Two Classes of Games on Polyhedral Sets in Systems Economic Studies

Abstract

Two classes of two- and three-person games on polyhedral sets of player strategies that appear in estimating fair shares of the market participants in a marketplace are considered. In games from both classes, payoff functions of the players are sums of linear functions of vector arguments or those of linear ones and a bilinear function. Games from the first class are those in which player strategies are connected, i.e., they cannot be chosen by the players independently, whereas player strategies in games from the second class are disjoint. For the games from both classes either sufficient or necessary and sufficient conditions of the equilibriums are provided, and these conditions allow one to calculate the equilibriums by effective optimization techniques. This fact contributes to making the equilibrium concept a productive approach to quantitatively analyzing conflicts in systems economic studies. Economic problems that appear in systems described by nonlinear mathematical models with linear constraints, in particular, by some network models, including (a) restructuring a company and positioning the restructured company in a market or in several markets, (b) forming a pool of regional clients interested in selling their products and in buying somebody else’s ones outside their regions via forward contracts offered by regional brokers, (c) determining initial prices for procurement contracts to be tendered by a public administration, (d) finding competitive transportation tariffs by a railroad company competing with tracking companies for providing transportation services both in a region and between two regions, (e) calculating an optimal volume of producing electricity by a base load power plant in a part of a country’s electrical grid under an uncertain demand in the corresponding network of the grid customers, and (f) forming a public–private partnership to develop a set of projects that a public administration needs to develop and implement, but does not have funds to finance on its own (partly or even completely) are discussed to illustrate how the games under consideration appear, and how they can be analyzed.