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Thousands of competing autonomous systems must cooperate with each other to provide global Internet connectivity. Each autonomous system (AS) encodes various economic, business, and performance decisions in its routing policy. The current interdomain routing system enables each AS to express policy using rankings that determine how each router in the AS chooses among different routes to a destination, and filters that determine which routes are hidden from each neighboring AS. Because the Internet is composed of many independent, competing networks, the interdomain routing system should provide autonomy, allowing network operators to set their rankings independently, and to have no constraints on allowed filters. This paper studies routing protocol stability under these conditions. We first demonstrate that ldquonext-hop rankings,rdquo commonly used in practice, may not ensure routing stability. We then prove that, when providers can set rankings and filters autonomously, guaranteeing that the routing system will converge to a stable path assignment imposes strong restrictions on the rankings ASes are allowed to choose. We discuss the implications of these results for the future of interdomain routing.