In a reputation-based trust-management system, agents maintain information about the past behaviour of other agents. This information is used to guide future trust-based decisions about interaction. However, while trust management is a component in security decision-making, few existing reputation-based trustmanagement systems aim to provide any formal security-guarantees. We describe a mathematical framework for a class of simple reputationbased systems. In these systems, decisions about interaction are taken based on policies that are exact requirements on agents' past histories. We present a basic declarative language, based on purepast linear temporal logic, intended for writing simple policies. While the basic language is reasonably expressive, we extend it to encompass more practical policies, including several known from the literature. A naturally occurring problem becomes how to efficiently re-evaluate a policy when new behavioural information is available. Algorithms for the various languages are presented along with complexity analyses.