Trust management is a form of distributed access control using distributed policy statements. Since one party may delegate partial control to another party, it is natural to ask what permissions may be granted as the result of policy changes by other parties. We study security properties such as safety and availability for a family of trust management languages, devising algorithms for deciding the possible consequences of certain changes in policy. While trust management is more powerful in certain ways than mechanisms in the access matrix model, and the security properties considered are more than simple safety, we find that in contrast to the classical HRU undecidability of safety properties, our primary security properties are decidable. In particular, most properties we studied are decidable in polynomial time. Containment, the most complicated security property we studied, is decidable in polynomial time for the simplest TM language in the family. The problem becomes coNP-hard when intersection or linked roles are added to the language.