Mathematics and Statistics Vol. 11(6), pp. 973 - 987
DOI: 10.13189/ms.2023.110613
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On Nash Equilibrium Solutions for Rough Differential Games


Abd El-Monem A. Megahed 1, Mohamed R. Zeen El Deen 2,*, Asmaa A. Ahmed 2
1 Department of Basic Science, Faculty of Computer and Informatics, Suez Canal University, Ismalia, Egypt
2 Department of Mathematics and Computer Science, Faculty of Science, Suez University, Egypt

ABSTRACT

The purpose of this paper is to investigate the Nash equilibrium concept for differential games when there is uncertainty in the available information for the players. Our study involves examining the problem of uncertainty in player information during the game using the "rough sets" concept, which is widely used for many such problems. Furthermore, we also explore the possible alliance between continuous differential games and the rough programming approach. Our primary aim is to ascertain the Nash equilibrium for a differential game in situations where the players have uncertain information, so they are exerting rough control, along with the trajectory of the system state being rough as well. We derive the necessary and sufficient conditions for the open-loop Nash equilibrium of the rough differential game. Additionally, we make use of the expected value operator and trust measure of rough interval to convert the rough problem into a crisp problem, allowing us to calculate the expected Nash equilibrium strategies and α-trust Nash equilibrium strategies for the game. Finally, a numerical example that outlines the steps involved in producing the rough interval of the Nash equilibrium and system state trajectory for the rough differential game is given. Moreover, this example demonstrates how to obtain each crisp problem from a rough one and then determines its Nash equilibrium and the corresponding state trajectory.

KEYWORDS
Nash Differential Games, Rough Set, Rough Intervals, Trust Measure, Expected Value

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Abd El-Monem A. Megahed , Mohamed R. Zeen El Deen , Asmaa A. Ahmed , "On Nash Equilibrium Solutions for Rough Differential Games," Mathematics and Statistics, Vol. 11, No. 6, pp. 973 - 987, 2023. DOI: 10.13189/ms.2023.110613.

(b). APA Format:
Abd El-Monem A. Megahed , Mohamed R. Zeen El Deen , Asmaa A. Ahmed (2023). On Nash Equilibrium Solutions for Rough Differential Games. Mathematics and Statistics, 11(6), 973 - 987. DOI: 10.13189/ms.2023.110613.