That's a clearly defective model from a game theoretical point of view. A certain rich person can do very little to preserve the stability of the society, so their actions would be concentrated on things that give them direct benefits, that may or may not help society. It's essentially an externality problem, everybody is marginally hurt by say pollution, but some have local gains that far outweigh the costs, motivating them to pollute, bribe politicians and regulators, skirt taxes and so on, leaving other rich or poor people to support the welfare of the society. When the scales tip in favor of the freeriders, you have a failed state that no rich profiteer could have prevented, working individually.
Working individually, yes. But realistically this is more of an iterative prisoner's dilemma than a straight prisoner's dilemma. If the rich don't hold up their part of the social contract, the poor tend to eat them (eventually). So the rich are only incentivized to screw the system if they can get away with it. The problems start happening when we let them get away with it.
