Optimal Feedback in a Dynamic Game of Generalized Shortest Path

Abstract

We consider the problem of generalized shortest path. The task is to transit optimally from the origin through a system Mi, i ∈ 1, m, of intermediate sets in R d to a fixed destination point (or set), under conditions that only one node in Mi can be chosen for passing. Any returns to the sets that have already been passed, are prohibited. The (combinatorial) cost function to minimize is either additive or bottleneck. The visiting nodes xi ∈ Mi, i ∈ 1, m, are either governed by an antagonistic nature or by a rational antagonist. For this multistage game problem both openloop and feedback settings are suggested. The feedback problem is posed in the class of feedback strategies which can change route during motion, depending on the current moves of the opponent. They provide, in general, a strictly better value of the problem, with respect to the open-loop minimax setting. The optimal feedback minimax strategy is constructed, and some (polynomial) heuristics are given.