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Two-Level Planning

J. Kornai
Th. Liptak

Abstract

The planning task may originally be formulated as a single linear programming problem of the maximizing type. This overall central information (OCI) problem may be decomposed into subproblems that can be solved by mutually independent "sectors," coordinated by the "centre" through having the latter allocate the resources to the various sectors. The original OCI problem is then transformed into a two-level problem, in which the "central problem" is to evolve an allocation pattern where the sum of the maximal yields of the "sector problems" will be the greatest. The solution of the two-level problem is achieved by setting up a game-theoretical model. The players are on the one hand the centre, on the other the team of sectors. The strategies of the centre are the feasible allocation patterns, those of the sectors are the high feasible shadow price systems in the duals of the sector problems. The payoff function is the sum of the dual sector objective functions. It is shown that if certain regularity conditions are satisfied, then the value of the polyhedral game which has thus been defined is the maximal yield of the OCI problem. In place of a direct solution of the polyhedral game, a fictitious play of the game is undertaken. The first part of the paper discusses a general model, within whose scope the symbols and definitions are presented and the mathematical theorems are proved. In the second part, the results of the first part are applied to a long-term macroeconomic planning model.