How did Ptolemy’s empire function?

Ptolemy synthesized Greek knowledge of the known Universe. His work enabled astronomers to make accurate predictions of planetary positions and solar and lunar eclipses, promoting acceptance of his view of the cosmos in the Byzantine and Islamic worlds and throughout Europe for over 1400 years. I don't know if that directly answers your question though.

council members were chosen randomly, and men could present laws for debate in assembly.

answer; the period of time known as the middle ages;