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Existence of Equilibrium Solution for Multi-Leader–Follower Games with Fuzzy Goals and Parameters

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Abstract

In this paper, we first propose the model of multi-leader–follower games with fuzzy goals involving fuzzy parameters and introduce its \(\alpha \)-FNS equilibrium. Next, we shift our attention to the existence of \(\alpha \)-FNS equilibrium and prove it by Kakutani’s fixed point theorem. Finally, we illustrate an example to show that the equilibrium existence result is valid.

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References

  1. Achemine, F., Merakeb, A., Hamaz, A.: A new equilibrium for an n-person game with fuzzy parameters. Ajmms. 32(2), 118–132 (2013)

    Google Scholar 

  2. Bector, C., Chandra, S., Vidyottama, V.: Matrix games with fuzzy goals and fuzzy linear programming duality. Fuzzy. Optim. Decis. Ma. 3(3), 255–269 (2004)

    MathSciNet  Google Scholar 

  3. Buckley, J., Jowers, L.-J.: Fuzzy two-person zero-sum games Monte Carlo Methods in Fuzzy Optimization, pp. 165–173. Springer, Berlin, Heidelberg (2007)

    Google Scholar 

  4. Bashir, Z., Watróbski, J., Rashid, T., et al.: Intuitionistic-fuzzy goals in zero-sum multi criteria matrix games. Symmetry. 9(8), 158 (2017)

    MathSciNet  Google Scholar 

  5. Bector, C., Chandra, S., Vijay, V.: Duality in linear programming with fuzzy param-eters and matrix games with fuzzy pay-offs. Fuzzy. Sets. Syst. 146(2), 253–269 (2004)

    Google Scholar 

  6. Cevikel, A., Ahlatcioglu, M.: Solutions for fuzzy matrix games. Comput. Math. Appl. 60(3), 399–410 (2010)

    MathSciNet  Google Scholar 

  7. Cunlin, L., Qiang, Z.: Nash equilibrium strategy for fuzzy non-cooperative games. Fuzzy. Sets. Syst. 176(1), 46–55 (2011)

    MathSciNet  Google Scholar 

  8. Ding, X.: Equilibrium existence theorems for multi-leader-follower generalized multi objective games in FC-spaces. J. Global. Optim. 53(3), 381–390 (2012)

    MathSciNet  Google Scholar 

  9. Dan, B.: Fuzzy games: a description of the concept. Fuzzy. Sets. Syst. 1(3), 181–192 (1978)

    MathSciNet  Google Scholar 

  10. Fang, S., Nuttle, H.-W.-L., Wang, D.: Fuzzy formulation of auctions and optimal sequencing for multiple auctions. Fuzzy. Sets Syst. 142(3), 421–441 (2004)

    MathSciNet  Google Scholar 

  11. Jia, W., Xiang, S., He, J.-H., et al.: Existence and stability of weakly Pareto-Nash equilibrium for generalized multiobjective multi-leader-follower games. J. Glob. Optim. 61(2), 397–405 (2015)

    MathSciNet  Google Scholar 

  12. Julien, L.A.: On Noncooperative oligopoly equilibrium in the multiple leader-follower game. Eur. J. Oper. Res. 256(2), 650–662 (2017)

    MathSciNet  Google Scholar 

  13. Kumar, S.: Max-min solution approach for multiobjective matrix game with fuzzy goals. Yujor. 26(1), 51–60 (2016)

    MathSciNet  Google Scholar 

  14. Kumar, S.: Piecewise linear programming approach to solve multiobjective matrix games with I-fuzzy goals. Jcd. 8(1), 1–13 (2021)

    Google Scholar 

  15. Kacher, F., Larbani, M.: Solution concept for a non-cooperative game with fuzzy parameters. Int. Game. Theory. Rev. 8(03), 489–498 (2006)

    MathSciNet  Google Scholar 

  16. Kacher, F., Larbani, M.: Existence of equilibrium solution for a non-cooperative game with fuzzy goals and parameters. Fuzzy. Sets. Syst. 159(2), 164–176 (2008)

    MathSciNet  Google Scholar 

  17. Liu, B.: Stackelberg-Nash equilibrium for multilevel programming with multiple followers using genetic algorithms. Comput. Math. Appl. 36(7), 79–89 (1998)

    MathSciNet  Google Scholar 

  18. Larbani, M.: Multiobjective problems with fuzzy parameters and games against nature. Fuzzy. Sets. Syst. 161(20), 2642–2660 (2010)

    MathSciNet  Google Scholar 

  19. Liu, Z., Wang, G., Yang, G.: Existence of equilibrium solution for leader-follower games with fuzzy goals and parameters[J]. Fuzzy. Sets. Syst. 473, 108731 (2023)

    MathSciNet  Google Scholar 

  20. Maiti, S., Roy, S.-K.: Bi-level programming for Stackelberg game with intuitionistic fuzzy number: a ranking approach. J. Oper. Res. Soc. China. 9(1), 131–149 (2021)

    MathSciNet  Google Scholar 

  21. Nishizaki, I., Sakawa, M.: Equilibrium solutions in multiobjective bimatrix games with fuzzy payoffs and fuzzy goals. Fuzzy. Sets. Syst. 111(1), 99–116 (2000)

    MathSciNet  Google Scholar 

  22. Nishizaki, I., Sakawa, M.: Equilibrium solutions for multiobjective bimatrix games in corporating fuzzy goals. J. Optmiz. Theory App. 86(2), 433–457 (1995)

    Google Scholar 

  23. Nishizaki, I., Sakawa, M.: Solutions based on fuzzy goals in fuzzy linear programming games. Fuzzy. Sets. Syst. 115(1), 105–119 (2000)

    MathSciNet  Google Scholar 

  24. Pang, J., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multileader-follower games. Comput. Manag. Sci. 2(1), 21–56 (2005)

    MathSciNet  Google Scholar 

  25. Stackelberg, H.: The theory of the market economy. Oxford University Press, Oxford (1952)

    Google Scholar 

  26. Steffensen, S., Thünen, A.: An explicit nash equilibrium to a multi-leader-follower game. Pamm. 19(1), e201900307 (2019)

    Google Scholar 

  27. Sakawa, M., Nishizaki, I.: Max-min solutions for fuzzy multiobjective matrix games. Fuzzy. Sets. Syst. 67(1), 53–69 (1994)

    MathSciNet  Google Scholar 

  28. Song, Q.Q., Tang, G.Q., Wang, L.S.: On essential stable sets of solutions in set optimization problems. J. Optimiz. Theory. App 156, 591–599 (2013)

    MathSciNet  Google Scholar 

  29. Song, Q.Q.: On the structure of core solutions of discontinuous general cooperative games. Int. J. Game. Theory 4, 1–21 (2023)

    Google Scholar 

  30. Vidyottama, V., Chandra, S., Bector, C.-R.: Bi-matrix games with fuzzy goals and fuzzy payoffs. Fuzzy. Optim. Decis. Ma. 3(4), 327–344 (2004)

    Google Scholar 

  31. Vijay, V., Chandra, S., Bector, C.-R.: Matrix games with fuzzy goals and fuzzy payoffs. Omega. 33(5), 425–429 (2005)

    Google Scholar 

  32. Watada, J., Chen, X.: Strategic decision-making from the perspective of fuzzy two-echelon supply chain model international conference on intelligent decision technologies. Springer, Cham. 637-648 (2017)

  33. Yang, S., Wu, J., Sun, H., et al.: Integrated optimization of pricing and relocation in the competitive carsharing market: a multi-leader-follower game model. Transport. Res. C-mer. 138, 103613 (2022)

    Google Scholar 

  34. Yu, J., Wang, H.: An existence theorem for equilibrium points for multi-leader-follower games. Nonlinear. Aanl-Theor. 69(5–6), 1775–1777 (2008)

    MathSciNet  Google Scholar 

  35. Yang, Z., Pu, Y.: Existence of equilibrium points for multi-leader-follower games under uncertainty. Control Decision. 27(5), 736–740 (2012)

    MathSciNet  Google Scholar 

  36. Yang, Z., Ju, Y.: Existence and generic stability of cooperative equilibria for multi- leader-multi-follower games. J. Global. Optim. 65(3), 563–573 (2016)

    MathSciNet  Google Scholar 

  37. Yu, J.: Fifteen lectures on game theory. Science Press, Beijing (2020)

    Google Scholar 

  38. Yu, J.: Game Theorem and Nonlinear Analysis. Science Press, Beijing (2010)

    Google Scholar 

  39. Yang, H., Yu, J.: Essential components of the set of weakly Pareto-Nash equilibrium points. Appl. Math. Lett. 15(5), 553–560 (2002)

    MathSciNet  Google Scholar 

  40. Zangeneh, A., Shayegan-Rad, A., Nazari, F.: Multi-leader-follower game theory for modelling interaction between virtual power plants and distribution company. Iet. Gener. Transm. Dis. 12(21), 5747–5752 (2018)

    Google Scholar 

  41. Zhang, Y., et al.: On Nash-Stackelberg-Nash games under decision-dependent uncerb tainties: model and equilibrium. Automatica. 142, 110401 (2022)

    Google Scholar 

  42. Zadeh, L.: Fuzzy sets. Inf. Control. 8(3), 338–353 (1965)

    Google Scholar 

  43. Zhang, G.: Fuzzy Value Measurement Theory. Tsinghua University Press, Beijing (1997)

    Google Scholar 

  44. Zhao, J.: The equilibria of a multiple objective game. Int. J. Game. Theory 20(2), 171–182 (1991)

    Google Scholar 

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Funding

This study was approved by Major Scientific and Technological Special Project of Guizhou Province ([2019]1067) and Guizhou Provincial Science and Technology Fund ([2022]168).

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Authors and Affiliations

  1. Guizhou University, Guiyang, 550025, China

    Zhenli Liu, Guoling Wang & Guanghui Yang

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  1. Zhenli Liu
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  2. Guoling Wang
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  3. Guanghui Yang
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Correspondence to Guanghui Yang.

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Communicated by Kyriakos G. Vamvoudakis.

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Liu, Z., Wang, G. & Yang, G. Existence of Equilibrium Solution for Multi-Leader–Follower Games with Fuzzy Goals and Parameters. J Optim Theory Appl 200, 585–601 (2024). https://doi.org/10.1007/s10957-023-02340-y

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  • DOI: https://doi.org/10.1007/s10957-023-02340-y

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