Nash-Stackelberg equilibrium solutions for linear multidivisional multilevel programming problems
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This is a study of decentralized planning problems that involve both sequential and simultaneous decisions made by independent and interdependent divisions of a multi-agent decision system. Those divisions can often be arranged within a hierarchical structure according to their decision sequence. Simultaneous decisions are made by divisions within each level of the hierarchy. The joint decisions at each level are then executed sequentially throughout the hierarchy. To analyze such problems, we present a mathematical programming technique called multidivisional multilevel programming . Multidivisional multilevel programming is an extension of multilevel programming and permits the modelling of multiple decision-makers at every level. For linear multidivisional multilevel programming problems, the decision at each level will be shown to be an n -person Nash game played on a sequence of polytopes each consisting of a subset of extreme points of the polytope for the entire problem. We develop a Nash-Stackelberg solution approach for this multi-agent sequential and simultaneous decision making process. We also define the Nash-Stackelberg hybrid equilibria for such problems and show the existence of the equilibrium solution. We establish the geometric properties of a class of linear d -divisional r -level programming problems and use them to develop a search procedure to find equilibrium solutions. A computational example of a linear multidivisional two-level programming problem is also presented.