Computational Aspects in Two-Level Differential Games

Summary

The basic principle of most planning and budgeting systems is an allocation of resources between a number of subordinate units who prepare a set of requests, and a central decision-maker who eventually approves or rejects the requests. Decentralized planning following primal or dual decomposition schemes are well known {2,3,7}; however, the lower units are here assumed to have full information at that level, and to act as to pursuing the objectives of the central unit

The first purpose of this paper is therefore to elaborate upon the way in which the planning problems at the supremal and infimal levels may be treated through constraint coordination, in the case where all units (including the supremal unit) have conflicting goals. Generalizing the approaches in {5,6,7,8,10}, the evolutions of the supremal and infimal units are to be governed by a global differential game of fixed duration, with state constraints and a given initial state (Sections l.,4.). Stochastic hierarchical sequential games, with discrete states at each level, inter-level and intra-level transition probabilities, have besides been studied in {4}

The second purpose of this paper is to show how projection operators from the global state and control spaces into the individual state and control spaces, may define the information structure in a two-level game(Section 1). This allows to describe situations where a large country (supremal player) has only partial control on, and information about, a community of smaller countries (infimal players). Lastly, a constraint coordination algorithm is studied for the purpose of approximating closed-loop Nash equilibrium strategies in a N-person game, via the computation of such strategies in an infimal (N-l)-person game played in a two-level system with the N’th player at the supremal level (Sections 2,3,5)