Class of differential games with no Nash equilibrium, but with equilibrium of objections and counterobjections

A nonstop stream of publications is devoted to the investigation of positive and negative properties of Nash equilibrium concept prevailing in economics (as solution of noncooperative game). Mostly they are related to non-uniqueness and, as a consequence, to the lack of equivalence, interchangeability, external stability as well as instability to simultaneous deviation of such solutions of two and more players. The game "dilemma of prisoners" also revealed the property of "ability to improve".The book Equilibrium Control of Multi-criteria Dynamic Problems (V.I. Zhukovskiy and N.T. Tynyanskiy, M.: MSU, 1984) is devoted to detailed analysis of such "negative" properties for differential positional games.The authors of this book com to the following conclusion: either make use of those situations of Nash equilibrium that are simultaneously free from some of the stated disadvantages, or introduce new solutions of noncooperative game. Such solutions having the merits of Nash equilibrium situation would allow to get rid of its certain disadvantages. The present article is devoted to one of such possibilities for differential games related to concepts of objections and counterobjections. The concepts of objections and counterobjections used in it are based on the concepts of objections and counterobjections well-known classical game theory. The papers of E.I. Wilkas [1973] are devoted to theoretical questions of this concept. The term "active equilibrium" suggested R.E. Smolyakov in 1983, the notion of equilibrium of objections and counterobjections in differential games was first used apparently by E.M. Vaisbord in 1974, and then it was picked up by the first author of the present article in the above mentioned book [1984], but this concept was applied and is being applied in differential games, in our opinion, insufficiently widely. This fact "called to life" the present paper. In it the class of differential games of two persons is revealed, where the usual Nash equilibrium situation is absent, but the equilibrium of objections and counterobjections is present.