Theater level campaign resource allocation

In a generic multi-stage multi-resource theater level military game, each of the adversaries must make decisions of two types: temporal (what amount of each resource should be allocated for each stage) and spatial (movement of resources around the battlefield). This note is concerned, to a greater depth, with analysis and modeling of a temporal aspect of a game. This consideration is of its own technical interest in case when the interaction of resources can be assumed to occur in one arena. In our model every resource of one adversary interacts in sequence with each of the other adversary's resources. The allocation decisions are made as to optimize a certain cost functional. When the monotonicity of the cost functional is not assumed, the structure of the optimal solution is not stationary and cannot be reduced to solving a single-stage resource allocation problem at each stage. We discretize the state and action spaces and apply dynamic programming techniques to find the optimal allocation strategies and the optimal value of the game. An example is provided to illustrate that the solution in pure strategies might not exist.