The welfare of the United States is highly dependent upon its critical infrastructures and key resources. The Marine Transportation System is critical to the flow of commerce. The United States Coast Guard is charged with facilitating the protection of the Marine Transportation System from acts of terrorism under the Port, Waterways, and Coastal Security Mission. The Coast Guard faces the challenge of providing essential protection strategies with limited resources. Optimizing limited resources to provide maximum protection from deliberate attacks is a complex problem. In this thesis we explore various analytic techniques that can be used to provide guidance in resource allocation for defense against terrorism. We focus on two techniques, risk-based analysis and game theoretic analysis. We review the fundamental mathematical concepts and philosophical assumptions necessary for these techniques to be applicable.