What was the position of Av Beis Din vs Nassi in Rabbinic Pairs?

Question

Traditionally, from Moses and on there was always only one head of the Sanhedrin, the president, the prince, the king, the Supreme Judge, Av Beis Din, etc. No other members or titles or names are ever mentioned IIRC, (to prove the point that all courts are equally authoritative).

After the Hasmonean revolt, the age of the Rabbinic Pair (Zugot) commenced and presented a sudden change in the pattern: two heads of the Sanhedrin functioning alongside each other - Nassi AND Av Beis Din.

What exactly happened here - was it the position of Nassi invented or of Av Beis Din? And what were their duties and responsibilities?

Answer 1

According to the explanation of Doros Harishonim quoted in this answer, it was the position of Nasi that was invented. And (as @interested pointed out in a comment) it was well before the Hasmoneans (it took place while the Ptolemies ruled over Eretz Yisrael, and they were ousted about forty years before the Hasmonean revolt).

The Nasi, then, was the temporal leader, replacing the Kohen Gadol, who had served in that role during the early Second Temple period but who now had abdicated that role to unworthy factions. The Av Beis Din was, as before, the chief justice of the Sanhedrin.