Two-Level Planning with Conflicting Goals
Within the scope of planning problems in two-level systems, we consider a single supremal player and N infimal players also called sectors. These planning problems are formulated as interacting optimal control problems with (N+1) different criteria. Constraint coordination makes this system hierarchical: some constraints on the state of any infimal sector are functions of the state and controls of the supremal sector, while the latter has state constraints depending on all lowerlevel states.
Generalizing classical planning schemes based on the decomposition of a single supremal criterion function, some resource allocation procedures are described taking explicitly account of the different goals under given equilibrium conditions. Initializing the procedure with a dynamic Nash equilibrium among the (N+1) sectors, this equilibrium is modified sequentially at the initiative of the supremal player by updating the technical coefficients in the interaction constraint at this level; this updating scheme improves at each step the utility for the supremal level, and can be used in negotiation practice in absence of optimization by the individual players. An application to an actual budgeting problem in a University is described.
The former model is lastly generalized by introducing the concepts of mixed, inter-level, and intra-level equilibria. For example, for any given supremal controls, the N infimal players reach an intra-level Nash equilibrium; a monotonic function of the indices of performance of these infimal players is selected as a criterion function for the lower level as a whole; an intra-level Pareto equilibrium is then defined between the lower-level intra-level Nash equilibrium, and the supremal sector.
KeywordsNash Equilibrium Optimal Control Problem Differential Game Average Annual Cost Technical Coefficient
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