Where there is a weak government you tend to have high inflation. So if you have no government the natural response is to have no money.
Actually it's not that simple. Zimbabwe as Tito's Yugoslavia show that under strong (rigid) government you have a crazy inflation.
The value of national currency is based on the international perception but also (mainly) on national perception of its value compare to external currency for non international exchange market currencies.

The natural response in a non state/government environment is, as Stan just pointed it, to go for the most stabe foreign currency.

By the way, dollars is the most appreciated curency and its easiness to counterfake is part of its popularity.