Lobbying
The seriously rich can afford to employ both public relationships and straight-up lobbyists to fight for their interests. A favoured tactic is to threaten to leave but their activity is often ruthlessly separated from the cities they live in by the layer of security they employ and the exclusivity of their preferred haunts which allows them to signal their wealth.
It is also unclear whether wealth is capable of delivering all the elements of a functioning nation state despite Randian fantasies of withdrawal to private societies. While isolated currently all basic provisions of services and goods come from the regular nation state. Wealth is used to differentiate within that system rather run it parallel.
Take the law for example, this is typically the friend and commonly used tool of the rich but in disputes between sovereign plutocrats who will have the authority to regulate disputes without people toppling out of windows.
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