On One Modification of Nash Equilibrium

Abstract

By the end of the last century there were four areas in the mathematical theory of positional differential games: non-coalitional positional differential games, cooperative, hierarchical and, finally, the least-understood coalitional positional differential games. In their turn, coalitional games are divided into games with transferable payoffs (games with side payments when players can split profits in the course of the game) and with non-transferable payoffs (games with side payments when there are no such distributions for this or that reason). The coalitional games with side payments are being extensively explored at the Faculties of Applied Mathematics and Management Processes of St. Petersburg University and the Institute of Mathematics and Information Technologies of Petrozavodsk State University (by Professors L.A. Petrosyan, V.V. Mozalov, E.M. Parilina, A.N. Rettieva and their numerous students). However, side payments are not always present even in economic cooperation; moreover, side payments can be legislated against. We believe that the research of the equilibrium of threats and counter-threats (sanctions and counter-sanctions) in non-coalitional differential games that we have carried out over the last years allows to also cover some aspects of non-transferable payoff coalitional games. The article considers the issues of namely the internal and external stability of coalitions in the class of positional differential games. The coefficient constraints were identified for the mathematical model of a linear-quadratic differential positional game with twin-coalitional structure for six persons where this coalitional structure is internally and externally stable.