What if every landlord but one raised the price by 1000 a month, and that one realized he could get more business by raising only 900 a month, so in response to that one landlord others lowered to only +800 so the one countered with +700... Etc. Etc.
This argument would be much more persuasive if each landlord could supply an effectively unlimited supply of housing. As things currently stand, a landlord reducing his rent would clear his limited inventory first, but the broader effects of the actions of a single agent would be fairly muted.
Sure, in theory you can eminent domain a neighborhood of low density housing and replace it with higher density housing. In practice though, its politically impossible most places.