This paper studies under what conditions congestion control schemes can be both effcient, so that capacity is not wasted, and incentive compatible, so that each participant can maximize its utility by following the prescribed protocol. We show that both conditions can be achieved if routers run strict priority queueing (SPQ) or weighted fair queueing (WFQ) and end-hosts run any of a family of protocols which we call Probing Increase Educated Decrease (PIED). A natural question is whether incentive compatibility and effciency are possible while avoiding the per-flow processing of WFQ. We partially address that question in the negative by showing that any policy satisfying a certain "locality" condition cannot guarantee both properties. Our results also have implication for convergence to some steady-state throughput for the flows. Even when senders transmit at a fixed rate (as in a UDP flow which does not react to congestion), feedback effects among the routers can result in complex dynamics which do not appear in the simple topologies studied in past work.