Differential game of three persons in which Nash equilibrium doesn’t exist but equilibrium of objections and counterobjection is present.

In opinion of luminaries in mathematical game theory the equilibrium asacceptable solution of differential game is characterized by the property of stability: the deviation from it of individual player cannot increase the payoff of deviated one. The solution proposed in [22], [23] by the 25-years old post-graduate of Princeton university John Forbes Nash (Jr) and later on called Nash equilibrium (NE) completely responds to this condition. NE certainly gained «the reigning position» in economics, sociology, military sciences. John Nash was awarded the Nobel Prize in economics in 1994 (simultaneously with John Harsanyi) «for fundamental analysis of equilibrium in the theory of noncooperative games». Actually, Nash created the basis of scientific method which played the vast role in the development of world economics. When opening fast any scientific journal in economics, operations research, systems analysis or game theory we certainly collide with publications concerning Nash equilibrium (NE). But «And in the sun there are spots»: the set of situations of Nash equilibrium can be internally and externally unstable. So in the simplest noncooperative game of two persons in normal form
$$\langle\{1, 2\}, \{X_i = [−1; 1]\}_{i=1,2}, \{f_i(x_1, x_2) = 2x_1x_2 − x^2_i \}_{i=1,2} \rangle$$
the set of Nash equilibrium situations will be
$$X^e = \{x^e = (x^e_1 , x^e_2 ) = (α, α)\,|\,∀α = const \in [−1, 1]\}, f_i(x^e) = α^2 (i = 1, 2).$$
For the elements of this set (the segment of bisectrix of 1st and 3rd quarter of coordinate angle) firstly, for $x^{(1)} = (0, 0) \in X^e$ and $x^{(2)} = (1, 1) \in X^e$ we have $f_i(x^{(1)}) = 0 < f_i(x^{(2)}) = 1\,(i = 1, 2)$ and therefore the set $X^e$ is internally unstable; secondly, $f_i(x^{(1)}) = 0 < f_i(\frac{1}{4}, \frac{1}{3})\,(i = 1, 2)$ and therefore the set $X^e$ is externally unstable.
Both external and internal instability of the set of Nash equilibrium (NE) is negative for its practical use. In the first case there exists the situation which dominates NE (including all players), and in the second one such situation is even NE. Pareto maximality of Nash equilibrium situation allowed to avoid consequences of external and internal instability. But such coincidence is most likely exotic phenomenon (we know at least three cases of such coincidence). So to avoid trouble connected with external and internal instability we must add condition of Pareto maximality to discussed below equilibrium of objections and counterobjections (EOAC). Let us pass to EOAC. First it occurred in books [13], [15]. The point is that 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 [10] is devoted to detailed analysis of such negative properties for differential positional games. The authors of this book come 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 concept of objections and counterobjections well known classical game theory. The paper [4] is 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 [2], and then it was picked up by the first author of the present article in the above mentioned book [10], 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. This concept of objections and counterobjections, as mentioned above, occurs in first publications on mathematical game theory. But in these publications either static variant of the game or differential game of only two persons were examined. Differential games of three and more players were not considered. This fact stimulated the authors to write the paper.
Keywords: noncooperative games, Nash equilibrium, active equilibrium, equilibrium of objections and counterobjections