Everyone for themselves. Anarchy.
I doubt it is possible for Anarchy to exist outside of a temporary situation between governments.
I hate to sound like a broken record, but the American model, if implemented world wide, would be the best bet.
Sovereign nations voluntarily entered into a union of smaller regional governments, with smaller areas inside of them, with the scope of government power inverse to the size of the area/ population it covers. Free-trade, no alliances, only defensive war, no standing armies, the bill of rights, the whole gambit. If the globe would adopt this idea you would basically peace. Nations wouldn't be guaranteed prosperity, but each, and the people living in them would have the freedom to take care of themselves.
In reality there will always be thirsts for power and nations using aggression and isolationism against others or their own citizens.