The Pareto Equilibrium of Objections and Counterobjections in Linear-Quadratic Games of N Person

Abstract

Publications on mathematical game theory with many (not less than 2) players can be conditionally distributed in four directions: non-cooperative, hierarchical, cooperative and coalition games. The last two, in turn, are divided into games with side and non-side payments and games with transferable and nontransferable payoffs, respectively. If the first ones are actively studied (St. Petersburg State Faculty of Applied Mathematics and Control Processes, St. Petersburg Institute of Economics and Mathematics, Institute of Applied Mathematical Research of Karelian Research Centre RAS), the games with non-transferable payoffs are not covered. The paper proposes the conception of objections and counter-objections. The initial investigations were published in two monographs of E.I. Vilkas, the Lithuanian mathematician (the student of N.N. Vorobjev, the professor of St. Petersburg University). For the differential games this conception was first applied by E.M. Waisbord in 1974, then it was continued by the first author of the present article together with E.M. Waisbord in the book Introduction to the theory of differential games of n-persons and its application (1980), and in the monograph Equilibrium of objections and counterobjections (2010) by V.I. Zhukovskiy. The paper proves that in еру mathematical model there is no Nash equilibrium but there are equilibria of objections and conterobjections and simultaneously Pareto maximality.