Coalition formation is a key aspect of automated negotiation among self-interested agents. In order for coalitions to be stable, a key question that must be answered is how the gains from cooperation are to be distributed. Various solution concepts (such as the Shapley value, core, least core, and nucleolus) have been proposed. In this paper, we demonstrate how these concepts are vulnerable to various kinds of manipulations in open anonymous environments such as the Internet. These manipulations include submitting false names (one acting as many), collusion (many acting as one), and the hiding of skills. To address these threats, we introduce a new solution concept called the anonymity-proof core, which is robust to these manipulations. We show that the anonymity-proof core is characterized by certain simple axiomatic conditions. Furthermore, we show that by relaxing these conditions, we obtain a concept called the least anonymity-proof core, which is guaranteed to be non-empty.