Altruistic behavior in a non-antagonistic positional differential game

A two-person non-antagonistic positional differential game (NPDG) whose dynamics is described by an ordinary nonlinear vector differential equation is considered. Constraints on values of players' controls are geometrical ones. The final time of the game is fixed. Cost functionals of both players are terminal ones. The formalization of positional strategies in the NPDG is based on the formalization and the results of the general theory of antagonistic positional differential games (APDGs)(see monographs by N.N. Krasovskii and A.I. Subbotin [3, 4]). Additionally, in the present paper we assume that each player together with the usual, normal type of behavior, oriented the maximization of their own functional, can use other types of behavior, introduced in the works [2, 5]. In particular, it may be altruistic, aggressive and paradoxical types. Further it is assumed that in the course of the game players can shift their behavior from one type to another. Using the players' possibility of such switches in a repeated bimatrix $2\times2$ game allowed in the works [5, 6] to obtain new solutions of this game. In the present paper, the spread of this approach on the NPDG leads to a new formulation of the problem. In particular, it is interesting how the players' outcomes at Nash solutions are transformed. An urgent task is to minimize the time of abnormal behavior subject to achieve a good result.
The paper proposes a formalization of the NPDG with behavior types (NPDGwBT). It is assumed that in an NPDGwBT each player simultaneously with a choice of positional strategy chooses also his own function indicator defined on the whole interval of the game and taking values in the set $\{$normal, altruistic, aggressive, paradoxical$\}$. The indicator function shows the dynamics of changes in player' behavior type, which adheres to this player. Thus, in this NPDGwBT each player controls the selection of the pair $\{$positional strategy, function indicator$\}$. The definition of the concept of a BT-solution of such a game is given. Expectedly, that in NPDGwBT the using behavior types which differ from normal one (so-called abnormal types), in some cases, may lead to more favorable outcomes for the players than in the NPDG. In the paper two examples of the NPDGwBT with simple dynamics in the plane, in each of which one player keeps to altruistic type of behavior over a period of time, are considered. It is shown, that in the first example on BT-solution the payoffs of both players are increased, in comparison with the game with normal type of behavior, and in the second example the sum of players' payoffs is increased.