Coalition Pareto-Optimal Solution in a Nontransferable Game

By the end of the last century, four directions had been established in the mathematical theory of positional differential games (PDGs): a noncoalition version of PDG, a cooperative, hierarchical, and, finally, the least studied, a coalition version of PDG. In turn, within the coalition, there are usually games with transferable payoffs (with side payments, when players can share their winnings during the game) and nontransferable payoffs (games with side payments, when such redistributions are absent for one reason or another). Studies of coalition games with side payments are concentrated and actively conducted at the Faculty of Applied Mathematics and Management Processes of St. Petersburg University and Institute of Applied Mathematical Research of the Karelian Research Centre of Russian Academy of Sciences (L.A. Petrosyan, V.V. Mazalov, E.M. Parilina, A.N. Rettieva, and their numerous domestic and foreign students). However, side payments are not always present even in economic interactions; moreover, side payments may be generally prohibited by law. The studies we have undertaken in recent years on the balance of threats and counterthreats (sanctions and countersanctions) in noncoalition differential games allow, in our opinion, covering some aspects of the nontransferable version of coalition games. This article is devoted to the issues of internal and external stability of coalitions in the PDG class. It reveals the coefficient constraints in the mathematical model of the positional differential linear-quadratic game of six persons with a two-coalition structure, in which this coalition structure is internally and externally stable.