Mechanism is an algorithm plus incentives.
A mechanism is a tool that takes inputs from multiple people and uses these inputs as a way to determine things about its participants' values, so as to make some kind of decision people care about.
In a well functioning mechanism, the decision made is both efficient (the best possible outcome given the participant's preferences, and incentive-compatible, meaning that people have the incentive to participate "honestly".
Private property is as effective as it is: not because it is a god given right, but because it's credibly neutral mechanism that solves a lot of problems in society, far from all but still a lot.
It's easier to agree that a neutral mechanism treats everyone reasonably fairly than it is to convince a diverse group of people that some particular blacklist of unallowed political viewpoints is correct.