Variations of the Flip-It game have been applied to model network cyber operations. While Flip-It can accurately express uncertainty and loss of control, it imposes no essential resource constraints for operations. Capture the flag (CTF) style competitive games, such as Flip-It , entail uncertainties and loss of control, but also impose realistic constraints on resource use. As such, they bear a closer resemblance to actual cyber operations. We formalize a dynamical network control game for CTF competitions and detail the static game for each time step. The static game can be reformulated as instances of a novel optimization problem called Adversarial Knapsack (AK) or Dueling Knapsack (DK) when there are only two players. We define the Adversarial Knapsack optimization problems as a system of interacting Weighted Knapsack problems, and illustrate its applications to general scenarios involving multiple agents with conflicting optimization goals, e.g., cyber operations and CTF games in particular. Common awareness of the scenario, rewards, and costs will set the stage for a non-cooperative game. Critically, rational players may second guess that their AK solution -- with a better response and higher reward -- is possible if opponents predictably play their AK optimal solutions. Thus, secondary reasoning which such as belief modeling of opponents play can be anticipated for rational players and will introduce a type of non-stability where players maneuver for slight reward differentials. To analyze this, we provide the best-response algorithms and simulation software to consider how rational agents may heuristically search for maneuvers. We further summarize insights offered by the game model by predicting that metrics such as Common Vulnerability Scoring System (CVSS) may intensify the secondary reasoning in cyber operations.