no financial sense on an individual, house-by-house level, what happened in practice is that the market itself got strangled by this collusion and despite the existence of many empty houses available for either purchase or rent, you ended up with a non-negligible amount of these places not ending up in the market organically and as such driving up the costs for everyone. The property owners win big with this scheme of course, and the way things were setup you'd be a fool to go against the grain and try outcompete on cheaper housing, since you can't beat an entire market getting strangled.

So yes, "the rich" aren't that stupid, they just so happen to have systems that make these sort of moves profitable.