Rough set approach to non-cooperative continuous differential games

  • M. G. BrikaaEmail author
  • Zhoushun Zheng
  • El-Saeed Ammar
Original Paper


In this article, we analyze the non-cooperative continuous differential games under rough interval environment. The combination of rough set and continuous differential games expresses a new class defined as rough non-cooperative continuous differential games. We develop an effective and simple technique for solving the problem under study; in this methodology, the rough non-cooperative continuous differential games are transformed into two problems with interval environment corresponding to the upper and lower approximation of rough intervals. Furthermore, four crisp problems of non-cooperative continuous differential games are derived as follows: upper lower problem (ULP), lower lower problem (LLP), lower upper problem (LUP) and upper upper problem (UUP). Moreover, the trust measure and the expected value operator of rough interval are used to find the α-trust equilibrium strategies and the expected equilibrium strategies. Sufficient and necessary conditions for equilibrium strategies of rough non-cooperative continuous differential games are also derived. Finally, applicability and validity of the models and technique proposed in this article are illustrated with a practical example.


Non-cooperative games Rough interval Differential games Trust measure 



This work was partly supported by the National Key Research an Development Program of China (No. 2017YFB0305601), the National Key Research an Development Program of China (No. 2017YFB0701700).


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • M. G. Brikaa
    • 1
    • 2
    Email author
  • Zhoushun Zheng
    • 1
  • El-Saeed Ammar
    • 3
  1. 1.School of Mathematics and StatisticsCentral South UniversityChangshaPeople’s Republic of China
  2. 2.Department of Basic Science Faculty of Computers and InformaticsSuez Canal UniversityIsmailiaEgypt
  3. 3.Department of Mathematics Faculty of ScienceTanta UniversityTantaEgypt

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