Multicast routing is a technique for transmitting a packet from a single source to multiple receivers without wasting network bandwidth. To achieve transmission efficiency, multicast routing constructs a directed tree that connects the source to all the receivers and sends only one copy of the packet over each link of the directed tree. When a packet reaches a branch point in the tree, it is duplicated and a copy is sent over each downstream link. Multicasting large amounts of data to large groups of receivers is likely to incur significant costs, and these costs need to be covered by payments collected from the receivers. However, receivers cannot be charged more than what they are willing to pay, and the transmission costs of shared network links cannot be attributed to any single receiver. Thus, one must design cost-sharing mechanisms to determine which users receive the transmission and how much they are charged. Recent work in economics (Moulin and Shenker, 2001) leads naturally to the consideration of two mechanisms: marginal cost (MC), which is efficient and strategyproof, and Shapley value (SH), which is budget-balanced and group-strategyproof and, among all mechanisms with these two properties, minimizes the worst-case welfare loss. Subsequent work in computer science shows that the MC mechanism can be computed by a simple, distributed algorithm that uses only two modest-sized messages per link of the multicast tree (Feigenbaum, Papadimitriou, and Shenker, 2001) but that computing the SH mechanism requires, in the worst case, that Omega(|P|) bits be sent over Omega(|N|) links, where P is the set of potential receivers, and N is the set of tree nodes (Feigenbaum, Krishnamurthy, Sami, and Shenker, 2004). Here, we extend these results in two directions. First, we give a group-strategyproof mechanism that exhibits a tradeoff between the other properties of the Shapley value: It can be computed by an algorithm that is more communication-efficient than the natural SH algorithm (exponentially more so in the worst case), but it might fail to achieve exact budget balance or exact minimum welfare loss (albeit by a bounded amount). Second, we completely characterize the groups that can strategize successfully against the MC mechanism.