The algorithm is not operating on public info, it is soliciting proprietary pricing and vacancy data that is not available to renters or regulators. That secretive information sharing is the basis of the whole scheme.
And the collusion doesn't stop with this information hoarding; the pricing recommendations are as profitable as they are precisely because many property owners in a market are enacting them with the knowledge that a known quantity of their peers have no intent to undercut. Sure, someone can renege on that, but collusion doesn't require that you have an airtight legal contract to bind all parties; after all, such contracts are inherently illegal!
And the collusion doesn't stop with this information hoarding; the pricing recommendations are as profitable as they are precisely because many property owners in a market are enacting them with the knowledge that a known quantity of their peers have no intent to undercut. Sure, someone can renege on that, but collusion doesn't require that you have an airtight legal contract to bind all parties; after all, such contracts are inherently illegal!