Altruistic and Aggressive Types of Behaviorin a Nonzero-Sum Positional Differential Gameof Three Persons
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truistic and Aggressive Types of Behavior in a Nonzero-Sum Positional Differential Game of Three Persons A. F. Kleimenov1,2 Received May 16, 2019; revised July 5, 2019; accepted July 15, 2019
Abstract—In this paper, the results of the author’s previous work concerning a nonzerosum positional differential game of two persons with different types of the players’ behavior are extended to a game of three persons. For simplicity, the paradoxical type of behavior is not used by the players. The notions of altruistic and aggressive behavior types are generalized. As in the two-person game, it is assumed that each player chooses not only a positional strategy but also an indicator program function. The rules for forming controls for each triple of behavior types and the definition of a BT-solution are clarified. An example of a game with the dynamics of simple motion on a plane and a phase constraint is considered. It is assumed that the players can show altruism and aggression towards their partners, and the case of simultaneous aggression from all the players is not excluded. A description of BT-solutions of the game is given. Keywords: nonzero-sum positional differential game, terminal payoff functionals, altruistic and aggressive behavior types, Nash equilibrium solution.
DOI: 10.1134/S0081543820040094 INTRODUCTION The present paper is devoted to the development of the theory of positional differential games [1, 2] for certain classes of nonzero-sum games (see, for example, [3]). Earlier, the author proposed [4–6] a formalization of a nonzero-sum positional differential twoperson game with different types of the players’ behavior. In these studies, it was assumed that each of the two players, in addition to the usual normal (nor) behavior type aimed at maximizing their payoff functional, could use other types of behavior introduced in [7, 8], namely, the altruistic (alt) and aggressive (agg) types of behavior with respect to the other player as well as the paradoxical (par) type. During the game, the players could switch their behavior from one type to another. It was assumed that each player chose not only a positional strategy but also, simultaneously, an indicator function defined on the whole game horizon and taking values in the set {nor, alt, agg, par}. A player’s indicator function showed the dynamics of changes of their behavior type. Rules for the formation of controls were introduced for each pair of behavior types of the players. The formalization of positional strategies in the game was based on the formalization and results in the 1
Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia 2 Ural Federal University, Yekaterinburg, 620002 Russia e-mail: [email protected]
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ALTRUISTIC AND AGGRESSIVE TYPES OF BEHAVIOR
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general theory of positional differential games [1, 2, 9]. The notion of BT-solution was proposed, and the procedure for constructing BT-solutions was illustrated with simple examples. In the present paper, we extend the re
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