Asshai and Stygai

[Mordor]

Who rules over these cities? Do they have a leader or king? 

[Macgregor of the North]

By what little is written about Stygai it wouldn't surprise me if there are no living humans who live, or rule there. 

Asshai on the other hand definitely has humans and is a busy port even though the population numbers are small. As to who rules it, it may come down to whose magic/sorcery/powers are greater, as they would command the most slaves and respect and with that in mind would probably control most of the gems and gold which are found there. 

In everything I've read on Asshai there does not seem to be a King or ruler of Asshai, it seems to be an anything goes kinda town and who has the greater powers may well rule the roost there.

 

[Maester of Valyria]

I always took the view that Asshai was basically a place without any ruler or laws, and people there do whatever takes their fancy, regardless of how reprehensible those actions might seem anywhere else.

As for Stygai...short answer is we don't know. One might speculate that it's inhabited only by the very brave and very foolish, or by demons, or the Children of the Forest, or Lizard Men, or Shrykes, or anything really. But I doubt that whatever human settlement may or may not exist there has a proper ruler.

[BalerionTheCat]

As far as I understand, Asshai is a walled city, accessed only by sea. No one but the most powerful magicians, the shadowbinders, can venture beyond its walls. There are no children, no animals. The place is only inhabited by the magicians and their servants and slaves. Who are there for sharing and trading knowledge, and conducting experiments impossible anywhere else. But not much interested in ruling or administrating a city.

Stygai seems to be at the core of the shadow. IMO, the place where the Black Stone from the Yi Ti legends fell from the sky. The shadowbinders may explore its ruins, but no one, no human anyway, live there.