I think kingdom's authority must to rise when his peasants use a well, travel along a road, dig for resources, sell or buy an item on a barter stand, craft something much better quality than materials used, etc, so kingdom needs to open for his peasants high quality soil/clay/sand/water nodes, build some good manufactures, open some protected marketplaces, etc (Sevillian Community Market - is a good example), and link all that points of interest by the roads.
Also kingdom must be rewarded in authority for acts of judgement and punished for crimes against his peasants.
To prevent all this things from botting, border cairns (that will collect an authority separately) may work something like beehives with their 50 uses regenerated over time, and players may not produce portion of authority each time they craft something or use things but generate it randomly depends of character's luck (and get some exp for themselves, btw)
Sorry for my english, it's not my native language.
