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A Proposed New Non-Linear Programming Technique for Solving a Mixed Strategy Problem in Game Theory

Received: 8 June 2023     Accepted: 7 July 2023     Published: 31 July 2023
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Abstract

This Paper explores a new non-linear programming approach for determining mixed strategies in non-zero-sum games. Our approach leverages the power of non-linear optimization algorithms to solve the mixed strategy determination problem efficiently. We formulate the problem as a non-linear programming model, considering the individual player’s utility functions and the strategic interdependencies among them. The proposed approach offers accurately represents strategic interactions by incorporating non-linear objective functions and constraints. The proposed non-linear programming technique offers several advantages for solving game theory problems. Firstly, it enables the consideration of complex and nonlinear relationships among players' strategies, allowing for more realistic and nuanced modeling. Secondly, the technique offers flexibility in incorporating various types of constraints, including capacity limitations, budget constraints, or regulatory requirements, enhancing the applicability to real-world scenarios. Lastly, NLP algorithms provide efficient and robust optimization procedures, ensuring reliable solutions within reasonable time frames. We use MATLAB to solve the Non-Linear programming problem which gives us more accurate results. To demonstrate the effectiveness of the proposed technique, it can be applied to diverse game theory problems, such as auctions, bargaining, pricing decisions, and resource allocation. The results obtained through this approach offer insights into optimal strategies, equilibrium outcomes, and potential trade-offs, facilitating informed decision-making in strategic environments.

Published in International Journal of Systems Science and Applied Mathematics (Volume 8, Issue 2)
DOI 10.11648/j.ijssam.20230802.11
Page(s) 17-22
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2023. Published by Science Publishing Group

Keywords

Equilibrium Concepts, Strategic Interactions, Nash Equilibrium, Pareto Optimal, Dominant Strategy, Mixed Strategy, Payoff Matrix, Saddle Point

References
[1] Nash, J. (1950). Equilibrium points in n-person games. Proceedings of the National Academy of Sciences, 36 (1), 48-49.
[2] von Neumann, J., & Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton University Press.
[3] Taha H. A., Operations Research An Introduction, Prentice Hall of India Pvt. Ltd, New Delhi, 1999.
[4] "Game Theory and Strategy" by Philip D. Straffin Jr. (Mathematical Association of America, 1993).
[5] "Game Theory: A Nontechnical Introduction" by Morton D. Davis (Dover Publications, 1997).
[6] "An Introduction to Game Theory" by Martin J. Osborne and Ariel Rubinstein (Oxford University Press, 1994).
[7] Maskin, E., & Tirole, J. (1990). The principal-agent relationship with an informed principal, I: the case of private values. Econometrica, 58 (2), 379-409.
[8] Harsanyi, J. C., & Selten, R. (1988). A General Theory of Equilibrium Selection in Games. MIT Press.
[9] Myerson, R. B. (1981). Optimal Auction Design. Mathematics of Operations Research, 6 (1), 58-73.
[10] Milgrom, P., & Roberts, J. (1990). Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica, 58 (6), 1255-1277.
[11] Harsanyi, J. C. (1973). Games with randomly disturbed payoffs: A new rationale for mixed-strategy equilibrium points. International Journal of Game Theory, 2 (1), 1-23.
[12] Osborne, M. J., & Rubinstein, A. (1994). A Course in Game Theory. MIT Press.
Cite This Article
  • APA Style

    Md. Golam Robbani, Md. Asadujjaman, Md. Mehedi Hassan. (2023). A Proposed New Non-Linear Programming Technique for Solving a Mixed Strategy Problem in Game Theory. International Journal of Systems Science and Applied Mathematics, 8(2), 17-22. https://doi.org/10.11648/j.ijssam.20230802.11

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    ACS Style

    Md. Golam Robbani; Md. Asadujjaman; Md. Mehedi Hassan. A Proposed New Non-Linear Programming Technique for Solving a Mixed Strategy Problem in Game Theory. Int. J. Syst. Sci. Appl. Math. 2023, 8(2), 17-22. doi: 10.11648/j.ijssam.20230802.11

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    AMA Style

    Md. Golam Robbani, Md. Asadujjaman, Md. Mehedi Hassan. A Proposed New Non-Linear Programming Technique for Solving a Mixed Strategy Problem in Game Theory. Int J Syst Sci Appl Math. 2023;8(2):17-22. doi: 10.11648/j.ijssam.20230802.11

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  • @article{10.11648/j.ijssam.20230802.11,
      author = {Md. Golam Robbani and Md. Asadujjaman and Md. Mehedi Hassan},
      title = {A Proposed New Non-Linear Programming Technique for Solving a Mixed Strategy Problem in Game Theory},
      journal = {International Journal of Systems Science and Applied Mathematics},
      volume = {8},
      number = {2},
      pages = {17-22},
      doi = {10.11648/j.ijssam.20230802.11},
      url = {https://doi.org/10.11648/j.ijssam.20230802.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20230802.11},
      abstract = {This Paper explores a new non-linear programming approach for determining mixed strategies in non-zero-sum games. Our approach leverages the power of non-linear optimization algorithms to solve the mixed strategy determination problem efficiently. We formulate the problem as a non-linear programming model, considering the individual player’s utility functions and the strategic interdependencies among them. The proposed approach offers accurately represents strategic interactions by incorporating non-linear objective functions and constraints. The proposed non-linear programming technique offers several advantages for solving game theory problems. Firstly, it enables the consideration of complex and nonlinear relationships among players' strategies, allowing for more realistic and nuanced modeling. Secondly, the technique offers flexibility in incorporating various types of constraints, including capacity limitations, budget constraints, or regulatory requirements, enhancing the applicability to real-world scenarios. Lastly, NLP algorithms provide efficient and robust optimization procedures, ensuring reliable solutions within reasonable time frames. We use MATLAB to solve the Non-Linear programming problem which gives us more accurate results. To demonstrate the effectiveness of the proposed technique, it can be applied to diverse game theory problems, such as auctions, bargaining, pricing decisions, and resource allocation. The results obtained through this approach offer insights into optimal strategies, equilibrium outcomes, and potential trade-offs, facilitating informed decision-making in strategic environments.},
     year = {2023}
    }
    

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    T1  - A Proposed New Non-Linear Programming Technique for Solving a Mixed Strategy Problem in Game Theory
    AU  - Md. Golam Robbani
    AU  - Md. Asadujjaman
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    DO  - 10.11648/j.ijssam.20230802.11
    T2  - International Journal of Systems Science and Applied Mathematics
    JF  - International Journal of Systems Science and Applied Mathematics
    JO  - International Journal of Systems Science and Applied Mathematics
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    PB  - Science Publishing Group
    SN  - 2575-5803
    UR  - https://doi.org/10.11648/j.ijssam.20230802.11
    AB  - This Paper explores a new non-linear programming approach for determining mixed strategies in non-zero-sum games. Our approach leverages the power of non-linear optimization algorithms to solve the mixed strategy determination problem efficiently. We formulate the problem as a non-linear programming model, considering the individual player’s utility functions and the strategic interdependencies among them. The proposed approach offers accurately represents strategic interactions by incorporating non-linear objective functions and constraints. The proposed non-linear programming technique offers several advantages for solving game theory problems. Firstly, it enables the consideration of complex and nonlinear relationships among players' strategies, allowing for more realistic and nuanced modeling. Secondly, the technique offers flexibility in incorporating various types of constraints, including capacity limitations, budget constraints, or regulatory requirements, enhancing the applicability to real-world scenarios. Lastly, NLP algorithms provide efficient and robust optimization procedures, ensuring reliable solutions within reasonable time frames. We use MATLAB to solve the Non-Linear programming problem which gives us more accurate results. To demonstrate the effectiveness of the proposed technique, it can be applied to diverse game theory problems, such as auctions, bargaining, pricing decisions, and resource allocation. The results obtained through this approach offer insights into optimal strategies, equilibrium outcomes, and potential trade-offs, facilitating informed decision-making in strategic environments.
    VL  - 8
    IS  - 2
    ER  - 

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Author Information
  • Department of Mathematics, University of Dhaka, Dhaka, Bangladesh

  • Department of Mathematics, University of Dhaka, Dhaka, Bangladesh

  • Department of Mathematics, University of Dhaka, Dhaka, Bangladesh

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