A common problem in risk analysis is to characterize the overall security of a system of valuable assets (e.g., government buildings or communication hubs), and to suggest measures to mitigate any hazards or security threats. Currently, analysts typically rely on a combination of indices, such as resilience, robustness, redundancy, security, and vulnerability. However, these indices are not by themselves sufficient as a guide to action; for example, while it is possible to develop policies to de...