The simple answer is that Peter was Bishop of Rome last. That's the last ecclesial province he had control of, and so his successor is whoever is Bishop of Rome.
The Antiochian argument, in my experience, is made by some bad-faith EO Christians. But the Patriarchs of the EO Churches even recognize themselves that they need the Pope for ecumenical councils, and that that individual is the successor of Peter in Rome. This is why they have not had any ecumenical councils since before 1054. See this article on Orthodoc Wiki for more info on this tangent. Some of their theologians try to work out a papacy-less ecumenism, but there is disagreement.