Aggressive behavior in a non-antagonistic positional differential game

This paper is a continuation of the research [3], in which a formalization of nonantagonistic positional differential two-player games (NPDGs) was offered for the case of different behavioral types of players (in short, the NPDGw$BT$) as follows. In addition to the usual (normal, nor) type of behavior oriented towards maximization of his/her own functional, each player can use other behavioral types described in [2, 7], namely, the altruistic (alt), aggressive (agg), and paradoxical (par) ones. By assumption, each player can switch from one behavioral type to another in the course of the game. Note that in [6, 7] such switching allowed to obtain new solutions in a repeated bimatrix $2 \times 2$ game. The formalization of players’ actions in the NPDGw$BT$ presented in [3] relies on the formalization and results of the general theory of antagonistic positional differential games [4, 5]. By assumption, in the NPDGw$BT$ each player chooses a closed-loop strategy simultaneously with his/her own indicator function that is defined on the whole game duration and takes values from the set $\{$nor, alt, agg, par$\}$. The indicator function shows the behavioral type dynamics of a given player. The strong and weak $BT$-solutions of the NPDGw$BT$ were defined in [3]. Expectedly, in some cases the abnormal types of behavior (the ones differing from the normal type) may yield better outcomes for players in the NPDGw$BT$ than in the NPDG. The main emphasis of the examples in [3] was stressed on the use of altruistic behavior by all players. This paper considers two examples of games with simple 2D motion dynamics and phase constraints in which each player can demonstrate altruism and also aggression towards a partner for certain time intervals, including the case of mutual aggression. In the first example, strong $BT$-solutions are constructed in which both players increase their payoffs in comparison with the game with the normal type of behavior. If the players are prohibited to choose the aggressive type of behavior, then the game has no $BT$-solutions. In the second example, strong $BT$-solutions are also constructed but, under the prohibition of aggression, they still exist and are induced by the altruistic type of behavior.