Application of linear programming in two-player zero-sum games

Research Article
Open access

Application of linear programming in two-player zero-sum games

Xinyu Wang 1*
  • 1 Boston University    
  • *corresponding author wxy1113@bu.edu
Published on 17 November 2023 | https://doi.org/10.54254/2753-8818/11/20230414
TNS Vol.11
ISSN (Print): 2753-8826
ISSN (Online): 2753-8818
ISBN (Print): 978-1-83558-133-9
ISBN (Online): 978-1-83558-134-6

Abstract

Linear programming is a useful optimization tool that has been utilized since the 1940s. It has been studied for a long time, and it can be applied to many different areas and fields. One of these applications is the two-player zero-sum game. In this game, each player has a set of possible strategies that is finite. While one player tries to maximize the net payoff, the other player tries to minimize this payoff. Since the problem involves optimizations, linear programming can be involved. This paper analyzes a famous and classic example of the game – the Colonel Blotto Game. This game is a military planning problem. Indeed, Linear programming was first developed to solve complex military and planning problems during wartime. This paper utilizes a payoff matrix and some mathematical models to build the mathematical problem of the Colonel Blotto Game. Then the results can be obtained by converting the mathematical problem into a Python Linprog problem. The paper finds the best strategy for Colonel Blotto to maximize the net payoff and how eliminating certain possible strategies will or will not affect the results. Moreover, the paper provides an understanding of the combination of the two-player zero-sum game and linear programming.

Keywords:

linear optimization, linear constraints, two-player zero-sum games, the Colonel Blotto Game.

Wang,X. (2023). Application of linear programming in two-player zero-sum games. Theoretical and Natural Science,11,239-245.
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References

[1]. Dorfman R 1984 The discovery of linear programming. Annals of the History of Computing 6(3) 283-295.

[2]. Shamir R 1987 The efficiency of the simplex method: a survey. Management science 33(3) 301-334.

[3]. Nash J C 2000 The (Dantzig) simplex method for linear programming. Computing in Science & Engineering 2(1) 29-31.

[4]. Williams, H. Paul. "Duality in mathematics and linear and integer programming." Journal of Optimization Theory and Applications 90 (1996): 257-278.

[5]. Komodakis N, Jean-Christophe P 2015 Playing with duality: An overview of recent primal? dual approaches for solving large-scale optimization problems. IEEE Signal Processing Magazine 32(6) 31-54.

[6]. Akpan N P, Iwok I A 2016 Application of linear programming for optimal use of raw materials in bakery. International journal of mathematics and statistics invention 4(8) 51-57.

[7]. Oberman A M, Ruan Y L 2015 An efficient linear programming method for optimal transportation. Cornell University Numerical Analysis. arXiv preprint arXiv:1509.03668.

[8]. Alotaibi A, Farrukh N 2021 A Review of Applications of Linear Programming to Optimize Agricultural Solutions. International Journal of Information Engineering & Electronic Business 13(2).

[9]. Cantwell G L 2003 Can two person zero sum game theory improve military decision-making course of action selection. Army Command and General Staff Coll Fort Leavenworth Ks School of Advanced Military Studies.

[10]. Aplak H, Mehmet K, Erkan K 2014 A two person zero sum game oriented to integration of objectives. Journal of Military Studies 5(2) 65-85.

[11]. Adler I 2013 The equivalence of linear programs and zero-sum games. International Journal of Game Theory 42(1) 165.

Cite this article

Wang,X. (2023). Application of linear programming in two-player zero-sum games. Theoretical and Natural Science,11,239-245.

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The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.

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About volume

Volume title: Proceedings of the 2023 International Conference on Mathematical Physics and Computational Simulation

ISBN：978-1-83558-133-9(Print) / 978-1-83558-134-6(Online)
Editor：Roman Bauer
Conference website: https://www.confmpcs.org/
Conference date: 12 August 2023
Series: Theoretical and Natural Science
Volume number: Vol.11
ISSN：2753-8818(Print) / 2753-8826(Online)

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References

[1]. Dorfman R 1984 The discovery of linear programming. Annals of the History of Computing 6(3) 283-295.

[2]. Shamir R 1987 The efficiency of the simplex method: a survey. Management science 33(3) 301-334.

[3]. Nash J C 2000 The (Dantzig) simplex method for linear programming. Computing in Science & Engineering 2(1) 29-31.

[4]. Williams, H. Paul. "Duality in mathematics and linear and integer programming." Journal of Optimization Theory and Applications 90 (1996): 257-278.

[5]. Komodakis N, Jean-Christophe P 2015 Playing with duality: An overview of recent primal? dual approaches for solving large-scale optimization problems. IEEE Signal Processing Magazine 32(6) 31-54.

[6]. Akpan N P, Iwok I A 2016 Application of linear programming for optimal use of raw materials in bakery. International journal of mathematics and statistics invention 4(8) 51-57.

[7]. Oberman A M, Ruan Y L 2015 An efficient linear programming method for optimal transportation. Cornell University Numerical Analysis. arXiv preprint arXiv:1509.03668.

[8]. Alotaibi A, Farrukh N 2021 A Review of Applications of Linear Programming to Optimize Agricultural Solutions. International Journal of Information Engineering & Electronic Business 13(2).

[9]. Cantwell G L 2003 Can two person zero sum game theory improve military decision-making course of action selection. Army Command and General Staff Coll Fort Leavenworth Ks School of Advanced Military Studies.

[10]. Aplak H, Mehmet K, Erkan K 2014 A two person zero sum game oriented to integration of objectives. Journal of Military Studies 5(2) 65-85.

[11]. Adler I 2013 The equivalence of linear programs and zero-sum games. International Journal of Game Theory 42(1) 165.

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