As the number of nations possessing space launch capabilities increases, Earth orbit inherently becomes a competitive environment. Furthermore, each competing agent possesses unique objectives that may or may not align with the objectives of other agents. The competitive dynamics presented by this system are well suited for the application of game theory: the study of rational competitors from a mathematical perspective. The presented work combines the disciplines of game theory, optimal control, and astrodynamics to form generic game solution methods. These solution methods are used to obtain optimal open-loop strategies for an interceptor and an evasive target. A game involving an interceptor, a defender, and a ballistic target is also studied. Parameter space analyses are performed across a wide range of initial conditions to identify and visualize trends in the solution spaces. Additionally, a framework for testing strategies in a closed-loop format is developed to evaluate the consequences of sub-optimal actions. The various trends and characteristics found in the solution spaces are discussed, as is the relevancy of the results to modern space security and contingency planning.