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Interdomain routing is essential to both the stability and efficiency of the global Internet. However, most previous studies focus only on stability, and only on a special class of routing protocols, namely BGP-type, path-vector protocols. In this paper, we conduct a systematic analysis of interdomain routing considering optimality and implementation in strategic settings. We adopt the novel perspective that an interdomain routing system is one which defines a social choice rule that aggregates individual preferences of all of the autonomous systems (ASes) in a network to select interdomain routes with a set of desirable properties. An interdomain routing protocol, then, is a mechanism to implement the identified interdomain routing social choice rule, when the ASes can adopt strategic actions. By pointing out the incompatibility among the desirable properties of an interdomain routing system and the requirements for strategic implementation in distributed settings, we reveal fundamental tradeoffs that must be made when extending BGP or designing the next-generation interdomain routing system. We also provide new insights into BGP, by "reverseengineering" its behaviors from the perspective of social choice and implementation theory.