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A study on two-person zero-sum rough interval continuous differential games

  • El-Saeed Ammar
  • M. G. BrikaaEmail author
  • Entsar Abdel-Rehim
Theoretical Article


In this paper, we concentrate on solving the zero-sum two-person continuous differential games using rough programming approach. A new class defined as rough continuous differential games is resulted from the combination of rough programming and continuous differential games. An effective and simple technique is given for solving such problem. In addition, the trust measure and the expected value operator of rough interval are used to find the \( \upalpha \)-trust and expected equilibrium strategies for the rough zero-sum two-person continuous differential games. Moreover, sufficient and necessary conditions for an open loop saddle point solution of rough continuous differential games are also derived. Finally, a numerical example is given to confirm the theoretical results.


Two-person zero-sum game Rough interval Differential games Trust measure Sufficient and necessary conditions 



The authors would like to thank the editor and the anonymous reviewers for their helpful comments for revising the article.


  1. 1.
    Von Neumann, J., Morgenstern, D.: The Theory of Games in Economic Bahavior. Wiley, New York (1944)Google Scholar
  2. 2.
    Hung, I.C., Hsia, K.H., Chen, L.W.: Fuzzy differential game of guarding a movable territory. Inf. Sci. 91, 113–131 (1996)Google Scholar
  3. 3.
    Harsanyi, J.C.: Games with incomplete information played by ‘Bayesian’ players. I: the basic model. Manag. Sci. 14, 159–182 (1967)Google Scholar
  4. 4.
    Xu, J.: Zero sum two-person game with grey number payoff matrix in linear programming. J. Grey Syst. 10(3), 225–233 (1998)Google Scholar
  5. 5.
    Dhingra, A.K., Rao, S.S.: A cooperative fuzzy game theoretic approach to multiple objective design optimization. Eur. J. Oper. Res. 83, 547–567 (1995)Google Scholar
  6. 6.
    Takahashi, S.: The number of pure Nash equilibria in a random game with nondecreasing best responses. Games Econ. Behav. 63(1), 328–340 (2008)Google Scholar
  7. 7.
    Espin, R., Fernandez, E., Mazcorro, G., Ines, M.: A fuzzy approach to cooperative n-person games. Eur. J. Oper. Res. 176(3), 1735–1751 (2007)Google Scholar
  8. 8.
    Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11(5), 341–356 (1982)Google Scholar
  9. 9.
    Pawlak, Z., Skowron, A.: Rudiment of rough sets. Inf. Sci. (NY) 177(1), 3–27 (2007)Google Scholar
  10. 10.
    Nasiri, J.H., Mashinchi, M.: Rough set and data analysis in decision tables. J. Uncertain Syst. 3(3), 232–240 (2009)Google Scholar
  11. 11.
    Weigou, Y., Mingyu, L., Zhi, L.: Variable precision rough set based decision tree classifier. J. Intell. Fuzzy Syst. 23(2), 61–70 (2012)Google Scholar
  12. 12.
    Arabani, M., Nashaei, M.A.L.: Application of rough set theory as a new approach to simplify dams location. Sci. Iran. 13(2), 152–158 (2006)Google Scholar
  13. 13.
    Rebolledo, M.: Rough intervals enhancing intervals for qualitative modeling of technical systems. Artif. Intell. 170(8), 667–685 (2006)Google Scholar
  14. 14.
    Liu, B.: Theory and Practice of Uncertain Programming. Physica, Heidelberg (2002)Google Scholar
  15. 15.
    Isaacs, R.: Differential Games: A Mathematical Theory with Application to Warfare and Pursuit, Control and Optimization. Wiley, New York (1965)Google Scholar
  16. 16.
    Megahed, A.A., Hegazy, S.: Min–max zero sum two persons continuous differential game with fuzzy controls. Asian J. Curr. Eng. Maths 2(2), 86–98 (2013)Google Scholar
  17. 17.
    Megahed, A.A., Hegazy, S.: Min–max zero sum two persons fuzzy continuous differential game. Int. J. Appl. Math. 21(1), 1–16 (2008)Google Scholar
  18. 18.
    Campos, L.: Fuzzy linear programming model to solve fuzzy matrix game. Fuzzy Sets Syst. 32, 275–289 (1989)Google Scholar
  19. 19.
    Roy, S.K., Mula, P.: Solving matrix game with rough payoffs using genetic algorithm. Oper. Res. Int. J. 16(1), 117–130 (2016)Google Scholar
  20. 20.
    Roy, S.K., Mondal, S.N.: An approach to solve fuzzy interval valued matrix game. Int. J. Oper. Res. 26(3), 253–267 (2016)Google Scholar
  21. 21.
    Mula, P., Roy, S.K., Li, D.: Birough programming approach for solving bi-matrix games with birough payoff elements. J. Intell. Fuzzy Syst. 29, 863–875 (2015)Google Scholar
  22. 22.
    Das, C.B., Roy, S.K.: Fuzzy based GA for entropy bimatrix goal game. Int. J. Uncertain Fuzziness Knowl. Based Syst. 18(6), 779–799 (2010)Google Scholar
  23. 23.
    Das, C.B., Roy, S.K.: Fuzzy based GA to multi-objective entropy bimatrix game. Int. J. Oper. Res. 50(1), 125–140 (2013)Google Scholar
  24. 24.
    Roy, S.K., Mula, P.: Rough set approach to bi-matrix game. Int. J. Oper. Res. 23(2), 1 (2015)Google Scholar
  25. 25.
    Bhaumik, A., Roy, S.K., Li, D.-F.: Analysis of triangular intuitionistic fuzzy matrix games using robust ranking. J. Intell. Fuzzy Syst. 33(1), 327–336 (2017)Google Scholar
  26. 26.
    Xiao, S., Lai, E.M.K.: Rough programming approach to power-balanced instruction scheduling for VLIW digital signal processors. IEEE Trans. Signal Process. 56(4), 1698–1709 (2008)Google Scholar
  27. 27.
    Fleming, W.H., Rishel, R.W.: Deterministic and Stochastic Optimal Control. Springer, Berlin (1975)Google Scholar
  28. 28.
    Yang, X., Gao, J.: Uncertain differential games with application to capitalism. J. Uncertain. Anal. Appl. 1, 1–17 (2013)Google Scholar

Copyright information

© Operational Research Society of India 2019

Authors and Affiliations

  • El-Saeed Ammar
    • 1
  • M. G. Brikaa
    • 2
    Email author
  • Entsar Abdel-Rehim
    • 3
  1. 1.Department of Mathematics, Faculty of ScienceTanta UniversityTantaEgypt
  2. 2.Department of Basic Science, Faculty of Computers and InformaticsSuez Canal UniversityIsmailiaEgypt
  3. 3.Department of Mathematics Faculty of ScienceSuez Canal UniversityIsmailiaEgypt

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